Season finale

So, as the season finale was approaching, with so many expectations and wishes coming with it, who would have thought it would turn out this way? What has been an incoherent year of jaded longing, stark disappointments, and overjoyed rapture was finally resolved in a more or less ultimate fashion. Or, at least, in a few aspects. And I am so immensely excited about what is going to ensue.

I have lavished so much of my energy and chiefly inner resources on transmuting my situation into an advantageous experience. Though unsure about the tangible outcomes, many a thing has changed, much to my delight at times but alas, way too often it was much to my horror. Irreversible though some of the changes may be, I am generally enthusiastic about and open to what the future might bring.

Everything happens for reasons I just don't know. 

Or so I have always remembered from my youth, all those distinct memories of my warming up and getting up to speed before the many confrontations. And my god was I eager to win. I fostered a burning passion inside me that drove me to my many triumphs, I was unstoppable and inexorable in my efforts to secure my position even further. What a surprise it was then when all this sensation had vanished, when I recognised I no longer cared about winning, when I doubted my own motives. It may be all long gone, but I trust I have managed to recover at least some of my former ambitions. Hence, I have created my homepage:

Behold!

https://sites.google.com/site/sotakovajanahomepage/

In the meantime, I also gave to talks in our student seminars, commentaries of which will follow in the next couple of days.

San Francisco and the like

Dear all,

it is me again. I am currently on a short break in San Francisco and this place is insanely wonderful again. I love where I am staying at and I am enjoying my sunny days to the fullest. Tomorrow, a tougher regime will start as I need to get going with algebraic geometry at last, however, with the amazing garden I can use and the quiet and easy neighbourhood I think I should succeed this time.

While this is not the view from my window, it is almost as pretty!

So, what has happened in my life since the last post? I gave another talk, this time as a part of my regular coursework in Regensburg and on geometric group theory, on the topic of Coxeter groups: these are abstract reflection groups, which means that they are generated by "reflections" (elements of order 2) and their Cayley graphs (with respect to the generating reflections) have some very nice geometric properties. For those interested, the notes for my talk are available here (I talked about various definitions of these groups and why these definitions are equivalent):

https://www.dropbox.com/s/ospztlbnz0vhkf9/Coxeter%20groups%20talk.pdf?dl=0

However, that is not all I have been up to. I have started looking into things I could be doing with my life and realised that there is a lot of math I enjoyed and somehow stopped doing. Also, I tend to perform much better when I have a long-term goal ahead of me, be it preparing for a seminar talk, writing down a set of notes of what I am entertaining myself with, preparing for summer schools I want to attend... So, I decided to find some topic for me at last and devote myself to a more systematic building of the theory, perhaps writing some things down.

And what could be better than things I have already thought about for some time and for which I do know the people to talk to and get new insights and ideas? Yes, I am talking about my thesis, the revision of which is long overdue, I could finally include the parts I didn't have time to write down before, I could restructure it. There are a few reasons for doing that:

• I changed a bit during the year since I wrote the thesis. I still want to tell a story when I write about mathematics, only I have become more aware of the preferences of other people and learnt how to write slightly differently, in a more structured way.
• I have since learnt and understood much more than was is included in my thesis. And then I have forgotten half of it. To remedy this unfortunate thing, I decided to write down the things I remember, go over them several more times and figure out the details that have (since) slipped my mind.
• As I said, having a long-term goal is a better incentive for me than just learning things randomly for the sake of continuing my studies (and I fear I have been slipping into that since the beginning of my semester).
• It is difficult to get things done when you do not need to do them, so by announcing my commitment publicly, I hope to put myself under just enough pressure to actually work on this task regularly, because I tend to work differently if I am supposed to deliver a result in the real world, not just in my head.

So, wish me good luck and please, do support me on this journey. I hope it will work out eventually.

Elliptic curves and complex multiplication: the talk

So, here comes the link for the Jupyter notebook I used during my presentations. The English notebook is already in its final location. Just a few words of caution, though:

• apparently Sage does not support opening Jupyter notebooks directly so please use either the non-interactive html version or save the file into any of your projects on Sage cloud,
• it includes both the Jupyter notebooks and my notes for the talk, which served as a guideline for me and an attempt how to summarise my thoughts on the matter,
• the notes are by no means in their final version, they were meant for personal use,
• feel free to do whatever you with with these resources and please, do contact me if you have any questions or have anything to say, even just "Hello, it was a fun read" would be great: then I would know that this format is good and perhaps I should finish and post some other endeavours in number theory I am toying with from time to time,
• none of the things include any references whatsoever: The standard reference for elliptic curves is Joseph Silverman's books, then I read some parts of some articles by René Schoof and Francois Morain, Reinier Bröker's thesis, good sources are past elliptic curve crypto conferences, I enjoyed Andreas Enge's slides and Ben Smith's and then, of course, the best source for me is everything written by Peter Stevenhagen, from whom I am learning how to enjoy algebraic number theory via playing with curves. And that's a lot of fun.

