- Submitted my thesis
People may be wondering why I was quite slow to do a post about Oberwolfach.
Long story short, the Friday before last I submitted my PhD thesis!
Me and a wad of paper Its a strange feeling submitting— a very complex day indeed. On the one hand its a tremendous achievement in and of itself, but also, before the defense you still might have some work to do.
The other thing is it marks the beginning of the end of such an amazing experience! Ive loved my PhD more and more as the years have gone on and have grown more and more appreciative of the guidance and conversations im able to have with my supervisor. You look back and think wow— I really did that but also forward and think its a shame it wont last forever.
I will defend at some point soon! Watch this space
- Oberwolfach
I recently attended Oberwolfach.
Me out in the German countryside Wow! What a place and what an amazing conference. The quality and range of interesting talks, plus the fantastic facilities made for a week to really remember. I want to especially thank the organisers for inviting me to such a great event.
The conference photo 
Me and the Boy surface I had such a great time hanging out with old friends such as Becky and Diego from Munster as well as making new connections and meeting some amazing experts in different fields of mathematics. I would love to attend one of these again but at the same time im glad to have ticked speak at Oberwolfach off my bucket list!!!
- PhD update!
I thought I should make a little post, its been ages since my last blog post. I’ve been busy writing my thesis– and have made good progress. This week I printed off my first draft, a big achievement.
This part of a phd is quite gruelling and hard work. There is a lot of formatting, understanding the weird parts of LaTeX and not so much mathematics itself in my opinion– my experience probably varies a lot from others since my thesis is essentiallly using content from papers that are already in the public domain, so the task is essentially to reformat my papers and unite them through an introduction and preliminaries.
I had a bit of a complex writeup period– I got really ill in January of this year, so had to take almost a month off but now I’m back to (almost) full health so its not been a threat to the PhD itself– just a bit stressful.
Next week is super exciting because I am going to Oberwolfach, the conference centre in the black forest in Germany. Im super excited. At the same time its a bit scary, because its so respected as a conference and (as far as I know) Im the only PhD student going to this one.
- Wishing everyone a calm restful break
This break we bought a kitten for my dads 60th so gonna spend most of the time just chilling with her
Conker the kitten - Visit to Orsay
The institut Last week I was in Orsay, visiting Matthieu Joseph at Universitie Paris Sud. I gave my first ever talk in France, about my recent work concerning Stein’s groups.
But this trip also had a larger research angle to it. Matthieu Joseph has many interesting ideas and I really enjoyed discussing with him different ideas 💡
Orsay itself is a beautiful campus, the mathematics building is really nice and fresh new building. Orsay perfectly balances city and town life, since you are so close to Paris and at the same time you are close to the countryside it feels that you can explore some of Paris and also have a quieter evening in Orsay itself.
Sightseeing in Paris 
Out for a crepe with Matthieu Special thanks to Matthieu and Camille for hosting me, I loved this trip!
- Stein’s group preprint
Today, I posed the preprint in which I have studied Stein’s groups on the ArXiv. https://arxiv.org/pdf/2312.07375.pdf
I thought I should explain some of the context behind this paper for my readers. The first thing to say is I have been working on this paper for many years at this point so I am very proud and relieved to have it finally out 🙂
I have been studying certain groups of piecewise linear bijections which were introduced by Stein, as generalisations of Thompson’s group V. In Thompson group V, the slopes are all powers of 2, but for Stein’s group it could be (any) subgroup of the positive reals. This includes groups generated by several integers, and group of slopes generated by a single irrational number, such as Cleary’s “golden ratio” group.
My first main result is that I show when the group of slopes is finitely generated, the derived subgroup of the Stein’s group is finitely generated and simple. Something that really helps me here is that the dynamical models for these groups are very similar, even though the groups can have very different algebraic properties, which makes the perspective of topological full groups very useful.
I then turn to homology of Stein’s group. Here, what was known previously is about the Higman Thompson groups, and also the abelianisations of a few key examples such as when the slope group is generated by integers, or Clearys group. But what is very useful is the homology of framework of my supervisor, Xin Li which translates the (much more computable) groupoid homology into the (much more interesting) group homology.
Then I show how you may adapt the TFG model to get other interesting groups, such as V like groups acting on all of R, rather than compact intervals, and to also include Brin-Thompson groups into this framework. I can then talk about the homology and finite generation of these groups also!
- Talk in Newcastle!
This Tuesday I had the pleasure of speaking in Newcastle University’s Algebra and Geometry seminar! I spoke about Steins groups and feel like the talk went well— its hard to tell with Maths talks but I got a bunch of questions which is usually a good sign.
I loved Newcastle as a city, it was my first time visiting and it actually blew me away how nice it was— the metro system is really easy to get around, I went to a pub or so and the pubs seem to be really nice independent businesses (many pubs in the UK are chain pubs at this point).
The University group seems also very welcoming and sociable. I felt very welcome both in the Algebra and Geometry group and also the Analysis group (which has a growing operator algebras emphasis). There is a very decent bunch of people who study geometric group theory and it was nice to talk about my research with them!
Thanks so much to the Seminar organisers and also to my good friend Christian Bonicke for hosting me. It certainly wont be my last time in Newcastle, I can say that for sure!
Our Seminar Meal— an amazing Thali! - First proper lecture
Today I lectured my first proper lecture. It was CORE skills to a class of 400. I showed them what a function was, how to graph and take derivatives.
I think it went well, but it was super scary!
- Boone-Higman for Hyperbolic Groups
Today, the paper in which Boone-Higman conjecture is resolved for hyperbolic groups is on ArXiV. It is a very significant paper in my research landscape.
https://arxiv.org/pdf/2309.06224.pdf
I wanted to explain why this is significant to me. First, what is the Boone-Higman conjecture?
The Boone-Higman conjecture is that every group G that has solvable word problem iff G embeds into a simple finitely presented group.
Its been open since ’72. It remains open, but this is a good step in the right direction. People still think its true. One direction is true, i.e. subgroups of simple finitely presented groups have solvable word problem. This is cause simple finitely presented groups have solvable word problem!Second, what is a hyperbolic group?
A hyperbolic group is a group that you can solve its word problem using Dehn’s algorithm. I.e. these are the groups with really nice solvable word problem.
Thirdly, why is this relevant to me?
This is relevant to me on a personal level because one of the authors has been in the last year some kind of informal mentor of mine, he is called Jim Belk and he works at Glasgow.
But its also relevant to me maths-wise because it represents another “win” for the topological full group community. The containers that they use in the proof very much use ideas and constructions that are familiar to me from my research.
The new constructions of groups come with interesting constructions of etale groupoids. I would love to hear from people in C*-algebras how they might prefer think about these groupoids. Are they related to the tight groupoid of the associated left inverse hull on the associated monoid? Do they fit into Spielbergs small category of paths or Xin’s Garside Category framework? Are they your favourite groupoids and now you have another reason to like them? What are the C*-algebras we can build from them? - Two papers I’m very happy to see today
The first is the following paper on the interactions between Steinberg algebras, C*-algebras and the natural representation of the TFGs in these algebras.
https://arxiv.org/pdf/2309.04817.pdfThis is one of the first papers that I have seen that studies the interaction between Steinberg algebras and TFGs, however on a historical note, its arguably true that the way that Pardo classified the Higman-Thompson groups was by studying the representations of them as TFGs in Leavitt path algebras. This shows there is a fertile relationship which would be great to explore deeper, as the authors do here.
The second paper is the paper of Kevin Brix, Chris Bruce and Adam Dor On. I have noticed Adam in Glasgow a lot this year!
https://arxiv.org/pdf/2309.04817.pdf
And they weren’t just drinking coffee, they were proving theorems. This paper is very strong, it simultaneously resolves an open question about C*-envelopes and answers a question my supervisor asks. The most satisfying application to me is the result of Theorem B. It demonstrates that the C*-envelope of the (nonself adjoint) operator algebra generated by a left-cancellative small category is the same as the boundary quotient C*-algebra. This result is strong because it covers many relevant constructions, from K-graphs to left cancellative semigroups. Even though this paper isn’t directly related to my research, I wanted to give the authors a shout out because I am extremely proud of them for these results ! - Been giving my talk a bit
Its also been recorded. I thought I should share, in case any blog followers are interested in it:
https://www.youtube.com/watch?v=lpxgl6G7RiA - Taking an online talk on tour
Hey, just a bit of self-promotion. I have a 1hr talk called “topological full groups by example” which is an introductory talk about topological full groups. I give this talk online, its a slides talk. I am talking already at a few places but I am trying to take it “on tour” i.e. give the talk to as many universities as possible.
If any of my blog readers would be interested in having me speak online at a seminar, I would very happily give this talk. Please feel free to reach out to me over email or similar.
I am also very open to give in-person seminar talks, although I will have a lot of teaching duties this semester so probably outside of term time. - Had a really nice YMC*A x3
Today, I am travelling back from YMC*A again. I really enjoyed this one, having now become friends with many people in the C* -algebras community, it had a very social aspect as well as a mathematical.
It was the first YMC*A where I had the opportunity to speak as well. I was very fortunate to have this opportunity, as there were also surely many other mathematicians who would have given great talks but were not able to. I hope that I made the most of my 20 minutes though, and I got some positive feedback from people saying they enjoyed hearing me talk about topological full groups.I enjoyed nearly all of the talks I attended, but will not be giving a summary of the talks like I did for the Edinburgh conference. This is because it is a young mathematicians conference and I don’t think its appropriate to pass judgement. But what I will say is that I was super nervous for my talk and that I am proud of everyone who spoke– it already takes a lot of courage to get up and speak in front of so many people.
The other thing I had mathematically was many interesting conversations with gifted young mathematicians. To name a few, I really enjoyed talking with Alon Dogon. We realised together that it was really an interesting question to ask whether being left-orderable was an obstruction for groups having property T. What was great about this was that (as well as Alon) two of the minicourse speakers were experts on property T (Adrian Iona and Narutaka Ozawa) and they had some insight about why this is an interesting and hard question. I might try to think about this some more.
I also spoke with Anna Duwineg a bit. Anna is an expert on groupoids, but also on Zappa-Szep products. Zappa-Szep products were of course discovered indepndently by Zappa and Szep, but they were also developed by Brin in order to understand normal-ish groups inside Thompsons group V. V is simple so it has no normal subgroup, but it does have a natural subgroup structure. For example, any element of V can be uniquely written as the product of a unique element of F with a unique element of the measure preserving subgroup of V. Group theorists call this sort of subgroup structure a Zappa-Szep product. I want to understand if this picture works well for other topological full groups and this discussion I found useful in understanding what would need to be done. Thanks Anna, for finding the time (she was an organiser, so this is a nontrivial thing)!
I also found the time to speak with Rachel Chaiser, who is an expert in the HK conjecture. Specifically, she has a lot of insight into the principal counterexamples found by Robin Deeley, who supervises her PhD. We spoke together with Ali Miller, my PhD brother. I learnt a lot from this conversation.
All in all, I felt like I learnt a lot this week but now I am very tired! Thanks for everyone (in particular the organisers) for making this week so great.I would like to end this post with a quote from one of the minicourse speakers (who remains nameless). We were discussing what makes a mathematician “young” in the name YMC*A. Their suggestion was that “a young mathematician is someone that wants to see more of the world, and a not-young mathematican is someone that wants to see more of their mum”. I think this is one of the loveliest things I’ve ever heard, and I wanted to share it on my blog.
- Gone a little bit not proving much…
…and thats okay and normal.
I kinda think that mathematicians often are seen from the outside to be constantly proving big theorems, making substantial progress with research. There are mathematicians that seem to be able to make some progress with research every day, but I would put it that most, like me, have “bad” days, weeks, even a month or two without proving much substaintial.
As a graduate student my assumption was always that in some kind of linear progression with mathematics, at some point I would stop learning the basics and instead just 90% of my workflow would be proving novel results. And this just isnt the case!
This doesn’t mean that we don’t make progress, or learn things on times like this. If anything, progress is kinda linear in mathematics even if it can feel from the inside quite stop-start. At the moment, even as I’m nearing the end of my PhD, lots of my work is actually reading graduate or even undergraduate texts in mathematics, in order to “fill in” some gaps in my knowledge that I need in order to prove some things.
As mathematicians, we certainly love to have days where we break through with an idea or something, and prove a bunch of things. But its important to be patient and kind to yourself before that day comes! Tbh, this is the part of my job I struggle with more often than not, I can have the habit of jumping in the deep end a bit early, which is unwise and loses time in the long run.
Specifically, at the moment, a lot of my time is going into understanding how to use a certain spectral sequence which has come up in research. Before I kinda knew what a spectral sequence was, and that I *should* care about them. But its not really until you are faced with a concrete problem that you start to really want to understand things like that, at least for me anyway.
