Academic FOMO

I am now in my fourth year of my PhD program in Amsterdam. Time flies! With the two years of working towards a PhD in Berkeley, that gives me 5+ years of grad school and quite a lot of stress and anxiety, and I want to share some aspects of it with you today. First, a clarification: I am working through some stuff now, but in general, I am actually quite hopeful!

I often feel like my life and academic experience is different from the people around me. Of course, everybody is unique and we don’t all do exactly the same things, but I think most people in Academia would agree what the “typical” or “average” (and in reality non-existent) trajectory is. We all understand that there isn’t one “correct” way of doing things, but without actively questioning our assumptions, we would probably paint exactly the same picture. Let’s call the imaginary person that has the average experience “Everyone Else”.

When I started doing math, I was still an athlete. I had to run to my trainings from my lectures, shower and eat quickly just to make it back, and kept at it for a few years. I was lucky to have received a lot of mentoring early on, so in my classes I was a bit more advanced than others, got to travel to summer schools early on and was exposed to wonderful math. So I did things Everyone Else didn’t get to do that much. When I got to Berkeley, I had already done a masters degree, and again had slightly different needs and expectations from Everyone Else. This sounds like by now, I should be used to being slightly different. But I still have to work with it. I am still often comparing myself with whatever Everyone Else is doing, and find it hard to focus on the things that are good for me instead.

This summer, I attended an amazing 3-week grad student summer school. I loved being there, learned so much, worked really hard, made many meaningful mathematical and human connections. But, I was also a TA for one of the courses – again, a bit different from Everyone Else. Of course, I ignored what that meant, the extra work, the extra mental load, and tried doing as many things as Everyone Else. Instead of taking a break, I was still answering questions and trying to help others learn. I loved doing it, but even at the time, I did recognize that it came at a cost. I ignored it, because I did not let myself be different from Everyone Else, did not admit that my work that week most mostly my TAing, and that I did not actually have to attend all the other things – but Everyone Else went to everything, so I tried to do the same.

Moreover, I am very lucky to work in a field – isogenies – that is so beautiful that many people want to talk about it. So that’s what we did. We set up extra meetings to talk math. But, rather than skipping one exercise session on a topic perhaps not interesting to most isogenistas, we did it in the… evenings, after a full day of math. I liked these meetings because I enjoyed the people and the math we did together. But, as my friends rightly pointed out, we were crazy to do it on our Friday nights. And, honestly, I went because it was the “one isogeny thing” that I did not want to miss, and also Everyone Else went!

I am now at home, and my life has been incredibly stressful in the last few weeks. Quite a rollercoaster ride, but I’m almost at the end of it. On top of that, I am still recovering from the mental damage the covid years have caused, I worry about the war in Ukraine, I was exhausted from the summer travel (it was very much worth it, but traveling isn’t instantaneous, and I was working hard during that time). But, when I look around me, I see Everyone Else is already back in the office and working hard and doesn’t seem to be affected by the same things… Wish me luck with focusing on what I need to do for myself, and let Everyone Else do their own thing!

How to isolate yourself because you don't feel good enough

I have been silent for a long time and today I want to tell you why. I will talk about how I've been isolating myself and why. 

The last two years had been difficult. I decided to leave my PhD program at UC Berkeley and jump into post-quantum cryptography. I was exhausted, close to being burned out. It was the right decision, things turned out okay, but dropping out - see how I internalize the negative narrative? - can mess you up. I've only ever heard encouragement and gotten a lot of support from everyone around me. But, because of the impostor syndrome, it feels like a big failure and that by leaving Berkeley, I am losing out. I am losing so much that my life will never be as good as it could have been.

This is, of course, absolute nonsense. But things did not improve when I left Berkeley. I changed fields, sort of. I still work in number theory and play with elliptic curves (topic for another time, perhaps?). But, suddenly, I found myself in a quantum algorithms group. And among cryptographers. I did quite some self-study before starting, but I was far from ready to understand the seminars or have any idea of what people would talk about over lunch. And whenever I had some questions that felt basic to other people, they just referred me to this one set of lecture notes...

