A long update after a long time

Time flies, especially when you are having fun... or when you are snowed under with homework. However, I still want to check in here before I seriously have to start studying for all my various exams and also have to catch up with all the work I didn't manage to get done this semester, which is, just like with anyone else, slightly more than feels comfortable.

In any case, I still have a few moments to take a deep breath and do things that I want. Therefore, what have I been up to? My last post is sometime in mid September, since then so many things have happened.

• I am applying for graduate school in the US. It is as much fun as it sounds. So far it involves an incredible amount of bureaucracy, writing and browsing, but the situation has been good so far. I got the Fulbright scholarship, which means that there is somebody doing most of the bureaucracy for me (applying to six universities on my behalf, five of which I would have happily applied to myself) and offering to pay for my first year of studies in the US. The results of my standardized tests have also been good (and now follows a detailed description of the tests I've taken): 
• I've demonstrated I can speak some English with my TOEFL test. That's just a normal English test involving some reading, some writing, some speaking and some listening. Nothing too ambitious, less absurd tricky grammar questions (compared to CAE&CPE), the format of the questions was easy to get used to, so if someone chooses to "train" for the test, it's more about improving English than just standardized useless questions. Surprisingly the only thing I cannot do very well is to speak about familiar topics. Well, perhaps I am turning into an introvert. Who knows.
• I've almost aced my GRE General score, if you don't look at the results of my essays. Those would have been superhard for me even in Czech. This test is a longish one, with some random English part, Math part and the aforementioned tricky essays. The Math part was easy to train for and also lots of fun, because the questions were just high school mathematics, occasionally sprinkled with questions resembling very simple Olympiad math reasoning. It was precisely what I liked: questions that were doable with hard work and preparation, but if one had the luxury of time, one could save lots of time just finding a smart solution. And the English part was fun, part of it was about reasoning about the text and careful reading, but a huge chunk was just these funny questions with 1-3 blanks and lots of choices for beautiful random English words: the task is to get a general feeling of the message of the sentence and then choose a combination of words that together make the most sense; typically all the words could be used to fill the blank, but only one combination would give a meaningful sentence. I like that a lot.
• And finally GRE Subject math, which is just a speed test in calculus. There are many past tests available and good books for review questions, so practice makes perfect I guess. Funnily enough, none of my practice tests ever went over 800 (on a scale to 900), but my final score was much better. I did enjoy revisiting first year mathematics and answering again all the questions about limits and derivatives, I also managed to use up a stack of scrap paper that I had been hoarding on my desk, so I guess it help. Of all the tests, this was the most important one, where preparation was essential. I think I did just enough to obtain a fairly decent score, more preparation probably would have helped to make me more confident, but that would have meant maybe a few months instead of 2-3 weeks. 
•  I am attending some courses. Some of them I am attending more often than others, sometimes there is simply too much homework to actually go to school. It sounds ridiculous but given that my classes are in Amsterdam and Utrecht, going to school means spending 3-4 hours a day on public transport, and sometimes I simply don't want to invest so much time into getting to school. But I have been quite lucky with my classes:
• I am immensely enjoying my class in commutative algebra. Both our lecturers are good in the class and if I forget about all the homework I messed up, there is just lots of math I enjoy: all the integrality questions and dimension of (affine) varieties is really dear to my heart. Also, most recently we talked about Hilbert series and Hilbert polynomials and understanding these brings so much joy into my life (because they just tell you so much about the ideal / variety / meaning of life). Except for the policy on homework that is ridiculous: it does make me a way worse mathematician. We are forbidden from discussing the (graded) homework with other students, we have a page limit on our solutions, which just damages the writing skills so much. I think imposing a stigma if someone wants to ask a question is the worst thing a teacher can do to their students and also encouraging people to write math in large blocks of unintelligible text full of shortcuts (because nice layout and indenting takes up space, you know, let alone diagrams, equations and god forbid careful case distinction) is also in the opposite direction to what should teachers do. Still, the material in the course is good and luckily there were only three graded assignments, so now I can happily talk to people about commutative algebra again...
• Another excellent class is my p-adics course. We are closely following the book of Koblitz (p-adic things and zeta functions in the name). The instructors are amazing and the homework is just the best homework I have seen: really engaging, interesting questions for which we have just enough tools to solve them, the problems do not take hours to type down and I've just felt so happy doing most of them (except for one part of the lastest homework sheet that stressed me out incredibly much). I don't actually like the things we are doing in this course, all the analysis and series and such, but it does make me feel like I am learning a lot and I am simply getting so much from this course.
• I am taking some more algebraic geometry, too, because there's never enough algebraic geometry. Some of the lectures have been a bit hectic, but the homework is fun, the things we are doing are engaging: we still only talk about varieties, but knowing a bit more from the viewpoint of sheaves just helps me appreciate the beauty of the classical theory even more. I think I am not precise enough in all my statements about geometry, so it is also a good drill having instructors that are far from lenient in mathematical reasoning. Just good practice and like I said, there is never enough algebraic geometry, especially for someone like me who has never really bothered learning it properly...
• Then there's the Galois representations. The course is fairly advanced and we are covering so many new topics that it is a bit over my head at the moment, the homework typically takes the whole week to sink in and understand what one is supposed to do. But I am taking my time, slowly getting through all this representation stuff. But it is getting too far into representation theory, in which I just have to take so much stuff for granted, that it doesn't make sense to me to try to understand it so deep: there are too many gaps still left. I will see how that goes further, I have already learnt so many interesting things from there so I hope I will be able to consolidate the knowledge at some point.
•  I would almost forget about algebraic topology, well, that's in Utrecht, so that doesn't happen to me that often. Also, most of the homework has been standard homological algebra, took too long to write down (seriously, typing down the homework shouldn't take twice as long as solving it). But it's a fairly standard course in algebraic topology so I hope I know what I am doing.
• And Dutch. I am almost fluent in Dutch now, well, at least in my personal dialect that includes a lot of made up words from English or German. People don't understand me very well but I don't understand them either, so I guess we are even. I still have a few months to continue learning, so hopefully the mutual understanding will only improve.
•  I've been also living, at least a bit. We started baking with my flatmates, so far only simple things, but brownies are always good and the marble cake we had was also amazing. We also dare to cook some more interesting dishes than pasta (which is what I would cook for myself basically daily), with the pinnacle so far being a leek quiche and spinach pancakes. I am running a bit, exercising and trying to get fit again.
• And I have mostly been in Leiden. I like it here. Seriously. This is an awesome place. Come and see for yourself!