This is an updated link to the folder with all the files:

https://cloud.sagemath.com/projects/d2b6ab7d-a1b1-4f3d-a4f7-3ae39b7ff889/files/

Elliptic curves and complex multiplication

So, the second semester has started already in Regensburg. I don't know what that feels like, though, as I am only going there tomorrow. I had my reasons for staying home at the beginning of the week and as I was still home, I took the opportunity to talk at the number theory seminar in Prague. And I think it was a great decision as it made me really happy.

But first things first, I had a practice round the day before, in Brno at a nascent student seminar or meeting group. The attendance of my first attempt was amazing and people said they enjoyed it, even though they did not manage to slow me down. I do acknowledge that it was not the best but I simply speed up if things do not slow me down. I was not pressed on time that much but maybe I just wanted to say too many things and took the wrong way to get there. But still, some people liked it and I am glad I had the opportunity to try things out.

It took the whole evening and the terrific travel to Prague today to put the talk into some presentable shape; second train accident in one week in my hometown resulted in a very stressful time for when it was not clear at all whether the train will come and I already knew that no other connection would take me to Prague on time... Well, the train did come and I did manage to calm down enough to focus...

But in any case, I talked about elliptic curves and complex multiplication. It is not the best topic for people who have never heard about elliptic curves nor algebraic number theory and since, in addition, I tried a more complex approach in Brno, it was a lost case no matter what I tried to do in Brno. So that was the wrong attitude. In Prague, I was more lucky with an audience who have heard about elliptic curves and seen some algebraic number theory so I could talk about more things and I got some very interesting questions from the audience and I think I managed very well. People were smiling in the end and I believe those smiles were not out of pity.

But here's the actual talk summary: I gave a talk using mostly Jupyter notebook as a long notebook with empty spaces to separate my "slides", I tried to give a lot of Sage examples and explain the things on what could be seen. I am not sure if that was more helpful or confusing but for me, it was very beautiful. The talk was in Czech but I am willing to give it a bit more time, fill in the details I wanted to mention and publish my notes and the notebooks in English as well.

But for those eager to see what I did and those willing to learn some Czech, I am going to upload the notebooks with some explanations and clarifications when I get connection decent enough for the cloud to work ;)

Update: for those eager to see the notebooks, please follow this link and read a bit of background info on the files.

This is an updated link to the folder with all the files:

https://cloud.sagemath.com/projects/d2b6ab7d-a1b1-4f3d-a4f7-3ae39b7ff889/files/

Another instance of my amazing life

Dear all,

I am doing fine. Or rather, I am doing amazingly well. I have been enjoying my long spring break to the fullest, frankly, even more than that, it has been some truly enjoyable time. My March was one of the best months in my life though it was very exhausting and demanding... But frankly, it was great to slow down towards the end and get some rest and easy time with my family and friends. I really needed both the excitement from travels and the easiness of my home.

So, this month I spent a great week in Mexico, a week in California, a week in Arizona, a week at home and some time in Bratislava. I have enjoyed the many amazing new people I have met during my travels, reunions with old friends, transforming some of my former acquaintances into good friends and most of all, coming back home was great and I took some time for a decent spring clean up in my room, including moving the furniture around to make a radical change to my room. All in all, I think I am doing very well.

Cacti, cacti, cacti...

There are still some minor issues clouding my days, as I am not sure I am heading where I want to: I know what I enjoy, I sort of know what that means in terms of studying, I sort of know what kind of things I want to do apart from my studies, I am just very unsure whether I am currently on the right track... And sometimes I get opportunities to change things radically with an evenly uncertain outcome, that makes for some hard thinking. Some of the opportunities are crazy but so appealing: who would not want to leave school and spend three sunny months in California with their beloved ones? Especially if not choosing so means to spend three months without them...

To be or not to be in SF? Is that even a question?