What can be hard on an emotional level, is due to the nature of research, you never know if all the hard work that you put into understanding something, in my case these spectral sequences, is going to “pay off” immediately. By which I mean, its a possibility that after I learn all this stuff, I might not actually be able to use it that much in my next paper, or even any paper in the future.
And its common to be under the impression that what matters to employers is whats in your papers, not really whats in your head.
But I think tbh, the whole thing is much more nuanced than that. Its certainly taken to be a good thing if you studied something, and it will come across in an interview if generally speaking you are a well-rounded mathematician who has studied somethings that maybe didnt come up in research yet.
Law of averages, its always good to be learning. And more than that, I would say its part of the pleasure of the job. When people ask why im in academia, certainly one reason im drawn to it is the idea that its cool if I just want to spend some time learning something im interested in at the time. Of course there are other benefits, like the travel and the people, and the flexible working hours. But I have to say that being paid to learn new things is pretty cool. - Very significant preprint in C* -algebras today
https://arxiv.org/abs/2307.06480
That is the “big paper” about classification.
This classification theorem, which classifies the (maximal) class of simple C*-algebras one could expect for by ordered K-theory and tracial data (the so-called Elliot invariant), was first announced in 2015.
It was a very significant result then because it “capped off” a huge program of classification, by many work from many people.
Since then, experts in the field have been working to “shorten” and “conceptualise” the proof of this. A significant piece of progress was made through results of Chris Shafhauser, who had a different way of thinking about some things.
And so many of us who work in C*-algebras, but maybe are not as “deep” into the classification program have been waiting for this preprint for a very long time, for instance I have waited the whole of my PhD in anticipation!
And now it is here, in the form of this preprint. It looks very nicely written and I will be sure to try to understand at least a very rough idea of how one proves such a result.
A significant moment in the history of C* -algebras. Congratulations to all of the authors. - Some open problems
- Groupoids people love this one weird trick (for computing the Elliot invariant)
I wanted to share with everyone this small trick, which I learnt from Xin, about computing the positon of the unit in K_0 of the reduced groupoid C*-algebra of an ample, (compact unit space), effective groupoid.
This little tip is very useful in practice.
Assuming the groupoid homology is fairly computable, which it usually is, there is a unit in H_0, just given by the whole unit space G^0. Then, there is a canonical map from H_0 to K_0– this is described for example in the following paper:
https://arxiv.org/pdf/2104.05885.pdf
They call it \mu_0
It turns out this map is actually unital- it maps the unit in H_0 to the unit in K_0. So in summary to find out the position of the unit in K_0, all you need to do is look at \mu_0(G^0).
I just wanted to share this little piece of insight, its a very useful trick for computing the Elliot invariant. Especially its useful for ample groupoids associated to UCT Kirchberg algebras, since over here all that the Elliot invariant is, is K-Theory where K_0 is ordered. In many cases, the HK conjecture is known for this type of groupoid, so really, all one needs to do is compute the (pointed) groupoid homology.
It also shows that you can get much information from groupoid homology. H_0 is actually an ordered group with this in mind, it makes it a better invariant for telling if your groupoids are different or not (and in my case, lots of topological full groups).
Another part of the Elliot invariant is the tracial data. There is also a similar relationship between “measures on your groupoids” and “traces on your C*-algebra”. I *think* (but would not be willing to formally conjecture) that these are also in one-one correspondence with proper characters on the derived subgroup of the topological full group. - Why group homology?
I wanted to give a bit of a post about why people care about group homology. I was asked this question a few times recently, and also became aware there was some confusion about what things we can find out about groups by looking at their homology. Id like to thank Becky Armstrong for giving me the idea to post this here.
I wont go into how higher homology groups are defined, just some facts you get about groups looking at their homology.
The first homology group/0th H_0(G), is also the most boring one. This is because it doesnt matter which group you put in, H_0(G) is the integers, Z. Nonetheless, its often important to remember this group in homology computations, so that you can double check you are along the right lines. If you get something other than Z, you have made a mistake.
The first interesting homology group is H_1(G). This has a lot of different perspectives to it. My perspective is the following.
A common perspective is the that this is the abelianisation. That is, let D(G) be the derived subgroup, in other words the smallest normal subgroup of G that contains all the elements of the form ghg^{-1} h^{-1}. When we have a normal subgroup, we can consider the quotient of a group by the normal subgroup ie G/D(G). This quotient is whats known as the abelianisation.
This perspective really helps with topological full groups, because we are trying often to study D(G), which is often simple, not our topological full group G, which has D(G) as a normal subgroup. So you can measure how far away you are from your normal subgroup. A group equal to its derived subgroup is called perfect, and this sometimes happens too, for example when you consider Thompsons group V.
Another reason its nice to view H_1 as a quotient is it allows for the following reality check:
If my group G is finitely generated, so is H_1(G) of my group
Because finite generation passes to quotients.
H_2(G) is my next favourite group. Something thats really hard to see, ie, a whole paper, is that if your group is finitely presented, H_2(G) must be finitely generated. So this helps us see when groups might be finitely presented (something better than finite generation), an important attribute in Geometric group theory.
But we can find homology in any degree.
A final thing to observe for anyone working in topological full groups that is easy to forget:
The derived subgroup of a topological full group of a minimal effective ample groupoid is always perfect, so H_1 vanishes.
- A flattering survey for people who work in topological full groups
I wanted to share on my blog, this survey of the recent progress towards the Boone-Higman conjecture, which is that:
Every group with solvable word problem embeds into a simple finitely presented group.
This is the survey: https://arxiv.org/pdf/2306.16356.pdf
What “flatters” those that work in topological full groups of etale groupoids is how many times the groups we are trying to embed into are constructed via the TFGs framework.We see for example, the proof of Belk-Hyde-Mattuci uses the language of topological full groups to show all countable abelian groups embed into a topological full group called VA.
Another amazing theorem, which uses topological full groups is a theorem of Belk-Bleak-Mattuci-Zaremsky, where they show if G is hyperbolic, G embeds into a certain simple, finitely presented topological full group. I am excited to read this paper in detail when it comes out.
It is natural that topological full groups might be useful to solve such a problem, for a few reasons:- They often have simple derived subgroup
- They often have finiteness properties such as being finitely presented.
- They provide a way, often, to blow up a smaller group into a big group that is finitely presented and simple.
- A short history of the homology of V and it’s generalisations.
Following on from my what is group homology, I thought I might give a short account of the homology for my favourite group, Thompsons group V, and its generalisations.
When Thompson wrote down what V is, he already knew it was simple; so it must be perfect, and so, H_1(V)=0.
In 1992, Brown showed that if you look at the homology of V with coefficients in Q, this vanishes in all degrees H_n(V,Q)=0. This doesnt leave a lot of things it could be, what it means is essentially in all degrees, H_n(V) is a sum of finite cyclic groups, since these are the only abelian groups that =0 when we tensor on Q.
It then took until 2018 for someone to pick up the reigns. This someone was an excellent paper by Syzmik and Wahl. They computed the homology in every degree and found out (wtf) that its 0 everywhere. Thats right, H_*(V)=0. They also showed that the homology of V_k,r does not depend on r (even though the isomorphism class does). And also that the homology of V_kr vanishes rationally. Another word for this property is that V is acyclic, or that V_kr is rationally acyclic (it means its homology vanishes).
But there are lots of other classes of groups for which it wasnt known. What for example were the homology of Steins groups, where we allow different types of slopes. And what about Brins groups, that were higher dimensional.
This is what my supervisor was able to start computing homology for. Xin computed the homology of Brins groups nV. And between me, him, and a bunch of other people working in the area we will write down what we know for all these other generalisations in coming months.
Something I found out recently was that I can find some examples that arent rationally acyclic, which is sorta cool. I am excited to share this research with the world!
- Some open questions about finitely generated simple groups
Following on from my post last Saturday about why people care about infinite finitely generated simple groups, I recently collected some open questions about finitely generated simple groups for a talk, and thought it would be a good idea to put it up on my blog. These questions are:
- Are there infinite, finitely presented amenable simple groups? (Juschenko-Monod)
- Is IET, the group of interval exchange transformations amenable?
- Is every finitely presented simple group generated by two group elements?
- Does every group with solvable word problem embed into a simple finitely presented group? (Boone-Higman)
Id like to know the answer to any of these questions, but I also have sources that ask these questions in maths papers— they arent just my open questions they are really big open questions in group theory!
- Irrational rotation algebras and simple finitely generated amenable groups
Following on from my previous post, I wanted to explain how people that work in C*-algebras can tell the following truth about groups:
There are more infinite, finitely generated amenable simple groups than there are finite simple groups.
The classification of finite simple groups tells us there are countably many finite simple groups. So it is enough to find an uncountable family.
Something that most people that work in C*-algebras know is that the irrational rotation algebras form an uncountable family of C*-algebras. This is a way to build a C*-algebra out of an irrational number, and the C*-algebras structure depends on the irrational number up to a plus or minus sign:
https://en.wikipedia.org/wiki/Noncommutative_torusThere is a related expansive Cantor minimal system called a Sturmian subshift which describes a similar C*-algebra as a crossed product. For example, I do this in my paper I put on the ArXiV:
https://arxiv.org/abs/2304.13691
(You need to choose Gamma to be Z+\theta Z where \theta is your favourite irrational number).
Jushenko and Monod were able to show that expansive Cantor minimal systems give rise to simple, finitely generated amenable groups.
Girodano-Putnam-Skau were able to show the following result:
If you have two dynamical systems and you build topological full groups from them, this is an invariant up to continous orbit equivalence of the dynamical system. Moreover, if your C*-algebras are different, certainly your topological full groups are going to be different.
The C*-algebras I build from Sturmian subshifts depend on theta up to plus or minus.
Because they give rise to C*-algebras that arent the same, the topological full groups are definitely different as well.
Let me please make the point that sturmian subshifts give rise to a classifiable C*-algebra with a unique trace. We can think of K_0 as theta Z +Z as an ordered group. However, K_1 is Z. So not exactly the irrational rotation algebra but its similar. - Why are simple finitely generated infinite groups interesting?
When you are an undergraduate your life looks like this, it is often said:
Its quite easy to get a decent social life and enough sleep, but you might not get good grades. Also, its kinda easy enough to get good grades and enough sleep, but your social life might suck.
What’s really hard is to get all three.
A similar thing is true for groups. A simple group is one that has no proper normal subgroups, an infinite group is one with infinitely many elements, and a finitely generated group is one that you only need finitely many group elements to generated the whole group.
Its not so hard to choose 2:
The integers is an example of a group thats infinite and finitely generated, but its not simple.
The cyclic group of order 2 is an example of a group thats simple and finitely generated. But its only got 2, not infinitely many elements.
The infinite alternating group is an example of a group thats simple and infinite, but it isnt finitely generated.
So can we get all three. Surprisingly, the answer to this is yes. But its really hard. One reason is the following result:
Thm: Let G be a subgroup of GL_n(C) for some n. Suppose it is finitely generated and simple, then it is finite.
So if your group is finitely generated, simple and infinite, it isnt linear.
The first example of such a group was given by Higman:
But its pretty complicated to define. What comes to more mathematicans mind is Thompsons group V:
https://en.wikipedia.org/wiki/Thompson_groups
Thompsons group V is actually even better than finitely generated. It is finitely presented. In fact, it is even better than finitely presented, it is type F_\infty.
What has been really exciting in recent years is we found a way to make groups that are finitely generated, simple and infinite from “dynamical systems”. This lead to some really important results like:
- There are amenable infinite, finitely generated simple groups:
https://annals.math.princeton.edu/2013/178-2/p07 - There are simple groups of type Fn but not Fn+1 for all n:
https://link.springer.com/article/10.1007/s00222-018-0835-8 - There are simple groups with intermediate growth:
https://annals.math.princeton.edu/2018/187-3/p02
These results are really amazing. One way you can see that mathematicians are interested in something is what journal people publish about it in. The annals is widely considered to be the #1 best journal, and inventiones is widely considered to be #2.
This way is called topological full groups. The framework is you start with a groupoid that is minimal, ample, effective, and expansive and study the group of bisections that have source and range being the whole unit space, with respect to composition.Im doing my phd about them, and I have found lots of interesting groups I’m excited to share with everyone!
- There are amenable infinite, finitely generated simple groups:
- Loved the ICMS conference!
I really loved this conference at the ICMS. I thought there were a lot of super interesting talks– some highlights were the following talks (apologies to those speakers who I did not have in my list, all of the talks were very interesting to me, but some were these ones:
- The very first talk was by Aidan Sims, and he spoke about Cartan pairs. This talk was great for many reasons, one was that it meant I could communicate about my research, which heavily uses this notion, with everyone. Another was that he was able to fit the notion of groupoid C*-algebras into the classical theorems that we see– polar decomposition, Gelfand duality, Gelfand-Naimark-Segals theorem. Especially the way in which operators’ polar decomposition works well within a groupoid C*-algebra fascinated me, and this was a perspective I did not consider before.