You can imagine how this made me feel. I left my relatively safe space in Berkeley, where I could be anyone who I wanted to be. And I was suddenly this new person at the department that I did not know with a very mismatching background. And I was teaching right from the beginning, I could not find housing in the city, so I had to commute from far, the paperwork at the start was far from smooth and took time to figure out.

Did I feel at home at my new institute? I don't think so. I did not have enough time to adjust to my new place and was still trying to get around the invisible walls that I put up around myself. When corona hit, I did not have the safety net yet, it was difficult to find people to reach out to. I have managed with some, failed with others. But the more you isolate yourself, the less sense it all makes. To give just one absurd example. I worked hard to catch up with all stuff that I thought I should know to belong to the group. But then I simply did not talk to anyone about any of these things for months.

You can't blame others for not reaching out to you. But if you don't, especially in this online world, people will not notice. If you interact with someone online, it takes a lot of energy to reach the human being behind the face on the screen. And the people who did not show up are so abstract nowadays that you don't even care anymore. And what about the people who showed up in the beginning and then stopped coming? In real life, you would notice the empty chair. But online meetings are all such a blur that you can show up suddenly after three months - and most people will not notice anything. 

Funnily enough, I have been such a misfit all my life. I usually find my way around, sooner or later. For the athletes, I was the weirdo mathematician. During my undergrad, I was among the weird mathematicians, but I always had to run off to my workouts. Then I moved abroad and moved several times, changing communities, meeting new people, losing contact with the old ones, over and over. Despite all the stress, I found myself in Berkeley, thanks to the I-House and the TAC workouts. Everyone in there was coming from a different place, it was easy to live free of assumptions.

So why am I making assumptions about myself now? I was not hired to be a cryptographer. I was definitely not hired as a quantum algorithms expert. My job is to be a number theorist who can learn enough about cryptography and quantum algorithms to do something cool with it. I am now focusing on doing things my way. I am organizing a reading group to force myself to discuss the things I am learning with other people. I am setting up my own social events, giving new chances to things I thought would not work for me. 

Like writing this blog: I am missing conversations about many topics now. Once I started talking to people, it turned out that I am not the only one. So I want to share with you. It already makes me feel less isolated. I hope you will, too. 

Some numbers for you

I took part in the wonderful All-comers meet today in Berkeley (http://www.rtspt.com/events/cal/12018/). I am now ready to share some numbers that I've managed to generate today:

shot put 10.67,

 javelin 28.16,

 60m 8.06 (-0.9),

 200m 25.80 (-0.1).

I started off with a combination of shot put and javelin. This was the first time I was holding a javelin in ages, and I wanted to finish all my javelin throws before my shot put, which didn't work out well - I had to do three shot put attempts in a row and then finish my javelin :) But still, 10.67 with a 4kg shot without a glide is not bad. Javelin was much worse, from the 10 or so throws that I did (4 in competition), basically, only 2 were measurable and none of them were anywhere near nice.

After that, I did some sprinting races. It was a funny order, but I don't get to pick anymore :) My god, it's been so long, I completely forgot how to warm up (was ready way too early) and so the running itself was very very far from good - which is great, as I am confident I can improve greatly. The times are basically what was expected, given that I've been training for some 2 months now, took a very relaxed 3 week winter break, and had my spikes on for the first time on Monday. Moreover, it was still a bit chilly and with the wind -0.9 it was far from ideal sprinting weather!

Now, you might ask, why am I doing this? In two months, I want to do a heptathlon. And I am still on my way to my goal of 5k, which would be a fantastic result. Please wish me lots of luck! But again, why am I doing this? I like it. And I need it. Just being a grad student would drive me insane. I need to train, I need to meet people who are not doing maths, who are not studying at Berkeley, I like being exhausted at the end of the day, happy about my training and being so hungry I can eat anything at our dining hall.

It takes a lot of energy to fit my trainings into my schedule. Especially now, as I am starting a 50% GSI position and the teaching will simply eat up a lot of my time. And with all the classes and seminars, travels for conferences and there being only certain times during the day that I can train, it's going to be rough. But today I loved it. It is definitely worth it, being so tired and stressed for time, having to wait the long lines in the university gym when that's the only time I can do my workout, dying on the stairs in Mark Jellison's TAC class or getting up for his 7.15am studio... It is not easy, but it is definitely worth it.