I have some good hopes about my future life, so let's see what comes out of it. 

Leidenlove

Greeted by the incredible weather in Holland, with lots of sunshine and heat and the hot sands on the beaches and water just warm enough to swim and play in the sea, I decided to finally take some holidays. I followed some maths and tried reading on my own, slowly started my courses and such, but mostly I have been spending the past month with friends, making new friends, swimming in the sea, biking around Leiden, relaxing, reading books and discovering new things. 

The sea.

After this month of mental hygiene, I am feeling relaxed and optimistic. Sure, I still occasionally feel I am not enough and should not even bother trying. Especially if I compare myself with my classmates, it is rather easy to start feeling second-rate. I am so fortunate to be surrounded by 10 other Algant students (albeit all male and in a tiny office), most of whom are pretty cool and a couple of them are a real treat. I am slowly meeting again all the students I met during my previous stay in Leiden, I have good flatmates and I even discovered the nice people from the other side of the building, doing dynamial systems. Or this is what I imagine them to be doing, to be fair, I only ever see them in the common room drinking coffee... But first weeks were difficult, moving, all the exhaustion from being at yet another place, but mostly since I was asked again and again to comment on my Regensburg experience and that is still very stressful for me.

Now I feel I am getting over that. My mentor in Leiden hasn't given up on me (and so many thanks for that!!!), I think my courses will give me the nice variety of topics I need, we are preparing a "workshop" on ranks of elliptic curves on Monday in Heidelberg, so I get to think about how to define and convey the Tate-Shafarevich group as a local-to-global obstruction in the easiest terms (which does exclude mentioning both torsors and Galois cohomology) and I feel happy about my life. I am heading to Prague now for an interveiw that could open some interesting Ph.D. options for me, then a week in Germany and then back to my beloved Leiden, back to studying seriously, back to all my homework, classes, long evenings working in the office, complaining about the Dutch weather, questioning the Dutch food and beverage choices... If only one could combine the Czech beer and cuisine with the Dutch beaches and their tranquil life...

But for now, I want to enjoy the few moments I get to spend at home, in my country, taking the train between Blansko and Prague, as I have done perhaps 40 times since last September. I still haven't got tired of the pale yellow fields and actual hills and deep green forests and meandering rivers all around. My life is good.

The end of my German year

I am going through a funny period of my life. Finishing my exams was no big thing for me as I took two exams at the end of the semester and then decided not to take the third (in Regensburg, you can get up to two passes from courses just for homework). So there was no huge relief, more like anxiety whether I am going to take the last exam after all or whether the professor will agree to my not taking it… Even my last three visits to Germany were problematic and extremely stressful so I was really glad I did not have to go for the last exam. There are still a couple of things that I should wrap up in Germany but those things can wait easily. So I never got any feeling of closure or anything. But it is behind me. It really is.

In the end, I am happy it is all over and I can finally move on, but I am disappointed with many things and I must say that the advice that I should grit my teeth and it will be over soon is among the worst things I could have been told. It did not work and it was not worth it. On one hand, I do look forward to doing some maths again but all my motivation has been ruined during this year, all my willpower and wishful thinking are simply gone and I no longer possess anything resembling studying habits and overall I think I have forgotten more than learned in many aspects this year.

Well, hopefully not all the damage will be irreparable and I will still recover, though I simply feel exhausted when it comes to maths and life and studying now. Well, some mistakes in life are very costly. But the things are beginning to turn. I got a nice opportunity to come back home and help with generating random numbers, so that would be a bit of statistics for me, but home is home. But if feels great to be able to come back home.

Having passed my driving exams yesterday, I am celebrating and enjoying my three days of holidays with due bureaucracy and packing arranging things and getting rid of the immense backlog I have accumulated over the past few months. My life is finally going to slow down a bit in Holland and by the end of September or beginning of October, I should be focusing on studying only. And then I will hopefully finish all the drafts of posts for my blog as well :) All the maths I owe you!

The year was exhaustive for me. I feel completely drained now. But a new beginning in a different place with different math is waiting for me not so far ahead. So thank you for all your support during this year. Let’s hope it will not be needed that much next year.

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.