- I really enjoyed Becky’s talk about conjugacy of local homeomorphisms. Actually, I sometimes think about these things too, but not as seriously as Becky and her collaborators. The talk was great because Becky defined everything one needed to understand the key objects, and paced it at a rate that was comfortable to take notes. It was clearly a talk that was made with the audience in mind. But the results she presented seemed quite deep and I will need to think more about them my own time.
- It was great to see Brita talk about her generalisations of Thompson group V. Last August, I unfortunately missed her talk because I was ill. So this was my first time to see her talk, and it was great. It was interesting to see the perspective Brita takes on these groups, that they are automorphism groups on Tree-like objects. This perspective is great, but from what I understand, the irrational slope Thompson groups that I consider do not always admit such a nice perspective, for example if you took a transcendental number, e, then the unit interval doesnt subdivide.
- Kevin spoke about ideal structure of C*-algebras, especially those that come from graphs or other dynamical sources. This was the first time I saw him speak about this research and I found this incredibly interesting.
And this was all in the first day!
- The second day included to start off with a talk by Lisa Orloff Clark. This was about these so-called Steinberg algebras, which I have a growing appreciation for. Lisa had an excellent visualisation of Steinberg algebras, groupoids and groupoid C*-algebras and how there were ideas moving from the analysis to the algebra and back again. I found this a great way to think about the cross-pollination of ideas in these three areas.
- Alejandra Garrido Angulo, who seems to be an amazing researcher that I hadn’t come across before, gave the second talk. She had a fascinating construction, generalising topological full groups, so that she could construct simple, compactly generated locally compact topological groups. I loved the ideas in this work and on a personal note, really enjoyed talking with Alejandra about mathematics throughout the week.
- Jim Belk gave an amazing talk, talking about his progress with collaborators on the Boone-Higman conjecture: that every group with solvable word problem should embed into a finitely presented simple group. Jim said in his talk that there hasnt been much significant progress for this conjecture, but I couldnt disagree more. When I think of a solvable word problem-group, I think of a hyperbolic group, they are the canonical examples. So the fact that this framework confirms this conjecture for this class is very significant for me.
- Benjamin Steinberg was the final speaker on Tuesday. He talked about his work with Nora, where they look at simplicity of Steinberg algebras and Nekrashevych algebras. I noticed that, being a humble mathematician, he called Steinberg algebras “groupoid algebras”– even though everyone else calls them after him! My favourite slide of his clarified one of the most annoying question for people that work in topological groupoids– “what is the relationship between topologically free, effective, and essentially free”.
- Brita gave a public lecture afterwards, no mean feat, and she did an amazing job!
Wednesday was a bit shorter as a day, but we still had some amazing talks.
- Jack Spielberg, who is undoubtably a big name in operator algebras– he constructed for example ample models for all UCT Kirchberg algebras. He has a framework of left cancellative small categories (more general than Xin’s garside category framework), and constructing C*-algebras from them. He then presented an amazing example of a groupoid that gives an AF algebra, but isnt the usual AF groupoid. I will do a blog post in some time about this example, because I think that it is amazing.
- Victoria Gould talked about the Howson property for inverse semigroups. This seems related to what groupoids people call finitely aligned, but Victoria comes at things from the perspective of abstract algebra. Her talk was great, I particularly enjoyed learning about Bands, a type of semigroup I hadnt heard of before.
We then had an amazing workshop meal. In general, we ate so well that week. Mark Lawson arranged the catering I believe, and he did an amazing job.
- Pardo kicked things off on Thursday. It was so great to see him speak, he is such an influential figure in operator algebras, especially semigroup C*-algebras. He surveyed results about refinement monoids.
- We had our first less senior speaker on Thursday, Malcolm Jones from New Zealand. He has some amazing ideas about groupoid models for higher rank graph C*-algebras when things are not finitely aligned. This seems to be very interesting, and a very pertinent question. It would be interesting to see what the lack of finite alignment means on the level of topological full groups.
- Then my PhD brother, Jeremy Hume gave an amazing talk about his K-Theory computations for complex dynamical systems, namely rational functions. I was so proud to see him give such an amazing talk, and impressed by how general his results were in this computation.
- Jamie Gabe also spoke on Thursday. Something that surprised me to see in his talk was topological full groups. He mentioned a very interesting sequence in K-Theory which seemed to have a connection with the AH conjecture in TFGs.
- We were blessed to have a Bruce-Li talk afterwards. Chris is an amazing speaker and I think his talk really had something for everyone in the room– if you are interested in Algebra, he had things about algebraic actions which were new to me, if you are interested in C*-algebras, he had a neat result at the end. He gave the joke that his main theorem was a Bruce-Lee theorem and everyone laughed!
- It was great seeing David Pask talk about Higher Rank Graphs, being one of the founders of the field.
This leaves Friday!
- Elizabeth Gillaspy kicked things off with an excellent talk about the Williams conjecture, in particular recent progress in this conjecture for a class of graphs called Meteor graphs. They are called Meteor graphs because they have a tail and a head- or in graph language, a sink and a source. The talk was full of examples, and great diagrams, which I really appreciated.
- I also loved Nora’s talk on Friday. I thought she communicated so well what the issues were in talking about Cayley graphs of inverse semigroups, and what was learnt when we consider their coarse geometry. She spoke about the significant progress made by her and Diego considering uniform Roe algebras of the analogues of Cayley graphs for inverse semigroups.
Overall, this was a conference that I wont forget anytime soon. I got to meet lots of my friends, mathematical heros and experience/talk about ideas at the cutting edge of mathematics that I am interested. All of this was in an environment where I felt comfortable to ask (potentially stupid) questions, drink a healthy amount of beer, and make stupid jokes.
- Interesting paper about Exotic group C*-algebras, nonamenability and the Von Neumann-Day conjecture.
I read recently with interest the following paper about exotic group C*-algebras.
https://arxiv.org/pdf/2305.01990.pdf
There were some ideas I really appreciated in this paper. Something I thought was an interesting interpretation (which I hadn’t seen previously) was that the existence of exotic group C*-algebras is a strengthening of nonamenability. Every C*-algebraist knows that a group is amenable iff the reduced and full group C*-algebras agree– so amenable groups do not have exotic group C*-algebras. But it can happen that nonamenable groups still just have a couple of associated C*-algebras, the full and the reduced.
Something that sorta fascinated me about this paper was that the groups the Gerasimova-Monod discuss are so-called von Neumann Day conjecture counterexamples. This conjecture is the conjecture that all nonamenable groups contain the free group on two generators, and has been known to be untrue for quite some time. Monod constructed some cool examples of counterexamples that are very tractable and in this paper they study the associated C*-algebras.
Such examples are really hard to find (hence why it was an open problem for so long). One way to think about why this is true is that the conjecture is true for linear groups. Another thing to think about is that Thompson’s group F, one of the first natural candidates for a counterexample, it is still a major open problem to determine amenability.
It turns out that — counterintuitively to me– these groups have a really ample supply of exotic group C*-algebras, they are in some sense very strongly nonamenable. I say counterintuitively for the following reason– when someone says nonamenable groups the example that comes to mind usually is the free group on two generators. But in many ways the nonamenability of this group is very weak, for example it is weakly amenable.
Something which I guess I had presumed is that von-Neumann day counterexamples would be very close to amenable groups– something between free groups and amenable groups or something. I guess this intuition came from thinking about Thompsons group F, which (if it was to be nonamenable) is in many formal senses very close to an amenable group.
But this intuition has kinda been made out to be very naiive by the paper– in some sense these groups are very strongly nonamenable, they have this rich supply of exotic group C*-algebras.
I’d love to understand more about this sort of thing! - Updated my talks page!
See here for more of my notes from my talks recently:
https://owentanner1997.wordpress.com/academic-talks/
These in particular include notes for my seminars in Glasgow, both for the analysis working seminar and the geometry and topology seminar.
Also, my seminar talks in Manchester and in Cardiff. - Nice to give a talk in Cardiff
Yesterday I gave a GAPT seminar talk in Cardiff University. The talk was about my new paper on interval exchanges and this was the first time I gave such a talk.
It was nice to be back in Wales and talking to an audience with people who taught me undergraduate mathematics, as well as some friendly new faces who I’ve met recently like Ian Charlesworth.
I think the talk went well overall, I approached it very much from the perspective of group theory.
Coming back over the severn bridge - New ArXiV paper!
Today I had a new preprint out. This is my paper on interval exchange groups.
https://arxiv.org/abs/2304.13691
There are lots of nice ideas to talk about in this paper, I thought I might make a blog post informally going through a couple of them. One is rigidity. Something that has been observed now by a bunch of people working in my area, is that for two “nice enough” groupoids G,H , the following are equivalent:- The full groups F(G)=F(H) are isomorphic as abstract groups
- The groupoids G, H are conjugate
- The corresponding Cartan pairs are isomorphic (in the sense of Cartan pairs)
Using this very broad idea, it is clear if the Cartan pairs are different, so are the groups. One way you might show that the Cartan pairs are different is by computing the Elliot invariant of the reduced groupoid C*-algebra. This would then give you some cases when the full groups are different.
This is something I do in the above paper, and it turns out you get a full classification of interval exchange groups in this way.
The other idea is transfer of homological information from groupoids to other objects. There has been much progress in understanding how homological information of groupoids transfers into information about K-Theory of reduced groupoid C*-algebras, for example in Matui’s HK conjecture, or in the PhD Thesis of my PhD brother Ali Miller. Recently, my supervisor Xin Li also found a way to transfer homological information from groupoids to their topological full groups. I use these ideas in the paper to determine lots of new information about interval exchange groups and also certain reduced groupoid C*-algebras.
A final idea I was thinking about is this. TFGs provide us with lots of interesting examples of simple, finitely generated groups. But can we describe the generators of these groups? This is in general quite a hard thing to do, but it turns out you can do this a lot of the time for interval exchange groups, or more generally, when your groupoid comes from the action of a polycyclic group (by using the ideas provided in a paper by Chornyi-Juschenko-Nekrashevych). From a group theory point of view, this is certainly worth the effort. It is great to have a concrete picture of generators.
Now for some cute point-set topology! One thing I thought was concrete most people could get out of this paper is a really convenient description of the Cantor set for my purposes. The construction is as follows. Let Gamma be some dense, countable, subset of the unit interval I. We form a new space I_Gamma. The recipe is as follows:- At each point that is both in Gamma and I, replace it with two points, p+ and p-.
- Otherwise, just have the one point x in I_Gamma.
- Topologise this with the order topology, where our order says that p+>p-, and points that are larger in I are larger in this new I_Gamma.
The order topology is gonna have a basis of open sets by (p-,q+). But hang on, these are actually clopen (p-,q+)=[p+,q-]. Moreover, I_Gamma is compact. So we have a compact space with a countable basis of compact open sets. This must be the Cantor space! I feel like this could be a good topology exercise or something 🙂
- First Preprint Out, With Eusebio
Last month my first preprint appeared on the ArXiv. I thought it might be nice to make a post where I describe informally some of the ideas that this paper came from.
For many years now, its been known that we can get Thompsons group V, as well as many of its generalisations of this group as topological full groups of certain etale groupoids.
Etale groupoids are quite known to people that work in C* -algebras. Even those that dont work with them probably know the definition from the countless talks about them! They give rise to interesting types of C*- algebras, for example classifiable C* algebras have (twisted) groupoid models. The groupoid that gives us Thompson’s group V also gives us the Cuntz algebra O2.
What is interesting to me is the interactions between these two areas, on the one hand someone can build a C* algebra and on the other hand an interesting group from the same object– a groupoid.
Me and Eusebio show that in some formal sense, that groupoids that give rise to similar C* -algebras to O2 also give rise to groups that are similar to V. In a way this is interesting, because people study these objects (V, O2) for very different reasons but they seem to be fundamentally connected.
An open problem that we comment on is the following question:
Is every simple finitely presented group generated by 2 elements?
To us, its shocking that such a fundamental question/easy to phrase question in group theory should have something to do with purely infinite C*- algebras. But it really does seem to– we have that many classes of these C* algebras coming from groupoids are so-called Kirchberg algebras. In some informal sense, whenever your groupoid gives rise to a Kirchberg algebra on the one hand and a finitely presented simple group on the other hand, it is actually a 2 generated group.
I hope my blog readers find this paper interesting. This paper was a lot of work and took many months (as does every maths paper). But it was also a lot of fun to work with Eusebio on it at the same time. I hope that someone enjoys reading it too!
- Talking today at the G&T Seminar
Today, I’m talking at the Geometry and Topology seminar in Glasgow. It was a very last minute thing, but I’m really excited to show some people outside of analysis in Glasgow what I’ve been up to.
It’s really great to be able to talk about my research with other groups– one of my favourite things about the work that I do is that relatively speaking, it is accessible as mathematics, and quite interdisciplinary in that most mathematicians have at least some interest in what kinds of groups there are!