But anyway, I still have some exams to do, I promised Vita Kala that I would speak at his number theory seminar in Prague and that is some exciting time for me, I am so optimistic about my talk and willing to experiment with a slightly different way of presenting things. Not just the chalkboard talk... On one hand, I want to talk about things I have been trying to learn myself recently, which is complex multiplication on elliptic curves. On the other hand, I should not assume that anyone knows about elliptic curves. And what I love the most about this topic is simply how much of my knowledge this topic is actively using, let alone how much it builds on to get the right picture. And I sort of like to switch between the various mindsets one can espouse whilst talking about elliptic curves.

After that, I should head back to Germany after some 2 months of being away. I wonder what that will feel like, how much dust has accumulated in my room, whether there will be more light during the day and whether my hallmates have somewhat transformed into considerate people. With my timetable again starting at 8.30 every day I would really like to get some sleep regularly. And after that, who knows. There are some amazing things potentially coming my way, so I hope time will only make things better and better.

So much for today's share of procrastination. My elliptic curves are waiting for me!

Winter semester evaluation, pt. 3

Some you might object: wait, where is part 2? The reason is, I still haven't come to terms with that happened to me last semester, how little I did, how poorly I performed in my classes, how bad simply everything worked out for me. I don't want to spend time on those very unhappy moments for me again, especially as I am feeling much better again, after a lot of hard work from me and everyone around me. I am so disappointed with my past semester and super-disheartened, so I am postponing my summary of the previous semester until I am confident it will not get me down again. Seriously, I very much prefer doing maths to crying all the time, so...

And now for the plans for the future. I still need to take all my exams: algebraic geometry I (schemes), elliptic curves and modular forms (self-contained course that allows you to define modular forms as sections of line bundles) and linear groups and heights (Breuillard's results on growth of linear groups and Lehmer's conjecture, a pleasant mix of height estimates, eigenvalues, logic and nice people). So there is indeed a lot for me to do. This is approximately the end of March and the greater part of April.

As for travel plans, I went to a school in Heidelberg, studying topics relating to L-functions and automorphic forms. It was more analytic so it did not suit my taste very well, but I survived and learnt some bits.But then I got to meet my mentor from Leiden in Amsterdam and that changed so many things for me. Simply coming back to Leiden made me so excited about mathematics again and having someone listen to me complaining and analyzing what was happening made me feel positive that I can change things. Or that I even want to change things...

Now I am in Mexico at a more algebraic school on moduli of curves. I love all the various facets of this theory: there is a bit (in fact, a great deal) of everything. Classical theory, minimal model program, geometric invariant theory, some tropical geometry. Lots to enjoy Guanajuato is an UNESCO World Heritage site, so I am excited about all the baroque collonial architecture.

Though, I will not move from the analytic topics for long as I am attending this year's Arizona winter school on analytics methods in arithmetic geometry. The reasons for this is that I had been interested in Sato-Tate's conjecture so spent a fair amount of time on more analytic topics this semester. Moreover, it ties in nicely with some of the questions I encountered in my cryptography endeavours, so I am rather excited about the school.

And then back and some studying for my exams. Yay!

Then, I am staying in Regensburg for the Spring semester, beginning mid April. And with a gorgeous selection of courses (not their official names, but presumed contents):

• Arithmetic of modular forms (as has been advertised/promised so many times when dealing with the analytic aspects and principal bundles this semester, as if it is going to be any easier), Shimura-Eichler (allegedly the goal of the course), L-functions
• More schemes and coherent sheaves in algebraic geometry
• Seminar on the Weil conjectures for elliptic curves, following Silverman (so the proof using the Weil pairing)
• Coxeter groups seminar (a big unknown for me, though I am giving the first talk of the seminar, seems to be a generalisation of symmetry groups including nice pictures)
• Representation theory and automorphic forms seminar. Having offended my professor with not knowing any functional analysis, we will also focus on the decomposition of representations for compact operators and other important topics every mathematician should know. We could also discuss Tate's thesis.
• Deja-vu's in local class field theory seminar (for which I chickened out again and decided to give one of the introductory talks on local fields, my favourite topics) with grou cohomology for me. Perhaps class field theory is still cursed for me, however, we will be reading Neukirch, so I hope it will finally work out!

So, lots of things to do for me! But I guess it is time for me to work hard, math is beautiful. There will be a bit more travelling for me to do, but in general I hope it will be a more relaxed and quiet semester for me than the last two. So, we'll see! I am excited. Sort of. But worried about my future semester... I don't want things to get all messed up like the previous sememster. I still don't know whether I am still falling into this abyss or whether I am finally climbing up again...