Thank you all for your support. It means a lot to me. Please help me push myself in the next two months to make the 5k. My trainings mean a lot to me and my progress with my training strongly correlates with my progress in my studies. I already have so many big goals and I am glad that today I got some actual numbers, and that these numbers mean that I am on the right track! In two weeks, I will test my hurdles. That's the big test for me!

PS: I've been learning some really cool math now. I will write about it soon.

The Paris-Berkeley syndrome

Hi all,

this has been some long time. I've moved to Berkeley in the meantime, I'm almost done with the first semester, basically, two more weeks to go, a few more homework sets to finish and a project paper, but that's it. It has been a semester full of things happening, full of emotions, full of frustration, full of stress...

To explain the title, apparently, Paris is so hyped up in Japan that sometimes, when Japanese tourists come to Paris and discover that the city is not just a romantic, beautiful experience, the culture shock kicks in really bad. Suddenly they're in a place the language of which they don't understand, the locals are not particularly excited about yet another tourist, it's a big city full of people, dirty and unpleasant in its own way... Some people just lose it and have to be sent back to Japan, where the symptoms gradually go away. It's a first-world problem and it is a fascinating read...

So yeah, the culture shock has been pretty bad, given that I had a perfect life in Leiden: living a quiet small life full of maths, biking around in the fields, cooking, and baking... Now I am in a very hyped-up (Bay) area, in a community of hard-working, success-oriented, incredibly smart and ambitious people. Berkeley is a big bubble, around the university it's difficult to see someone who's not an undergrad, let alone outside of the university (actually, maybe I am adjusting well, looking down on undergrads ;) sorry!). I'm living in conditions that are very suboptimal for me, I am definitely not satisfied with my I-House experience, and though I am grateful to all the donors whose generosity helps make living at I-House an excellent experience for many people, I do not think the environment is a very fostering one for me. It's just not for everybody.

Math-wise it's been a bit of a misfit, too, I am still trying to find a topic for me, I tried learning a few new things, had a few shots at algebraic geometry again, but so far it's not working that well for me. But I did get very excited about Iwasawa theory, so I am looking forward to the Arizona Winter school already (even though that's in some 3.5 months). There is still maths that I love, maths that makes me happy. However, being burnt out the way I am, it's not easy to keep studying. There are days that I can do it, there are days I do just a little, but there are days in which even a bit costs me too much energy that I just can't do it anymore. But I'm learning to deal with that, slowly, too.

Having many people around me who are always studying, always doing great things, getting so much done, being so excited about being at Berkeley, that is perhaps the hardest part of it all. I tend to simply shut down and as a result, I can't do much, not even the little bit that I was hoping I could get done. So yeah, if you feel like you're not doing enough, just look at me, you are still doing much more. And I am still fine. I am slowly getting my stuff done, I'm slowly getting where I want to be. And all of this is improving. These three months have been big and challenging, but I have come a long way since the intial shock of moving here. I've had a lot of help from many of my friends and I am also grateful for the unwavering support of the man dearest to my heart, who's gone through a lot of very bad days with me and without whom I would have gone back home already.

But otherwise, the Bay area is fun. And I love driving around. And I love being away from Berkeley. Finally not hearing the city, finally not hearing all these millions of fire trucks and ambulances always driving around. The sunsets are unbelievable, the campus is nice to just walk through, it's not all that bad. I also like American coffee and burgers and so at least I can feed myself here.

Also, what really helped me and made a big difference is that I decided to start training again, this time it's not just me, I'm being Totally Athletically Conditioned by Mark Jellison (TAC is a class in our sports center) and we're also doing really awesome gyms together. It's been some two weeks now and I still like it. I think my body is responding quite well to this training and all these newly found endorphins are doing miracles for my mind as well. So I think I have finally found some sustainable exercise regime for myself and if life goes well, I should do some races in the Spring. So wish me luck and persistence :)

Anyway, I hope to be checking in more often now. I think I am over the worst part of it and though there are still some pretty tough decisions left to make, I think I am back in control of my life.