The talk is gonna be called “Two simple groups of dynamical origin”. It will cover some of my work but present it as topological full groups of partial dynamical systems as opposed to etale groupoids, which should help with accessibility.
By the end, I will show some concrete applications of the classification of C* -algebras, which should help to advertise one of the strengths of our field in recent year. This is that I am able to classify entirely a broad class of topological full groups using the Elliot invariant of the crossed products of the associated dynamical systems.
Thanks a lot to Jim Belk for organising this and giving me this opportunity.
- Excited for Eusebio visit
I have Eusebio visiting next week, where we are planning to put up a preprint and also think about some new stuff. I’m really looking forward to hanging out with him and doing some maths, we havent had the chance for a week of this yet and I think we could do some really good work.
Also it’ll be nice to show him about Glasgow a bit. I’m going to take him to Eusebi Deli, one of my fave spots in the west end that also happens to have the same name as him! - Talk in Manchester went well
I feel that my talk in Manchester went well, I recieved a bunch of compliments regarding it. It felt v good to talk to algebraists about my results in group theory and to know that there is a paper on the way.
I loved also being in the Alan Turing building, Turing being one of the most iconic british mathematicians of the last century, maybe ever. They have many posters and artefacts displayed around:
Some of the machine turing codebreaked 
Poster of turing I met with Nora already a bit today, to talk about maths and options going forward. She was very helpful. I particularly liked about this crowd was they were coming at things from a semigroup background, a perspective that I dont share but is very valid when talking about topological full groups. Manchester is probably one of the biggest hubs for semigroup theory in the world. These sorta of perspectives and connections really fascinate me.
- Talk in Manchester 1 week from today
I was asked a while back to speak in Manchesters semigroup seminar. I’m really excited to give this talk for all kinds of reasons!
- It’s my first time presenting my work to an audience that many are interested in geometric group theory. So I hope they will appreciate the interest. They also might be coming at things from a different perspective so could have some interesting questions for this reason.
- I have a very generous amount of time (almost 2 hours). So I will be able to give a nice overview and a bit about my results. I hope to talk some bits about my research with Xin, some bits about my stuff with Eusebio, and a bit about just TFGs in general.
- Manchester University is pretty cool. I’ll be speaking in a building named after Alan Turing, one of my mathematical idols. There’s lots of people there I hope to talk to, not least Nora who I met in Munster briefly.
- Finally, it’s my first ever talk in England. I don’t know much about England apart from that it has dodgy politics (likes this whole Brexit nonsense). It’s so exciting to be able to travel to a place where culture is so different to my own 😉 — who would have thought a PhD would take me to these weird and wonderful places. (Sarcasm)
- It’s my first time presenting my work to an audience that many are interested in geometric group theory. So I hope they will appreciate the interest. They also might be coming at things from a different perspective so could have some interesting questions for this reason.
- Very nice new preprint by Matthieu
I was excited to see this new preprint by Matthieu Joseph. I first heard about this result at Davids conference and I thought I would give a bit of a blog post in order to explain the significance in a bit of an informal way.
The preprint is available here.
https://arxiv.org/abs/2301.07616
Now that the classification program of C* -algebras has concluded, we know that the C* -algebras we can classify are unital have finite nuclear dimension, simple, Z-stable, satisfy the UCT and are separable. But this begs the questions, how can I tell if a C* -algebra satisfies all of these complicated conditions.A way that we have been building C*-algebras is by actions of groups. There is a whole bunch of research that makes the question of when these so-called crossed products are classifiable. If they act on a compact space, they are unital. They automatically satisfy the UCT because they have Cartan subalgebras. They are automatically nuclear whenever the group acting is amenable etc etc.
In an ideal world, minimal actions of amenable groups on the Cantor space should be classifiable.
However, one of these adjectives stands out as very hard to detect on the level of groupoids and group actions. That is Z-stability.
However, we have a line of attack. In David Kerr’s seminal work, he showed that the property of almost finiteness seems to be related to Z-stability, this is his dynamical analogue of the Toms-Winter conjecture.
So, are minimal amenable group actions always almost finite?
Matthieu says no. He found an amenable group (quite a few in fact) with minimal actions on the Cantor set that are not almost finite. In fact these examples fail to be essentially free for some invariant measure. This is a condition that is stronger than almost finiteness in this context.
To me the significance of this paper is finding an example which should a-priori be a classifiable C*-algebra — the crossed product of an amenable group on the Cantor space with a whole bunch of niceness conditions, where our methods to determine Z-stability just completely fail to work– without almost finiteness we are in the dark.
I really enjoyed speaking to Matthieu about this great preprint, and I hope it gets the love it deserves from our community X
(Disclaimer: this post contains mathematical lies for the purposes of exposition– forgive me) - First week back
Nice to get back into the office in Glasgow and see some familiar faces, this week I’ve been trying to get my head/priorities straight for what to work on going forward. It feels a bit like there’s a whole bunch of different things I could be doing and I need to work out how to organise my week to make sure they all get done.
One thing I really wanna do this coming year is to get some work out there and written– it’s felt for a little bit like I needed to do this. Luckily I got a couple things that feel like they are in a writing up stage.
Wishing all my readers a happy new year! Hope everyone has a productive and enjoyable start to the year - Back to the uk
So had my final week at Munster and now im returning to Glasgow and starting my holiday! I dont know where to start, perhaps with a funny note which is that I think?!?! I might be banned from the EU for a few months because I stayed in Germany for 90 days. But im sure I can sort this out!
In my time I feel like I’ve made good connections both professional and new friends with tonnes of people at Munster. The workgroups super welcoming and full of young people, would surely recommend it to anyone considering phds or postdocs in the near future.
I particularly liked all the lunches at the mensa (anyone whos eaten at the mensa would know me saying this is all about the company and nothing about the food!). It was great to check in with everyone once a day and hear how things are going.
The Seminars were another highlight. I really got a lot out of Becky and Diegos Cartan subalgebra seminar. I learnt a fair bit from giving my talk and listening to everyone elses. The kleines seminar was also great, and the quality of talks was really good there. The oberseminar featured some talks by really interesting international speakers like Tatiana Shulman, Andre Toms, Nigel Higson to name a few! Meeting and hearing about their research was really great. It was also cool to see Sergio give a really nice talk this last week.
I think Davids been getting more into topological full groups recently, so another thing I’ve really enjoyed is talking to him and some others at Munster that are thinking about full groups in a serious way. In particular, it was great to meet Spyros this early in his project— he is going to do some projects on tfgs and is in his first year. Its the first time ive had a relationship with someone earlier in their academic career than me doing stuff thats very closely related to my research, and I really liked the feeling of being able to answer questions that really bugged me when I was reading certain texts for the first time/trying to get my head around the literature.
It was also great having David Sherman about at the same time as me. For those that dont know, he was also visiting. It was nice to be hanging out with someone that also was visiting for a short period and share that experience together!
Much thanks to Wilhelm, David and Xin for helping to organise this whole visit! Its been a really great experience alltogether and I will remember it forever 🙂
- Final week of Munster
This week is my final week in Germany. I’ve gotten so much out of this experience, both personal and professional. I can’t thank the department and Wilhelm enough for making this visit possible.
I feel like there are very concrete reasons why this has been good for me, for example I’ve been able to give a bunch of talks and broaden my name recognition therefore. I’ve also been doing a bunch of projects & having deep mathematical discussions with professors here. But some of that’s too sensitive for the blog 😉 I thought I’d share a few of the maybe more subtle advantages of doing a trip like this:- Independence from supervisor: In coming to Munster, I essentially tried for the first time to become an independent researcher. It’s not like I was completely on my own; I’m sure if I really needed Xin he would have been there but it felt like trying this out in a safe way. It’s like a postdoc on stabilizers. I feel a lot more confident to carry out work these days now.
- Experiencing academia in a different country: each country, each department even, has its own take on how academia does and should work. Broadening your understanding here I think helps you to become more professional and respectful at conferences.
- Working out if the postdoc system is for you: in a PhD, there is a constant dilemma whether to continue to postdoc or move to industry. This placement helps me to make an informed decision. This is good for both me and my job prospects, good for me because I’m not just sleepwalking into a postdoc without knowing what its like to move to a new country and start working. It’s good for my job prospects cause then they know I’ve done this before; it proves I can move to a new department and be productive and it doesn’t all go to shit!
- Being able to advertise your field: this one for me was actually quite counterintuitive. The type of maths I’ve been doing recently is TFGs, and there are a whole bunch of people in Glasgow that really are interested in TFGs. However, because of this, it means I am not necessarily selling my field to new people, everyone who would care, does care. In Munster, there are plenty of people who first saw the definition of a topological full group when I gave a talk on my research. So this is really nice opportunity to spread the love!
- Finding new fields: the flipside of this is that you also hear a lot about new areas of maths and new researchers you haven’t heard of before. For example, there was a lot of discussions and talks about property (T), almost finiteness/classifiability of crossed products for example. This cross pollination makes for a really fertile research environment.
- Name recognition: goes without saying but if you are in some new department that is “talking worthy”. Name recognition counts– there are so many papers on the ArXiv so being a familiar face really counts for whether your research gets some eyes on it. I try to read arxiv based on titles not names, but have to say when you see someone you met at a conference or heard things about, you click.
- New ways of thinking about the same concept: something I think I’ve really developed whilst being in Munster is thinking geometrically/pictorally about concepts in TFGs, there are a few reasons for this. One is that I’m talking about my research with people that might be unfamiliar and as they say, pictures say a thousand words, so if I can draw something I tend to. The other is much more human– I would argue that Xin is quite algebraic generally in the way he thinks about mathematics, and David on the other hand is very geometric. So having David assuming a supervisionish role here in Munster, its challenged me to think that way a bit more too.
In case you didn’t get the gist of this post. I strongly recommend the idea of doing some sort of research placement to anyone working on their PhD, if its at all possible.
- Copenhagen in the snow
Had a great time this week talking maths with Soren and seeing what Copenhagen is all about. The group there seems really nice, and there is some interaction between the group theorists and operator algebraists which I always love to see!
Coming there I was nervous about my talk but I think it went okay in the end. Ive realised board talks are probably how I should present— I think Im surely better than slide talks. If I get asked to talk somewhere again it’d be definitely board talks.
Besides talking maths and giving my talk, i also met James Hyde, who I’ve also been reading a bunch lately. It was cool to hear him talk a bit about the work Im citing with Eusebio.
I also got to attend the famous Dansk christmas lunch. This was so crazy! I loved seeing everyone in the department get a bit jolly and interacting with oneanother. Plus I got to try a bunch of danish food which is always a bonus!
Speaking of food I ended up eating super well whilst I was there. I had hotel breakfast which I really appreciated. Then around the department there was decent lunches. On top of this though I managed to get a meal at Norrlyst on Thursday which was really standout.
Fish course at Norrlyst On Wednesday after my talk, I had a curry with James. This was the first decent curry ive had in months, since Munster unfortunately is a bit tame spice wise (for british standards— but im willing for someone to prove me wrong here). This was a huge relief for me 🙂
Today I checked out a coffee shop Chris recommended me and also tried Smørrebrød for the first time— both of which were great.
Aamanns— if you know, you go Besides food, I saw a decent amount of copenhagen too. I went to loads of parks, which were beautiful in the snow and ice. Also the architecture museum which was very good and short trip around for anyone with an hour to kill.
Me & an icy lake 🥶 Overall Copenhagen was a really great place to visit both academically and outside of work— I cant discuss too much about maths 🧮 so thats why this post is a bit travel-vloggy but yeah just felt like it haha
- In Copenhagen
Resuming the trend of cafe pics in nordic countries I am getting excited to give my talk and enjoying exploring the city a bit! This morning im aiming to answer the most pressing open question in C* algebras— who has better coffee shops nearby the department, Glasgow or Copenhagen. Whilst doing this im also trying my first Tebirkes!
One thing I already have noticed is that there in an emphasis towards filter coffee in Copenhagen that might push the dial in the wrong direction for me— I’m a sucker for a Cortardo.
However the coffee I’ve had so far has been exceptionally good. My feeling is that both have advantages and disadvantages, I think a coffee/pastry purist no doubt would go copenhagen any day of the week. However there is no denying the accessibility in Glasgow as well as how relaxed most Cafes are to work in, perhaps giving Glasgow an edge when working remotely but Copenhagen the edge on a weekend. The pastry game as well undeniably Denmark takes the win again. Of course, Glasgow is cheaper, even adjusting for local relative prices. The milk in Glasgow is definitely better than Copenhagen, sorry to say. I guess there is more proximity to good dairy farms in Scotland so this is also to be expected.