Winter semester evaluation, pt. 1

So, my semester in Regensburg is over. I think it is time for a small evaluation of my study process, some of which has been implied in my previous posts. First I would like to focus on the human topics, with follow-ups on more specific issues, like math and future and traveling.

Perhaps the most important realisation for me was that I cannot do mathematics without substantial support of my social group and perks of the first world society. I need to talk to people to avoid falling into depressions and possibly to people outside mathematics to realise how amazing mathematics really is. In this respect, my German class was immensely helpful, I met several amazing people there who have been very supportive and fun to be around.

Also, I have come to appreciate the role of my family and a place one truly belongs to without question. It feels great to have an amazing home one can come back to whenever needed. I had been feeling out of my place many times last year and it has negatively impacted my well-being. Long travels home might be time and money consuming, but even if so, this is certainly a way one can buy happiness. I don't know anything more beautiful than a walk in the winter forest, breathing the sweet fresh air and feeling at peace. And feeling like a part of the particular place, not just a visitor.

For various reasons, I have not been able to relate to my own classmates as much as I would have liked, but one cannot force math onto themselves and my math appetite has moved towards the evening hours. Also, as I have not been in the best of my moods basically all the semester, I had been grumpy and careless and frankly happy just to be holding my head up at times, so I hope the following semester will be better.

There have also been some unfortunate news from my personal life, some of which shook me more than others. But I am lucky enough to have people around me who helped me overcome these shocks. And compared to my previous semester, life has been rather nice to me.

Still, by the end of the semester I was feeling so overwhelmed that I decided to just stop pushing and go for holidays. This was a great decision and I am happy my partner has been incredibly understanding and flexible enough to drive me to the mountains and spend an amazing week with me. It was a good decision as now I am full of energy and willpower to work hard and since I've got back, I have already done better math than in the week before. And I am still ready and eager for more!

So, that's the human discoveries. As for math, it will be the contents of my following post.

Serre duality

So today, in Regensburg, at our student seminar on Riemann surfaces, I gave a wonderful talk on Serre duality. It was a rather long talk (80 minutes), we had to prepare some sort of detailed handouts for people and we were actually supposed to prove things rigorously. I had already given one such a talk on simplicial topology earlier this semester and it went very well, so, at least, the format seemed to be suitable for me.

I think I managed just fine. It is difficult to give a long proof in such a limited time (at least for me), especially if you need to build up all the theory beforehand. The book my talk was based on is the "Lectures on Riemann surfaces" by O. Forster, in which the proof is, to my taste, rather convoluted and unmotivated. And, admittedly, too long to present in public. And except for the last five minutes of the proof, in which I had to use a certain diagram for one sheaf and then, in reality, used a different one,  I am quite confident that I could give people more than just the book.

Moreover, I used a slightly different approach than the book, because I could not afford the luxury of the clean, uninterrupted, yet unmotivated exposition. I put a great emphasis on how the various objects I defined will be used in the proof or why we care. Unfortunately, the audience was traditionally ranging from dispassionate to impassive and had it not been for the experience gained for me and some details falling into place in my mind as I spoke, I would have preferred to spend my afternoon differently.

For those of you interested in my impression of the Serre duality, here is a link to my notes (beware of the last part, which is a bit inaccurate and I relied more on my hand-written comments from my own perusal of the proof)
https://www.dropbox.com/s/2frsw8ywxm3xl79/Serre_duality.pdf?dl=0

And bonus points to those who correctly identified which meaning of perusal I used there.

Regensburg yesterday.

Germany in the new year

So, a happy new year! It is still 2016 so I am not late at all. I have been getting a lot of amazing rest and truly enjoyed my Christmas holidays. Study-wise I was mostly concerned with the metaphysical questions and seem to have come to some conclusions regarding my own motivation for studying and the purpose of my life.

Admiring the beauty of my homelands.

With some exceptions and taking into account my exceptional amount of bad luck, which has persisted since I've got here, I am getting used to being in Regensburg. I am particularly amazed by my elliptic curves and modular forms course by professor Guido Kings and also my topology experience with Clara Löh is rather positive, so there is interesting content for me to study, but somehow I am still unable to arrange the conditions for me here to actually study and, you know, learn things?

So I am still struggling with finding any kind of solution to this problem: I sincerely acknowledge that some people are happy and productive here and it is a motivating and supporting environment for them, on the other hand, I honestly feel a significant decrease in convenience from my previous places. And as much as I am trying to get things done, I am not feeling I am accomplishing anything of that sort.