The sunset, the Bridge in the distance, the City...

My master's thesis

Dear all,

my master thesis is done. That's actually not that accurate: I submitted already on Friday, June 16, on Monday the 26th I had my defense talk and I still need to give a talk during the Algant graduation days, (July 24-26) which could potentially change my grade :) But yesterday I sent the final version and I no longer have to write anything or change anything.

It's so great to have such a big chunk of work done. It went rather well: I was lucky that my advisor gave me a problem that was interesting to think about and solvable within the few months I had for my thesis and the math was beautiful and challenging as always. The defense talk was also lots of fun. My boyfriend forced me to prepare well - he even went to the department with me on Sunday afternoon to practice with the chalkboard. I also chose to give a talk that was a bit different than my thesis, telling a different story: more of a motivation and a bit more coherent. In short, I think I sold my thesis there much better and I am very happy with my talk.

Even though that after the talk I still had some time to fix the small mistakes and improve the exposition, I didn't really find it in myself to sit down and rewrite it to reflect my slightly different point of view on my thesis that I formed during the talk preparation. It does help to clear your mind when you try to summarize in 45 minutes what you'd been doing the past half a year or so. Even more so when you try to explain it to someone from the beginning, say to your fellow master students, not just to people who know already what you will be talking about. So there would be many things I could write better in my thesis, I've received some very helpful comments that perhaps should have been included in the thesis, but yeah, it's finished now.

Even more so, as there are other things I need to be doing now. I moved out from my house last Friday, I need to secure my degree also from my second university, Regensburg, which might be a bit bureaucratically challenging again, I'm moving to Berkeley in less than six weeks and need to get my visa and finances sorted out... Lots of things to do now. And the weather is finally a bit nice again, so there are also things I want to do, like biking around, swimming in the sea, genuinely enjoy my summer :)

But for those of you who are spending their summers more actively, here is a link to my thesis and also a link to the notes from the preparation for my talk. The talk notes are without any references and I didn't follow them that closely, so some bits might be imprecise, if not completely wrong.

https://sites.google.com/site/sotakovajanahomepage/master-s-thesis---class-invariants

Wow, it's April already.

So the time has come. I am now happy about what I have learnt and computed for my thesis and now is the time to start writing things down. It is a very exciting endeavour, so exciting that I am turning back to my almost-forgotten blog!

Now, seriously. I have been a bit busy, after the Christmas break I had my exams (5 of them), then I visited Czech, worked on my thesis a bit, then I had to go to Czech again, then I worked on my thesis a bit more and here I am now: happy about my maths, less into writing things down but it is the last step in my masters' that I need to complete. Well, I still have one more exam to do and possibly I will choose to finish my Dutch course, but that is now somewhere at the back of my mind.

The weather in Leiden has improved magically, it's sunny so often, warm and not-windy, almost as if I were in California already. Oh, that's the big news, I've just accepted a place at Berkeley, so starting from August I will be hanging around mostly in the Bay area. For the summers I will likely be coming back to Holland, as long as I have such wonderful reasons to come back, but I will also spend some time back home. Maybe I will even do some math-related travelling, who knows?

So now that I am mostly bound to my desk and typing things down in our dark Algant office, not seeing all the amazing weather outside, I decided to take up running again. I am spending an hour or two on the track almost every day, provided the weather is nice, partly sunbathing, partly running around, almost like the good old times. If everything goes smoothly, my mother will even bring my old spikes to me and I want to run my first race in two months, here in Leiden. Probably not hurdles: though my technique was basically unaffected by those two long years of chilling on various couches all around the world, running-wise it's still a misery.

So that's it. Almost. There are tulips everywhere, seagulls all around the department make me feel like it is some sort of a beach resort, occasional trips to the beach and naive attempts at wading through the water remind me that it is still the Netherlands, I am grateful to the people around me for putting up with me and making this such a great time here. Hopefully writing will go well, training will still be fun, my Dutch will improve enough so that I won't panic anymore and life will in general be good.

Kijk! And this is just the beginning of the season :)

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.