Im surely too young and naiive to try to answer this question alone, I hope someone with more experience in the area might one day collaborate with me
- A thanks to David and Shirly…
On organising such a great conference. I really enjoyed the diversity of speakers this week, and got a lot mathematically from these talks. In particular, I was surprised just how often topological full groups came up, and just how often people discussed Thompsons group V. It was that I had defined such a group to the Munster crowd in my Kleineseminar before! although every time someone said thompsons group V, i would get a knowing glance from someone in the audience 🙂
I was also really interested in a lot of talks outside the scope of my research, for example there were several nice talks about property T, notions of isometric orbit equivalence. This is reflective of the diversity of dynamics and its interactions as a hole, which for me has been eye opening not just at this conference but from my meetinngs with David throughout my time in Munster.
That said, I also felt the relevance of my research in a couple of instances— I could provide examples of groups that fell inside the frameworks people were interested in. Several things worked for classes of groups that had an analogue in the C* setting, for example in this excellent talk on Self simulable groups, Xins garside category work immediately came to mind as a framework capturing many known examples. It was great to feel I was contributing something and could make interesting comments at certain points.
As always it was great just to make contact with mathematicians I had read but not met, as well as find new things to read. I enjoyed greatly to have a chance to speak with Nicolas Matte-Bon, Francois Le Maitre, Artem Dudko, Robin Tucker Drob and Mattheiu Joseph. All of whom have put out some very interesting papers I’ve enjoyed a lot. Talking about potential projects also really helped me to prioritise what I should focus my time and effort towards as I go forward into my PhD.
Next stop Copenhagen! And then its my final week in Munster!
- Talks update
Hi blog readers! I wanted to give a bit of an update on my talks section of my website. Any keen eyed readers would have noticed I’ve been speaking a lot recently, and I have a bunch of slides and notes up from this. Namely:
- In early October, I gave a series of two board talks for the Kleineseminar in Munster introducing TFGs.
- Later in October, I gave a talk at the Leuven Methusalem Junior seminar about O_n and V_n. This was for a general maths audience so a bit of a challenge, and was a slide talk.
- Because of my participation in the Cartan subalgebras seminar, I gave this week a talk on non-uniqueness of Cartan subalgebras here in Munster. This was a board talk.
- Next week, I’m talking in Copenhagen. But my notes are already there!
See the page on my talks for more information.
- Maths update
People might be interested to hear how my own work is going these days! I thought I might give a v rough outline of what I’ve been working on (of course one has to be a bit subtle with the details!):
- Since I’ve been in Munster, I’ve been dedicating some time to thinking about a question David Kerr asked me concerning amenability. It’s been really great working with David, and I think I’ve really honed a lot of my geometric intuition in my work because of how visual he is.
- I’ve been discussing a lot with Ulrich in Cardiff and Ali about the recent paper by Xin on homology of TFGs. We are hoping that together we might be able to use the ideas from there to compute homology for some interesting TFGs.
- With Eusebio, we have learnt a lot of interesting things from discussing topological full groups. I’ve learnt a lot from working with him, and excited for him to come visit us in Glasgow sometime next year. I think we have enough ideas to probably squeeze a couple papers out, so the plan is to start writing up our first results and then move to some new interesting things. So watch this space! I think it’s fair to say this project has watched the two of us together go on a journey from not knowing so much about the topic to producing some decent research, which has been really great to share with someone.
- Of course, most importantly, I have my project with Xin. Strangely, I’ve not been working on this so much since being in Germany. I’m excited to get back and stuck in because this work has been super interesting. But over there we have some answers, but just as many interesting questions! My biggest hope over here is that I would be working towards something that will catch the eye of people working in generalised Higman-Thompson groups, since that is the scope of the project.
- Since I’ve been in Munster, I’ve been dedicating some time to thinking about a question David Kerr asked me concerning amenability. It’s been really great working with David, and I think I’ve really honed a lot of my geometric intuition in my work because of how visual he is.
- Invited to Copenhagen!
This week I think it’s safe to say I had a big concrete achievement 🙂
I got invited for the week of the 7th December in Copenhagen to give a talk at the seminar there, and also to talk maths with some people there. I have to say it feels very good to be asked to go. I’ve also never been to Denmark so that is super exciting too.It’s also exciting, my project with Eusebio is now reaching a later stage, so I’m going to talk about that. Another reason this is exciting is somehow it connects the research between a couple of people working in Copenhagen at the moment!
- How is Germany going?
I realised today that I’ve gone a bit quiet on this blog recently, and probably it’s overdue an update!
I’ve been in Munster for about 2 months now, and so far it has been a busy and productive time for me. I’ve enjoyed meeting with David a lot, and learning about some of his perspectives on full groups. This meeting has lead to me thinking about a new question from time to time.I’ve also been working on something with Eusebio since ~April that has come to some kind of head in the last couple of months, and I think we are now in a writing-up stage. So that’s exciting!
Finally, I’ve been participating (and speaking!) in a whole bunch of seminars. I gave a couple of kleineseminars to introducing TFGs, as well as a talk in Leuven, and finally, I’m due to speak in the Cartan subalgebras seminar next month.
It’s been interesting to see how other Universities “do it” as well. It seems there are a lot more times that are timetabled here in Munster, a very active seminar schedule which I think I’ve taken full advantage of. Having a dedicated seminar on Cartan subalgebras has really helped me understand a bit more about them.
It’s also been great working with such a big group, as predicted! Everyone in Munster is super nice and welcoming, and I like how we all go to lunch together and coffee most days.
Living in a new country has also been a great experience for me. I think its certainly took me out of my comfort zone, my german maybe isn’t the best! But I think its been broadening in that respect. Munster as a city is really pretty and has a lot going for it. I particularly love going to the markets and the pub culture round here. Anyone who knows me also knows I’ve been a bit of a gym rat whilst I’ve been here– I splashed out on a really nice Gym and think it was a great shout.
- Xin’s new paper on homology of TFGs
I wanted to go a bit into a blog post that’s a bit more mathematical in nature about Xin’s new paper, available here:
https://arxiv.org/pdf/2209.08087.pdfThis paper concerns two main objects which are in some sense dual to each other. On the one hand, we look at ample Etale groupoids which can be thought of informally as group where instead of a single unit, we have a locally compact totally disconnected Hausdorff space on which the groupoid acts. This generalises the notion of a group acting on a totally disconnected locally compact Hausdorff space, so is thought of as dynamical in nature. For more on groupoids see for example my talk “What is a groupoid” here:
https://owentanner1997.files.wordpress.com/2022/04/groupoid_talk-2-1.pdfWhen considering a groupoid, one can employ the theory of noncommutative stone duality in order to understand how large subsets of the groupoid, called open compact bisections form an inverse semigroup. Moreover, this correspondence is in some sense functorial. There are many methods/formalisms for thinking about this.
But crucially, a very interesting subset of this inverse semigroup to look at is the unitary subgroup, also known as the topological full group. This is the open compact bisections whose source and range are the entire unit space. The topological full group is the second main object of study. For some interest on topological full groups see the talk I gave in Gothenburg here:
Topological full groups are a complete invariant for conjugacy for a large class of groupoids. This means that the groupoids are the same iff the full groups are isomorphic as groups. This implies that a lot of information can be determined about these groups by looking at the groupoids. For example, if you know the groupoids are different then you also know the groups are.
The reason that this is interesting is that topological full groups are weird. There is no way to formalise this statement, so in papers people say that topological full groups give the first examples of groups with strange properties– for example infinite simple groups with intermediate growth. Or infinite simple amenable groups. But the point is more that these groups are weird. They don’t fit into neat categories like amenable groups, or even linear groups. So they are really hard, rewarding and interesting to analyse from a perspective of geometric group theory.
Something that people really want to do when they find weird groups is compute homology. This is because homology is an invariant for groups– it helps us understand what kind of object we are(n’t) looking at, and relaxes us into being able to somehow attach nice objects we understand well (abelian GPS) to horrible objects we don’t (TFGs).But even computing homology for these guys has been mysterious. A very common example of a TFG is thompsons group V, which is associated to the full shift on two generators. This groups been about and studied since 1965, but the homology was only just computed– it was shown to be acyclic in 2014 (meaning the homology vanishes).
For all sorts of generalisations of V, as well as the mysterious new groups people have found using topological full groups, we had no real leads on how to compute homology.
That’s where Xin’s work comes in. What he asked was okay, so we can’t compute the homology of these groups directly, but perhaps it has some relation to homological invariants on the groupoids.
Indeed, he manages to work this out, with several results– a lot of homological information passes from groupoids to their topological full groups. This shouldn’t come as a huge shock, like we know that these groupoids and topological full groups are very closely related.
Theres a series of results here, but ill pick out my two favourites and some of their consequences. One is his “vanishing result”. This says that if the homology of the groupoid vanishes, then the homology of the topological full group vanishes. This is really useful in practice because it shows a large class of groups the homology vanishes. E.g. for the first time you can now show:
- V_{2,l}, the Thompsons group on an interval of length l is acyclic
- Steins groups are acyclic (this uses some observation of Britas that is work in progress)
- The higher dimensional thompsons groups nV are acyclic.
This is super interesting to me. Acyclicity is a very interesting gp theoretic property because it implies the possibility of all sorts of imbeddings. Everything can be multiplied by 0 to get 0, so you can usually find that all sort of other groups can be found inside acyclic groups. Fundamentally, this is one of the reasons that thompsons group V is a natural object to look at.
The second really interesting result to me is that a morita equivalence of groupoids gives an isomorphism in homology of the TFGs. This is cool because it shows how if the groupoids are morita equivalent (therefore have the same homology) then the TFGs also will have the same homology. Its a sort of weaker rigidity result– it says instead of the result I talked about above (the groupoids are the same iff the topological full groups are the same), that the groupoids are similar iff the topological full groups are similar.
For the casual blog reader– apologies for all the maths! But just wanted to share some thoughts on this paper because I find it really exciting. It definitely makes me proud to be Xin’s student (as if I weren’t already 😉 )!
- Arrived in Münster
This weeks my first week working in Munster, I got here on Friday!
My office Everything is going really well so far. The move was smooth and everything worked perfectly.
Im getting very excited to be stuck into the department in Münster. Its probably not an exaggeration to say its the largest operator algebras group in Europe, perhaps the world even. And now I’m here!
Theres a few things I’m doing whilst im here work wise. One is participating in 3 of the seminars which are namely a Cartan sub algebras working seminar, the kleines seminar (working seminar) and the oberseminar. I also hope to maybe give a kleines seminar or two about topological full groups!
Im also meeting with David Kerr a bit, to see if we have any ideas for something to work on. Having just met him at the Glasgow conference where he gave a really nice minicourse, this seems like a great opportunity.
Just hoping to make the most of it all!
Outside of maths, ive been practicing my german a bit, conversationally im struggling a bit with listening but starting to get an ear for it.. the a1 course definitely helped!
I managed to get a bike for rental and also find a really nice gym, so im set to stay fit too.
Also, I got the bumble app for friends. So meeting up with some people this week after work for a drink hopefully meet some people outside of maths too 🧮
- GLASdGOW!
Conference Photo (that I’m not in because I was ill ): ) Bit overwhelmed by how much I would like to write about our conference on this blog, but it’s high time I wrote something!
For starters, I think I just want to thank publically everyone that came along and all of the speakers for making the week so enjoyable and interesting.I’m really proud of the conference we made together as a team, it felt like there was a good balance between talks, discussions and networking. For me the whole experience was extremely valuable– from being an organizer I think it has added an important organisational set of skills to my PhD, but also with the opportunity to invite many interesting mathematicians to Glasgow I got a lot out of the week mathematically.
Someone in particular I was excited to meet that many people are aware of is Brita Nucinkis who was an invited speaker. She gave a talk very tightly related to the work I did, but also I was able to speak to her in more detail privately. It was a great opportunity to speak specifically to a group theory expert about the work I am doing with Xin, since this is the audience I am really aiming to impress through my work. She had many interesting and relevant comments for my construction, as well as several questions.
It was also great to see the minicourses by Kerr, Nekrashevych and Eilers. All three gave really excellent minicourses that I got a lot out of, and was consistently surprised by how relevant my studies so far could relate. I took it as a compliment as an organizer to see how well they had prepared and thought about their minicourses.
Of course, a conference is not a conference without some stresses– I unfortunately got food poisoning on Tuesday, which really sucked. There was also some annoying things going on such as an unfortunately timed fire alarm (if you were there you would know)! However, I think all things considered what went wrong were broadly minor and I’ve left the week feeling very proud of our team of organisers.
On our website I think there will soon be some slides/notes, as well as there will be a synopsis from our open problem session. I was happy to be able to get up myself and talk about some open problems I’m interested in! Check that out here:
https://sites.google.com/view/glasgow2022/home
Thanks again for everyone that helped make it a special week!
- YMC*A 2022!
I wanted to do a blog post to thank and write a bit about YMC*A 2022– the biggest conference for young mathematicians in operator algebras. Of course, I had an amazing time– Oslo had tons to offer culturally and culinary-wise. I’ll intersperse some of the maths stuff with some photos of me having a laugh out there.