Eagerly awaiting snow also in Regensburg!

So, the big question still remains and I do apologise to all my poor friends who have suffered from me being singularly annoying and disheartening with bringing it up over and over again, as the advice is most strikingly divided between "go now" and "definitely stay". It makes it especially difficult as I am rather convinced that I do want to do math and do not end up feeling resigned that I am just going to suffer through this period, hoping that it will improve in some vaguely defined future.

As for the happy things, I have been learning Serre duality and I am rather impressed with the beauty of the ideas there. I have a bit of a difficult time trying to structure my talk but I am getting to grips with it. Wish me luck, I will share more next time.

Home versus the rainy Regensburg. Well, whatever.

Windberg

So, another post? Yes. The name of this blog involves both my life and math and recently there has been more life and less math in my days, but I've just come back from a sweet math meeting and I want to share with you again.

Happy Nikolaustag! Better late than never!

So, how have I been? Decent enough. Things are getting better for me in Regensburg, I am attending my classes, doing my homework, occasionally preparing at home, the usual, I am even studying bits of German and I like it. I haven't got used to the morning temperatures in my classrooms, though, so usually after my first class I am frozen to the bone, however wrapped-up I might be. Afterwards I usually go back to my room to drink some hot tea and snuggle up under the blankets.

Overall, I like some of my classes, I am making progress in my studies and I try to study some of the books I have long longed to read. Silverman and Cox have been on my bedside table for some time and I even get round to opening the books and marvelling at the wondrous elliptic curves.

Anyhow, Windberg. The annual meeting of the Regensburg Graduiertenkolleg "Curvatures, Cycles and Cohomology". The people there were so nice and lectures fun and accessible and yesterday we had a funny Cohomology evening: a two-hour panoramic view of the various cohomology theories one might encounter in geometry. And it was awesome. Though, we might have enjoyed it more had it not been at 10 pm. 

Windberg Monastery. Sort of looks like Špilberk in Brno.

So, that's it for now. I could not stay in Windberg for longer because of other travel plans but I loved it there and I am feeling postiive now that more math will come and things will get even better for me!

***

And now for the people who are interested in how the evening went: I have not yet decided how to look at cohomology as I am mostly home schooled in this aspect, but for me it is both about invariants of some geometric spaces (for one, the well-behaved cohomology theories are usually homotopy invariant), encompassing geometry of the space (like Chern classes) and obstructions to some constructions (I think of Ext and splitting of short exact sequences, though) in terms of abelian groups. But many smarter people have thought longer about this. So, what have I got out of the talks? (All the misunderstandings are mine, remember, 10 pm.)

So as a prelude we already saw some chain complexes and simplicial homology in the morning and in the evening we started off with de Rham cohomology. For that, you need a smooth manifold and smooth differential forms with the exterior derivative. Surprisingly enough, the cohomology groups do not depend on the complex structure of the manifold as they are the same as singular cohomology with real coefficients. 

Then we moved on to sheaf cohomology, more categorical approach but for me also more appealing: one studies the geometrical objects by means of the functions defined on them. And this approach is absolutely natural because usually objects come with a nicely defined notion of functions on them. And functions describe the world (Tom Garrity). It also paves the way for all the abstract algebraic geometry. But the naive intuition of working with functions works, which is sweet.

The following bit was about Čech cohomology. For that I believe you play with real manifolds and glue functions on open coverings of your space. That is also a very useful concept as so many things are easily defined locally but one would like to piece the information together to get a global thingy. 

The most difficult or advanced topic of the night was crystalline cohomology. Toposes and cycle maps were mentioned (Thank you, Bernard). Anyway, it all comes from the celebrated Weil conjectures (counting points of varieties over finite fields), which would follow from constructing a suitable cohomology theory for varieties over finite fields. To be even more vague and imprecise and silly, you need to lift your curve defined over a finite field to something defined over a field of characteristic zero. Apparently, one can do this locally and then attempt to glue and then I was lost.

We wrapped up the evening with a discussion of how to actually work with cohomology. If we have a class in cohomology, how to decide whether it vanishes or not. The nonvanishing can be done finding a pairing in which the class does not vanish. So we played with compactly supported de Rham cohomology and bounded de Rham cohomology and paired the groups with homology, integrating things and checking well-definedness with Stokes and using the axiom of choice to get some nice limit from a bounded sequence which supposedly amounts to chosing a free ultrafilter on the integers. 

All in all a sweet evening!

Morning Windberg on the day of the cohomology evening. The following morning it was foggy and bleak.