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. 

Math holidays vol 2

So, Mainz, stacks, fun, amazing weather, little Leiden. What was next?

Takže Mainz, stacks, spousta zábavy, nádherné počasí, můj malý Leiden. A co bylo dál?

I visited family friends in Darmstadt and they took me to the Frankenstein Castle (among other nice places we visited that weekend)! They saw me grow up, remembered me from times my English constituted only a few questions about their favourite colours (which I had asked everyone around me nevertheless) and we had some very fascinating discussions about modern academic standards at universities, the future of the humankind and my perception of math as philosophy rather than technical, computational science.

Navštívila jsem rodinné přátele v Darmstadtu, a ti mě vzali k Frankensteinově hradu (mezi všemi těmi krásnými místy, co jsme ten víkend viděli).  Znají mě odmala, pamatují si mě z dob, kdy jsem se anglicky uměla zeptat leda tak jaká je jejich oblíbená barva (což jsem se lidí okolo stejně ptala) a vedli jsme spolu několik fascinujících diskuzí o moderním akademickém světě, budoucnosti lidstva a také mému pohledu na matematiku jako spíše filozofii než vědu technickou či výpočtení.

Indeed, what else are we dealing with in maths? Of course, we prove theorems, solve equations, compute expressions. But that is not all of it. An integral part of doing math is thinking about the definitions: what are the properties defining my object? What really makes them what they are? The concept of object really is a fundamental topic in philosophy. And we really spend a lot of time finding the right definition that fits our level of abstraction: that accurately describes our toys and can possibly give us a new insight into what we are playing with, but it can't be too general to allow too many obscure cases that do not allow us to deduce anything specific.

Však vskutku, co jiného v matematice děláme?  Samozřejmě, dokazujeme věty, řešíme rovnice, počítáme výrazy. Ale to není vše. Nedílnou součástí matematiky je rovněž debata o našich definicích: co jsou ony vlastnosti, které určují můj objekt? Co je opravdu dělá tím, čím jsou? Koncept objektu je jedním ze základních oblastí matematiky, kterým se slavně zabýval i Jára Cimrman. A opravdu trávíme spoustu času hledáním správných definic, které jsou dostatečně obecné, aby zahrnuly i nám nové objekty, ale dostatečně přísné, abychom se vyhnuli patologickým případům, které nám zabrání říct o mnoho víc, než říkají naše definice.

Another huge chunk of our time is consumed by justifying our claims. What is truth, anyway? If not the objects, defining, establishing and understanding truth is at the very core of philosophy. And what else do we do than trying to understand what is truth, what constitutes a proof of some statement? But the truth in itself is not our sole interest, we rejoice ourselves in creating hypotheses and following our thoughts processes often ad absurdum... Indeed, many a headache arise from the truths that cannot be proved and many other wonders and paradoxes of logic.

Mnoho času věnujeme taky dokazování našich tvrzení. Co je to vlastně pravda? Pokud ne objekty, definování, rozhodnutí a porozumění pravdy jistě leží v jádru věd filozofických. A co jiného my děláme, než že se snažíme porozumět tomu, co je pravda a co je to důkaz nějakého tvrzení? Ale pravda sama o sobě pro nás není vším, těšíme se z formulování domněnek a následného sledování našich myšlenových proudů až ad absurdum... Vskutku, pravdy, které nejze dokázat a jiné zázraky a paradoxy logiky nám často přivádí bolest ega a hlavy.

And I like to picture us more like ancient philosophers, slowly strolling through olive groves and discussing our thoughts than labouring over the monstrous computations in the solitary confinement of our dark rooms. My imagination might be wild but I am lucky enough to live my dream...

Nesin math village was even closer to my dreams...

A taky si nás ráda představuju jako antické filozofy, jak se pomalu procházíme olivovými hájky a rozmlouváme spolu, sdílíme své myšlenky a představy, nikoliv jak o samotě v temném pokoji toneme ve všech těch šílených výpočtech. Mám možná dobrou představivost, ale taky mám to štěstí žít ve svém snu...

Nesinova matematická vesnice byla ještě blížeji mému snu...