I enjoyed the sculpture park! Even if it was a bit hot (excuse the vest) So the conference was divided into a whole bunch of different sessions. We had a minicourse from Lyudmilla, which was all about nonlocal games, correlations. This stuff was great to learn about, I’ve always been interested in how this seemingly quite unrelated stuff lead to progress on the Connes embedding problem. It was really well presented in my opinion.
No complaints food wise! Neshveyev also talked about subproduct systems, associated C* algebras and their symmetries. I picked up a few decent definitions that had always been mysterious for me previously, and definitely noted his presentation style.
Island hopping and the fjord! 
Xin also gave a talk, where he presented some entirely new theorems about topological full groups he recently proved! It was a bit of a shock to me to see all these new theorems for the first time– I didn’t even know he was working on this stuff.
For me it was also particularly nice to hang out with Xin in the evenings, I felt like we had a really good chance to get to know each other a bit better outside of work.
The response to his series was great as well– obviously this is closely related to the work I’m enjoying doing at the moment so to see lots of young people bouncing ideas off each other was great. I remember having some great conversations with a bunch of people about how they might be bringing TFGs into their research.
Then I wanted to talk a bit about the contributed talks. Firstly, there was so many great contributed talks I have to say, the standard was amazingly high. I couldn’t choose my favourite but have to say that there was just a whole bunch of amazing research on display but I was especially appreciative of how accessible everyone made their talks. It didn’t feel like everyone was just showing off their best results, it felt like most people were more thinking about the audience. Which was great!
There were a few people I met whose research interests/general vibe made me feel we had a bit of a connection. A lot of Europeans I had met already at either the last YMC*A or Gothenburg. I would say though especially I loved talking maths with a few people– Samantha Pilgrim, Arturo Jaime, M. Ali Asadi-Vasfi and Mathias Palstrom I all had interesting conversations with. I’m hoping to maybe zoom chat with Arturo and Mathias in particular, since their work is closely related to some questions I’m interested in.
Sculpture There were also some special sessions. I was involved in the fun and casual lightning talk session, where I talked about my research in 3 mins- a bit of a challenge. Everyone was really great in this!
Your title matters for a three-minute talk! There was also an ask-anything session, a hugely successful new initiative that was launched by the organizers. I hope they do the same next year! I got roped into being the “expert” on purely infinite C* algebras, which I lead with Robert Neagu at Oxford. Despite me feeling a bit nervy, I think we did well since I was able to offer up some examples whilst Robert was amazing at giving the classification basics and why they are interesting.
Time to sign off just by saying thanks so much to all the organisers!
try to find me! - Ducks, rabbits and duality
The duckrabbit One thing I’ve learnt to appreciate a lot in mathematics, especially working in operator algebras where this is incredibly important is duality– this is the ability to think about the same mathematical object in two different ways.
Switching perspectives is an incredibly important part of mathematical research, from different angles, the same question can either be incredibly complicated or simple.
In my work this comes across in many different ways. One such thing is to consider different perspectives and notations for the so called Higman-Thompson groups. One perspective is that they are certain automorphism groups of trees, one uses ‘tables’ and one uses certain maps on the interval. For quite some time I was thinking to use tables before I realized the geometry of the maps on the interval made a lot of things easier to communicate and understand.
Perhaps the most important in our field is the so-called Gelfand duality, which describes commutative C* algebras as topological spaces, and commutative von Neumann algebras as measure spaces. This encourages the philosophy of thinking of operator algebras as “noncommutative” versions of topological or measure spaces, a philosophy that has arguably lead to some of the most important research in the field.
A final type of duality I wished to talk about is how there are many ways to describe a certain C* algebra. For example, one C* algebra might be thought of as a crossed product, as a groupoid C* algebra, as a graph C* algebra and as a partial crossed product all at the same time. For example, O_2 is one such example of a C* algebra. However, this duality can be absolutely essential for research and also pedagogy of operator algebras.
An area of contemporary research in operator algebras asks “what adjectives on a groupoid correspond to adjectives on a C* algebra”. For example, being minimal and effective on the groupoid level gives you something called simplicity on the C* level. These differing perspectives become easy when you have one example you can pass through– I know all the adjectives for O_2, and all the adjectives for the groupoid underlying it– this can really help my intuition.
For anyone else in the field, I recommend thinking about all of these dualities, it is almost the core of the whole of OA. - “You don’t look like a mathematician”
I don’t know how many people that work in maths get this comment but it’s so annoying. I’ve heard anecdotally from not-male colleagues they get this a lot, probably more than myself, but kinda felt like going on a rant on behalf of myself and them….
I’m not sure why certain people think it’s a compliment to say you don’t look like you are good at your job. I know there are stereotypes around maths people being nerdy, having bad dress sense etc, but I cant help but feel a lot of this is wound up in a lot of toxic ideas of what an academic in general should be like– a friendless, unfashionable guy that can’t look after himself and spends every waking minute working! That ain’t me, but also I frankly don’t know very many people that work in maths that fit that description either.
Surely we are at a point in 2022 where we are encouraging more different people into different occupations, and this means kinda leaving superficial expectations about what someone should look like in a certain career in the bin! Otherwise, we are creating social awkwardness for people that don’t “look” like mathematicians.
I know Seun gets this sorta comment too as someone working in Chemistry, which also pisses me off.
I don’t need to look like I work in maths, I just do work in maths! - Finally graduated
Last week, I returned to Cardiff for my graduation, which had been put off due to the pandemic. Unfortunately, the graduation itself I missed due to travel issues, and certain series of unfortunate events things didn’t quite go amazing for the weekend either. However, I thought I would share some pictures of myself, my mum, and Seun celebrating this achievement.
No doubt, I’m looking forward to a much more straightforward Ph.D. graduation!
Me and my Mum! 
Me and Seun! - Against decimalisation
Something quite strange that intersects with my research quite a lot is actually quite accessible. Namely, the idea of representing real numbers in different ways other than decimal.
For example, I construct some things by thinking about binary representations, or even by expanding real numbers in terms of a series of irrational numbers, such as the golden ratio.
It’s made me think a bit about how we ended up with the decimal system being the “universal” way of representing numbers. It is clear through computing that a lot of people need to consider other ways, and actually through constructions like mine that mathematical ground would be much more fertile if people had an understanding of how to move between these things.
In thinking about this I had a look a bit into the history of when and how this came to be, which I found to be very interesting. It seems that before 1800, many currencies including the British pound were not decimalized. In the advent of the industrial and the French revolution, this diversification became looked down upon by the European establishment as unscientific, or uncivilized even, which led to many countries decimalizing their currencies. It’s strange that mathematical culture has arguably maintained this colonial attitude, it’s not uncommon for people to visibly cringe when I say that I have to work in bases other than base 10!
Food for thought 😉
PS something I also found out about which to me is super crazy is that in France they once tried to decimalize time, so that there were 20 hours in each day. In a similar fashion, the UK tried to decimalise angles, so that for example a right angle was 100 degrees. I’m glad that at least in these examples at least we are happy to work outside of decimals– it shows that it can be good to use some other bases sometimes 😉 - Soft skill I’m starting to get the hang of
Something thats quite hard in mathematics is prioritising research ideas that turn out fruitful, and letting things die that arent so fruitful. Im something of an optimistic researcher, so have a bad habit of spending a lot of effort trying to prove something above my reach.
However, I’m starting to learn how to prioritise the results that I reckon I can do, and get a fair volume of work done. Sometimes quantity does get an edge on quality, especially if you are an early career researcher.
I think there’s an important lesson that I’m starting to learn within this, which is basically, its quite easy to conjecture things that you have no idea how to prove and then spend ages trying to prove them! But more reasonable is to conjecture something smaller connected to methods you know a fair bit about and build up from there. - Working Seminar Over!
This week was the last working seminar of this semester and the last one that I organised as well.
It was a great opportunity to organize this working seminar and I kinda got to lean the topics toward things I’m interested in, so certainly for me, this was one of the best years of seminars that I have already seen. We had Mike talking about Smale spaces, Sam on Set theory, Runlian on approximation properties for groups were all highlights for me, but also I got a lot out of the groupoid homology talks by Ali and Eduardo towards the end of the seminar, since my research is kinda needing that bit of homological algebra now!
All that said, it’s nice now to have the summer free. Between my project with Xin, my project with Eusebio, and the conference we’re organizing, there’s a whole lot of work to be done. - The collab with Eusebio is starting to get me excited
Feel like we are making some ground on understanding these purely infinite groupoids as per Matui, and some cool properties both on the C* algebraic level and on the groupoid level. Watch this space!
Also just getting quite excited at how I’ve got a bit of a golden choice between my work with Xin or with Eusebio depending on how I feel when I get up in the morning. Both projects have subtly different flavours, I find my work with Xin I’m learning a lot more definitions as I go, but with Eusebio there’s a whole bunch of algebra to do most of the time. - Starting to get quite excited for Germany
Today was my last german lesson, so I now have the EU A1 qualification in German language– in practicality this means I can string a sentence together if I know the relevant vocab, but not quite there with all the vocabulary.
Nonetheless, I’m getting pretty hyped for my trip to Muenster. I think I’m gonna be able to really experience a bit of a different culture, and also see some great maths. I noticed David Kerr/Shirly Geffen are organising a really interesting conference whilst I’m there, which is surely another bonus.
Also I’m feeling like it’ll be quite a natural time to head off then, think my project with Xin should hopefully be kinda wrapped up by then! So be quite a clean break. - Haha, we are all up in arms
Today, we (as in most PGR in Glasgow) are getting kicked out of the office for a bit. We have some temporary makeups but I can’t lie, it’s a shame to just get to Glasgow and now then to be not working in the maths building shortly after.
However, the building does need work so I see it from an admin perspective, so the title is only in jest.
Coincidentally, this was also when I’m taking a coupla weeks off so its really no problem for me, at least for a bit. Guess we will see where we are at next month or so! - Annual Review today
Today was my annual review, where I get to check in with some other academics on how the project is going and all that…
It went well! Annual reviews are basically nothing to worry about, for anyone thinking about a Ph.D. or in their first year, etc… it’s just a chance to raise if you have an issue with your supervisor or vice versa, and maybe get a bit of constructive feedback on how the work is going. - Today, Elke sorted out my accommodation
Just wanted to do a rare s/o on my blog. Today, Elke Enning, who works at WWU Muenster, managed to find accommodation for me, something I was pretty nervous and stressed about. So no more worries, and just to say thanks publically to Elke, and all the other administrative staff in mathematics who really make the whole thing work smoothly and stressless– very underappreciated part of our community sometimes.
- Working Seminar!
Today, I gave a working seminar on a remarkable result– basically, everyone’s favorite groupoid is actually a transformation groupoid. This would be the full shift on two generators.
I find this result interesting for all kinds of reasons. Culturally, it’s interesting, because it appears like this result, used to be quite well known but isn’t so well known in our generation of researchers. Another thing that’s interesting about it as a talk topic is it begs the question– what other groupoids are transformation groupoids?
This is a hard question that is wide open in dynamics and would love to investigate it a bit with someone. It might be a challenge, but we could surely find some necessary conditions! - Starting German lessons
Hallo! Ich heibe Owen.
Today I had my first German class, ahead of my trip to Muenster. Managed to get funding through the school of science and engineering, so I got professional help…
Needless to say I need it– I’m ashamed to say like most Brits, I surfed through the education system whilst basically ignoring foreign languages… to a huge regret. So languages aren’t a strong point of mine– its interesting being the bottom of the class for once haha…
But gonna try my best and to practice. If you are reading this as a German speaker, and see me from time to time please try say hallo! And langsaman, bitte! — I need all the help I can get 😉 - Muenster is happening!
Today I found out that for sure, I am able to go to Muenster from ~ Sep to ~ Dec this year. This is so exciting– on a personal level I’m really feeling great about the idea of living in Germany for a few months– going to learn some German and try to really embrace the culture a bit. I loved Muenster as a city as well, and just to be on the continent at the moment feels like the right place for me.
But then on another level– professionally OMG! Muenster is definitely the centre of OA in Europe, for those who don’t so much have a feel for the community yet. So, there are plenty of well known professors, not to mention incredibly talented PGR that I can’t wait to work alongside for a few months. I can’t believe this is happening but it’s surely the best thing that could be happening to me just now at this stage in my academic career. It’s not just that but I also think I could contribute some things to the community, for example I’m getting to know well tfgs, and some interesting questions that arise from them as a subfield of dynamics relevant to C*- algebras.
- Gothenburg Working Seminar
Today was a watershed day in my mathematical career– I gave my first talk to an international audience, and my first talk outside of a home institution all at once. Fortunately, the setting couldn’t be more perfect. Eusebio G, messaged me last week saying there was a vacancy to talk in the working seminar, and they’d been focusing on groupoids the past semester. So of course I had to say yes! The luxury of giving this course was the whole audience already knew what a groupoid was, so I was able to both talk about interesting results in general of TFGs, as well as walking them through the O_2 and V example!
- PGR Seminar
Today, I attended and gave my first PGR seminar. It was a really brief introduction to groupoids, where I gave the basic definitions and intuition behind the stuff. I was most excited that the talk was accessible for people outside of the analysis group, in particular, I brought some people from the topology group in by talking about how the fundamental groupoid is a good example of a groupoid, and how they generalize dynamical systems. Because of this, there’s more people in Glasgow who I can share my (mathematical) problems with.
- Had a great time in Gothenburg!
Now im back in Glasgow reflecting on what an enjoyable and productive week Ive just had in Sweden. I saw some amazing talks from great speakers of all academic ages, some highlights relevant to my work were Karen Strung, Diego Martinez, Julian Kranz, Wilhelm Winter, Mathew Kennedy, and Makoto Yamashita. Some talks less relevant to my research but that I still loved were Hannes Thiel, Adam Skalski, Sanaz Pooya and David Kyed.
I loved chatting to people as well. In particular I learnt a lot from talking with Karen on her restricted class of twisted groupoids, and from conversations with Jono and Ali about noncommutative Cartan subalgebras.
Thought I’d end by thanking the organisers for an amazing experience, and some photos from my time there!
Goats and seals of Gothenburg 
View over the city. Thanks for having me, Sweden! - Had a great working week in Glasgow
Had a great working week this week in Glasgow. My daily routine is really undergone a makeover! My walk to uni goes through Kelvingrove, and is one of the nicest walks in Glasgow.
In my lunch breaks ive been able to visit the gym which is really great— has a sauna and a pool 🤯. I also managed to visit the kelvingrove museum on wednesday! Here are some pics:
Kelvingrove museum 
The uni gym is gonna keep me very fit! 
My daily walk to work could be worse for sure - Back in Glasgow
Very excited to have moved back to Glasgow. Getting stuck in with the office life. Is great to not be working from home anymore!
My new desk - Last day working from home!
Today marks the end of my working from home in Cardiff section of my phd. Done a lot of great work from here but would be lying if I said i wasnt over the moon to be getting involved with the school of mathematics in Glasgow and to not be cooped up away from the operator algebraists.
We had some good news this week which was a grant accepted from lms for the workshop in glasgow. Im hoping we get another bunch of money so we can invite lots of early stage researchers which would make the conference even better!
Research is going well still, just keeping things quiet cause the work is at the stage where it could be pinched if i posted it publicly.
Excited to be attending all the seminars and meetings in person from now on.
- Announcing GLADSGOW!
Thought I would do a blog post to say I am on the organising committee for the conference GLADSGOW (Glasgow Late August Dynamical Systems, Groups, and Operators Workshop), a conference with very close links to my interests.
Anyone that might be interested should register here ASAP:https://sites.google.com/view/glasgow2022/registration?authuser=0
- Feeling really on top of things ahead of my move to Glasgow
I wanted to put a blog post just to say that at the moment I’m feeling that things are going really well. Research-wise, things are going well and at a point where things are a bit more sensitive so can’t go into too deeper detail about that. Then on top of this, I’m excited for the conferences coming up and also glad to have done some decent outreach this week.
Our conference is now at a point to be announced so I’ll do a blog post about that soon, it’s very exciting though.
Just feel like I will get to Glasgow feeling very good towards the next couple of years and felt this blog needed a bit of a happy message after some of the darker things in previous years! - First Talk!
Just did my first Cardiff Science Festival talk about symmetry! This was a big success, and you can rewatch it here:
https://www.youtube.com/watch?v=wzxwhMTrE90
The slides will also be available on my blog! - BBC Interview is up!
My BBC interview is now live on the science cafe show. The program starts at 18:30.
Hear me talk a bit about symmetry here:
https://www.bbc.co.uk/programmes/m0014dcl - First time in the abacws!
Today for the first time I am visiting the new maths building in Cardiff called the abacws. It’s really nice.
Im catching up with Simon and Ulrich as well as announcing my Cardiff science festival talk about symmetry to some undergraduates sitting topology!
The new building is very nice, certainly makes a change from the old building I did my undergraduate in! - BBC Radio Wales Interview
Here is a little peek behind the curtain! BBC Wales is interviewing me today about my Cardiff science festival talk for their Science Cafe show.
This is a prerecorded interview, so I’m not sure when it goes out. Nonetheless, this is clearly a big opportunity so I am excited for spreading the word about my talk. I’ll post again when it’s live!
- Homotopic Minds!
Just finished a really great meeting with Bambordé Baldé at homotopic minds, a branch of the Zaiku Group that aims to link together people from pure mathematics with deep tech.
Bambordé was a hugely likable character and really enjoyed learning about the program and am keen to get involved. I hope this is the start of something exciting. For now though, just felt I should do a blog post in case my readers are interested in signing up to be part of their cohort as well! - Bit on the research the last couple of weeks
Research this year started pretty slow, with maybe one or two useful noticings every other day. But the last couple days the floodgates opened and I’ve for some reason been pumping out theorems like its my job!
Quite a weird feeling!
Im also making my way through the amazon rainforest with all this paper im getting through— my paper bin was empty this morning 🤯
Rough notes I get through fast! - My Cardiff Science Festival Link!
In February I’m going to be talking at 10am, Feb 21st, and 2pm Feb 22nd about symmetry and its applications to science!
The link is here:
https://youtu.be/wzxwhMTrE90
It’s my first very big public outreach talk so I’m very excited about it. - Quite hard shifting gears to research
It’s pretty hard shifting my gears for research these days. Back when I was reading I could do 9-5 days, and be pretty sure that most of that time would be productive. At the moment, I would say I only tend to get a couple of hours decent work done a day!
I hope that as I go on I end up with more reasonable productive hours, or just that I fill up my working day with other things. - Really happy with how my article turned out!
Really happy with how my article turned out, available on pages 20-23 of GIST’s most recent issue:
It just ended up coming out so nice! I like how the background has cool abstract maths things going on, and the placement of the pictures etc.
To top it all off, the cherry on top was a lovely comic at the end by Gerard Mooney:
- Emotional Labour involved in a PhD
Okay so a PhD is hard work. But in what kind of way. Yes, there are a lot of deadlines and all of that. But something that is not yet discussed so much is the emotional labour involved in a PhD.
Weekly, I expect to have to convince myself things are going well for motivation. This is standard. I then discuss with my supervisor, and talk to him about things Im struggling with, which takes a fair bit of confidence and vulnerability. But is essential.
Then when you get your work together, you expect to be rejected. From postdocs, from journals. Not your 1st or 2nd but 3rd choice. Thats an emotional hits.
People describe sometimes a PhD as lots of ups and downs you have to surf. But I disagree. On these downs you really have to pull yourself up. Thats part of the work, and what I think of as the emotional labour of a PhD. - My 6 daily PhD habits
Being a Ph.D. student means your style of work is changing quite a lot of the time– sometimes you have many teaching commitments, another day it might be research only. However, one thing I have is some daily short tasks that I do daily which serve as little anchors into my daily routine.
- Check MathStackExchange. Even though I’m not much of an active member, I regularly look through some of the discussions in operator-algebras. Somedays are more active than others but I notice there is often some decent discussion and you see some of the faces you recognise from the worldwide operator algebras community!
- Check ArXiV. There are now two tags I look through daily– math.oa and math.ds. I try to do this every day to read at least the authors, titles and abstracts if it is especially relevant to something I care about. In doing this, I’ve gained quite a lot of awareness for what different people in the operator algebras community are interested in, and means that im more likely to know what to expect when I see someone talk. Another thing that’s nice is seeing someones research pop up that you know, and being able to congratulate them over email.
- Check my Emails (Once!). This is the hardest to stick to, but I really do try now to not procrastinate by checking my emails all the way throughout the day. Instead, I get all my emailing out the way at the start of the day and that way I’m not so distracted by them when doing research. (This is definitely theory and not practice!).
- Exercise at the start of the day. I’m not some superobsessed guy that does a 2 hour gym session everyday, but I try at least to get half an hour of exercise out the way in the morning. I find it gets my brain working that bit better and also feels good knowing I’ve already looked after myself a bit. Since we get to do hours we want as PhD students (most of us) I find that even a late(r) start due to exercise will end up with a more productive day than even an early start where I’m super tired and my brain isn’t even awake.
- (Other type of) Exercise at the start of the day. Bit of a weird one, but I try to do one math exercise (usually something not to do with my research at all) a day just to get my brain going a bit and my confidence up. This usually means working through a dover book such as the Moscow Puzzles or challenging mathematical problems with elementary solutions very slowly throughout a semester.
- Meditate. The benefits of meditation are too many to list and not my expertice but if I need ten minutes off, its pretty common that ill start up headspace and do a quick meditation. This doesn’t help immediately or something but on the long term it makes you calmer and more able to focus your thoughts on a problem at hand. Similar to exercising, keep the brain happy and the rest follows.
- Drink LOADS of water. Self explanatory! See above.
- Check MathStackExchange. Even though I’m not much of an active member, I regularly look through some of the discussions in operator-algebras. Somedays are more active than others but I notice there is often some decent discussion and you see some of the faces you recognise from the worldwide operator algebras community!
- Back again (but a bit later than expected)
This is my first week working in 2022, I’m getting my timetable together, etc. Things are a little later than I was originally planning since I got the dreaded coronavirus. That said, it was quite a minor illness for me; enough to take a week off but not enough to seriously affect my health (thank goodness).
Hope all my readers had a good festive season and a happy new year! - Timeoff time!
Taking my annual leave now til January which is great! Excited for some nice rest 🙂
- En route back to Cardiff
That trip was v successful! I had a couple of really decent meetings with Xin where we covered some TFG stuff. Generally the ideas are flowing atm so from a maths perspective going well.
Additionally, I managed to find a flat (finally) to move back into Glasgow. Its right by the Maths Dept! Im very excited to be doing it right this time round rather than in the peak of lockdown.. but nervous to say the least given that last time it caused a sizeable dip in my mental health!
However, its definitely nicer this time — I have a housemate I know, I have a office I can actually visit and plus quite a few friends have moved up. This also made this last trip much more enjoyable and crucially more balanced work/life wise.
- Very productive and lovely Monday
Today I met my secondary Chris Voigt for a proper meeting for the first time. It was quite good getting a bit of an outside perspective on how my research is going from someone a bit senior.
It was also interesting to hear from him about what research things he has been doing with Quantum Groups and how it relates to my masters thesis on the kz equations. I think this could be a great opportunity to broaden my research if I find some time later in the year.
Then I met Xin, we first had a look a at some lecture theatre (?!?) 😎then had a discussion about a few concrete problems related to tfg which was really great. We met up afterwards in the seminar room to congratulate Jamie Antoun on passing his viva with flying colours! Had a skype and then just a chat between some of us.
After work we had our department xmas meal I organised at Sichuan house on Sauchiehall Street. The place was Xins choice and certainly a treat. Good chinese food in the uk is hard to come by generally so was surprised by the quality.
Not to mention the company too— there was 8 of us in total: Runlian, Xin, Ali M, Ujan, Jacek, Francesco and Chris V. If only David was there we woulda had a full GB spread and then various people from all over the world. It was really interesting hearing a lot of peoples perspective on current events, particularly Ujan’s “Indian warning” on nationalism ;)!
But yeah feeling very at home in the department at the moment which is definitely a good feeling.
- Nice trek near loch lomond
Good friends 
Good view - Three of Simons Masters Students in 230
So to my surprise, I found that as well as myself, not just one other (Anna Clancy) but two other (Tudur Lewis) PhD students which share an office with me all did their masters thesis with Simon Wood at Cardiff 🤯 how crazy is that?!?
One after another, from 2017-2021, Simon has been sending us off to bonnie Scotland. - en route for Glasgow
I’m on my way to Glasgow for the second of my research trips this semester.
Feel excited to get chatting in person with everyone about maths, as well as catch up a bit with people in the department. Since I’ve been teaching I feel like my face might be a bit more recognizable this trip, which is always welcome!
Snow topped black mountains 
Sunrise over south wales - Essentials of a good PhD
Someone asked me what to consider before going into a PhD. I thought I would list a few bits of what I know by now are essential to a good PhD, rather than just “good”. I think good PhD is subjective. In order to have a PhD one needs to be/have:
- First and foremost healthy both mind and brain. This comes before all else and serves as a foundation for the rest. Without this you have nothing.
- Good relationship with supervisor.
- Paid enough that you think is fair. In this: your stipend, tutoring pay, and finally travel stipend. The ability to get more money through a grant or something.
- Have decent rights, like any other job. Check your holiday rights. See how many sick days you can get. This is important but neglected by so many!
- At a uni where you can make friends. Travellings great, but probably not a great idea to do a phd in Korea if you can only speak English and a bit of GCSE french, for example. Being able to speak to the locals is pretty key. (I have known someone on the wrong side of this…)
This list is by no means exhaustive but hope it helps people out here looking for January applications!
- Do you have a job?
My partner’s parents, some randomers at the pub, my family members, and my landlord all asked me this question over the course of my PhD so far. Even someone from my bank asked me, its a pretty standard question to ask someones employment status. However there is a big problem with it as a PhD student.
….but the truth is I don’t know the answer.
The truth is I *kinda* have a job, but I *kinda* don’t. For example the following things are kinda worky:- I do work
- I get paid a regular amount
- I can pick up more hours in the form of teaching
- I get holiday/annual leave
- I have a contract
- I have annual reviews
- I have some of the rights afforded to UK workers
But at the same time, these things aren’t:
- Im working towards a
- Im my own boss/dont have a boss
- I choose my hours and dont have to work at a particular time if I dont feel like it
- I have a stipend not a salary
- I dont have all the rights afforded to UK workers
- I can quit and wouldn’t have to work my notice
- I dont know whats expected of me each week/month/year
That said, I usually say yes. I know people with less “hobby” jobs than me! If I was at the pub, I would say something along the lines of “it’s a bit like being self-employed, but you don’t really know what you’re doing or how it makes money”.
Sidenote: Something I also want to note is that in Germany for example, the answer is a clear yes. You are employed by the uni and have a standard working contract. So this varies from country to country. This is not to say the German system is better or worse (it’s better!) but just that it’s a political choice that the UK system is set up in this way. - Upon reflection
As you might tell from this blog, I’m someone that does a lot of self-reflection. I know a lot of other people that do self-reflection, but some people are genuinely quite bad about it. To illustrate I have the following example.
“I spent 3 hours trying a particular approach to a problem and got nowhere, then suddenly realised a much easier way and got it in 5 minutes”
This is a common scenario in mathematics. I do it weekly if not daily. It IS frustrating. But how should we reflect upon it. Some people would have the following self reflection:“I wish I didn’t spend as long on the dead end. If I wasn’t so confident in that approach I would have spent less time on it.”
I think this is good. But it misses the potential benefits. So ask yourself are you in situation A:
“When struggling for ages on that problem.. I think I kinda learnt something about the maths I didn’t understand, so it wasn’t time wasted”
Or B:
“When struggling I just got really angry and went in circles. I decided I hate maths and started crying.”
These should be handled very differently. In situation A: you did the right thing sticking by the problem. Perhaps in hindsight, you woulda tried some different approaches but chances are the problem is there to aid learning not so you can solve it. In B: you weren’t really working at all. You should notice this and make a note.
Just a little analogy but its just to say that self reflection is a part of the pedagogy of mathematics. Its not just about wasting time, at the forefront should be “what did I get out of this work” and if the answer is a lot, its good work. If the answer is fuckall, its not work at all! - The motivation short circuit
Has anyone experienced the following pitfall in a PhD?
One knows that employability is important and a little shaky. So, not just work matters but also how the work is presented. However, what this means is one spends a significant amount of time in work thinking about how to massage your achievements to look more impressive. However, at this time you are not yourself doing work that might actually lead to something much more impressive like a paper or some new maths. Instead of being motivated to do something great, one is motivated to *look* like they are doing something great, which is a very different thing!
I call this the motivation short circuit and by god do I have this problem.
Like, look at this blog for example– so slick and has loads of content. However, if I look at my academic CV it’s a little wilt perhaps. The other week, I spend almost 4 hours looking through website formats to make my blog look nice, but didn’t change the content at all. In the process, I ignored some work set by Xin that was arguably much more important for progressing my Ph.D. but would have nowhere near been at the “curative” step yet.
It’s hard because one has very little to show from all of our hard work, especially in the first and second years. So we make something to show out of nothing, and settle with nothing to show from our work! - Cool “philosophy” behind Gabor’s talk
Just on something quite C* algebraic here. I really like this pretty neat idea of Gabor Szabo and Jamie Gabes classification program for actions of Groups on C*- algebras.
Its essentially to generalise the Elliot program which describes a pair of groups, k0/k1 that classify a broad range of algebras called classifiable C* algebras. For those outside of C* algebras, this methodology has been very successful, and has somewhat concluded.
The idea is to consider this whole program as just the case of the trivial (G=1) acting on an arbitrary C*- algebra. Then to aim to generalise the methods obtained throughout the classification program for arbitrary G.
Just something from the analysis seminar I picked up and thought was neat.
- New text
Those of you that are keen-eyed might have noticed a new text in my informal texts folder. This is “groupoids by example”, a short text introducing groupoid C* algebras and constructing groupoids such that the corresponding C* algebras in Davidson’s book “C* -algebras by example” are isomorphic.
- Mindful Maths
In the last year or so, I have tried to keep up a regular mindful meditation practice. This really has helped my mental health, productivity, and my personal relationships. All of this is to be expected, but what I wasn’t expecting was it to change my relationship with mathematics so much!
At the risk of butchering the topic, mindfulness as a concept is meant to bleed into everything you do– how you eat, how you work, and how you sleep. It serves to mirror how one feels, and lets you *be* in your feelings. Stop, acknowledge, and let go.
This can be an incredibly useful approach when approaching mathematics problems. Let me give you an example. Recently, I was given the task by Xin to find a groupoid that generates the Toeplitz algebra as a C* -algebra.
My first reaction was anxiety and a bit of panic, to be honest. It seemed like I didn’t know how to approach the problem and this was a bit of a cause of distress. Of course, this is quite the usual feeling to get when you are thinking about something for the first time. At this point, however, I have now got a very different approach to this than I used to.
In the past my reaction would have been to come back when I feel ready. Perhaps to do a bit of background reading, try the easier cases, and let it come to me. This approach is fine.
But, what I do these days is quite different, I prefer to take a bit of time to sit and let these “bad thoughts” come and go. Ive realised one can’t really avoid them, so it’s best to just get it all out in one go. It’s a bit like taking the emotional trash out. I find once I’ve sat with them, when I come back to the problem I’m much less likely to get these thoughts again.
When doing maths, this has affected me in a manner of minute ways. Im quite conscious of how I am breathing or deliberate in my approach now will sometimes write what I am doing as I do it. When focusing on maths, I make sure to focus on smaller details rather than the problem as a whole, which is surely more productive. - I blog!
But why?
Honestly, people have asked me this and I have no idea. I have no social media, so this is a decent place rather than facebook to put a bunch of thoughts and express myself a bit. But I think its more than that, to me its a bit like a diary. I quite like looking through like a diary, and seeing my PhD and personality grow as I move through it. Its also a great place to chuck up random bits of maths which aren’t ArXiv worthy but I’m still proud of. - Speaking tips picked up from my recent workshop
First of all, massive shout out to the doctoral school at COSE for putting together such a good workshop. I got loads out of it, and wanted to share some of these tips (as much of a note to myself as to anyone else!). So, in no particular order, I learnt about the following do’s:
- QR codes. They work so well for linking to a paper, virtual handout, and are scarily easy to make. This means that people can take a part of the work from home and increases interactivity. It could link to your blog so people could see more of your personality or all number of things.
- Handouts. Give your audience something physical such as a mini-poster, or essential notation for during the talk. Just by having them, they are forced to pay attention, and can bring something away physically from the talk even if they arent mentally there (say, a long conference)!
- Make it a performance. Dont be afraid to stand out, the stage is yours!
- Call to action. Don’t be afraid to ask the audience for collaboration if thats what you are looking for.
- Repeat, repeat, repeat. Whatever is a key point for your presentation, repeat it at least three times. Its worth presuming especially in the age of virtual conferences that your audience is not paying 100% attention. So its on you to repeat enough times that the majority of the audience gets the key points.
- Active communication of your needs. At the end, perhaps explain an open problem you think is relevant and explain who you would like to work with and why (e.g. I like the idea of working on the HK conjecture but would need to collaborate with someone that knows K-theory better than me!).
- Name open problems. People love a question!
- Ask your audience a question. People might hate a question here, but it will at least make a few of them wake up!
- Put your nerves to rest. You will be anxious. Have a plan to chill yourself out.
- Technical aspects should be thought about. Get a nice mic, people will love it. Im lucky enough that my partner does podcasts so we have a really nice usb mic– but I never used it even for big talks. I shoulda, and will from hereon. In the same vein, make sure there is strong lighting from BEHIND the laptop in order that you are well lit, and outside of there that the lighting is fairly dim. Position your laptop head on and with your eyes a third of the way up the screen. All of these are important points I’ve seen missed a million and one times!
…and the following do-nots!
- Bad beginnings. We talked a lot about the cringey starts to zoom talks where one says “can you hear me….” or waits around for ages before beginning. The talk begins when people can see your face.
- Have no story. Every good talk, no matter the subject, has a story.
- Have too many points. The maximum takeways should be 3 points, unless you are doing a miniseries. Make the points multiple times.
- Basically, talk the same as everyone else. It won’t make you stand out. Be yourself and let your personality shine through!
- Speak at a million miles an hour, so that no one can understand a word you are saying.
- Lots of admin; lots of work
Does anyone else get days in their PhD where it feels like all they did was admin. Im not here talking about fun stuff like arranging speakers, I’m talking seriously boring stuff. I spent a few days at the start of the semester like filling in forms and enrolling (for the second time) on my PhD. I also have to mess about loads on the GTC forms and enrollment so that I can pass through graduate school.
It’s important to remember even though this is a shitty part of a PhD, it is still a part of a PhD. You shouldn’t overwork yourself in weeks like this cause you feel guilty not doing enough project work. Also, if you hate admin, frankly, a PhD isn’t for you because you have to do a fuckton. When I tell my mates with industry jobs I do a PhD they generally seem to think I don’t have to worry as much about replying to emails and sorting things with HR, but if anything I have more. The uni is truly a mess of knotty systems of rules and you gotta jump through all kinds of hoops and work out exactly what everyone’s jobs are. Since academic staff have all kinds of admin duties themselves there’s a lot of trying to work out who even is the right person to contact when you have an issue!
But this isn’t a hate-post just to say yeah, that’s a part of a PhD and quite an annoying one haha - “You, not your thesis are the product of your PhD”
Before I did my Ph.D. I had already heard the above phrase so many times that I kinda didn’t even think about it.. but I think its actually a really important guiding piece of advice on a PhD and I kinda wanted to do a blog post about it. Cause I think sometimes as well, people are still thinking of their PhD as a series of papers and not learning/shaping who you are as a person. I think its really important to reflect pretty deeply on it… so here are some thoughts I had about this phrase!
Your papers/thesis are never going to show off everything you learnt in the last 4 years from a technical perspective. This is the first perhaps quite obvious point. If all you had learnt was directly related to your thesis then the only paper you would be able to write is your thesis. Your interests would be wayyyyy too specialised. On the flipside, people sometimes regret perhaps reading a paper or book that didn’t end up being relevant to their thesis. Don’t do this, be happy. The whole point is you learn stuff and most of it isn’t so relevant but it might be in the future.
The next point is besides technical knowledge, there are also all kinds of softer skills you gain from your PhD. The first for most of us is that its our first proper job, so we have to work out how to email professionally, manage our time and commitments and things like that. But also what you gain is the ability to learn mathematics in a really strong way, a lot of willpower and overcoming struggles. There is a lot of working on your own and a lot of interpersonal skills too. For me, I’m learning how to travel and things like that for the first time because I’m not very well travelled. I also learnt a bunch of other things, like a bit of how to code and how to teach, which is in no way related to my thesis but will surely make me more employable. The stipend funds you to become a doctor, not for you to write a thesis.
The final point I wanna make is though how good your thesis is very related to how good you are as a researcher, it’s not the be-all and end-all. For instance, you can have people that spent way more time on their background than others and have the potential for much more work at the end of their research, but perhaps fewer papers. This is another comfort but also warning of the saying. Getting the perfect thesis is okay, but it’s important that you get all the other stuff around it too. You want to be feeling like you embed in the community, and know some people. Make sure your PhD is rounded and that always you are motivated more by what kinda person you want to be when you graduate rather than anything else.
Something I remind myself often, which seems a bit of a counterintuitive thing or maybe demotivating but actually is quite a good thing to reflect on, is whatever thesis I end up writing, Xin could have probably written it as well. It would probably be a lot less effort for him, and the thesis would pass unconditionally since Xin has loads of experience writing proper texts etc.
The point of this is that it’s not so much about the research. Because if it was all about the research, Xin woulda just has done the research. It’s about me! And this is a good thing too, it kinda can take the pressure off yourself and help you prioritise. - Blog Revamped and Rebranded!
As you might have seen, today I’ve done a little bit of reformatting and rebranding my blog. It needed the dust blown off it a bit since it was still using the same old stale template from 2018. Also, was very hard to find all the texts and stuff that I had written.
About Me
Hi, I’m Owen. I blog about maths
C*-Diary 