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Joachim Kock (May 24 2020 at 12:14):Rongmin Lu said:
Hi Joachim,
I'm glad you appreciate this, and would be honoured if you could also tell us about the ideas in your career as well. I believe you have an interesting story to tell, particularly since you've studied in Valeria's home country of Brazil.
Ha, ha! I guess I count as old :-)
Joachim Kock (May 24 2020 at 12:19):I first arrived in Brazil -- Recife (Pernambuco) -- in 1992, as a kind of exchange student, to take a couple of courses for my masters degree back in Aarhus. One was a course on schemes (Hartshorne Ch.2, page for page) and one was on elliptic curves (following Silverman). This was a wonderful experience. Brazil is a fantastic country -- nice people, rich culture, good weather. The courses were also very good, in the spirit of the people: hard-working without being hurried.
Joachim Kock (May 24 2020 at 12:23):I also got a taste for being a foreigner. It's a great excuse for being different. Instead of people thinking I was weird, they could attribute it to the fact that I was a foreigner. And then I picked up some social skills. I also learned not to wear socks in my sandals, not to cross my legs like a woman, and not to drink beer directly from the bottle.
(=_=) (May 24 2020 at 12:27):Joachim Kock said:
Rongmin Lu said:
Hi Joachim,
I'm glad you appreciate this, and would be honoured if you could also tell us about the ideas in your career as well. I believe you have an interesting story to tell, particularly since you've studied in Valeria's home country of Brazil.
Ha, ha! I guess I count as old :-)
Thank you for taking up the invite. As I've said before:
Age is not part of the criteria. A willingness to do an AMA and an interesting (broadly defined :sweat_smile:) career are more relevant.
I should probably put up a FAQ somewhere.
Joachim Kock (May 24 2020 at 12:28):Back in Denmark, I passed the exams for the courses, and then I had to write my masters dissertation, which was supposed to be about flag manifolds in characteristic p. In 94 I went back to Recife to write the dissertation. I finished it in 96. (I never got to the flag manifolds, only complete intersections.) It took a bit longer than expected, because meanwhile I met the girl who is now my wife -- actually my first girlfriend. That was quite distracting.
(=_=) (May 24 2020 at 12:29):Joachim Kock said:
I also learned not to wear socks in my sandals, not to cross my legs like a woman, and not to drink beer directly from the bottle.
You should've come to Australia. People wear socks in sandals and drink beer directly from the bottle here. I didn't know Brazilians were so proper.
Joachim Kock (May 24 2020 at 12:33):I applied for a PhD in Aarhus, but I was not accepted :-( I suppose they were a bit worried it had taken me three years to write my masters thesis. But then luckily I got a grant in Recife to work with Israel Vainsencher. It was the same project I had presented in Aarhus, on enumerative geometry via quantum cohomology. I had mostly copied it from the description of the special programme that had just run at the Mittag-Leffler Institute, where some of my friends had participated.
Joachim Kock (May 24 2020 at 12:42):Rongmin Lu said:
You should've come to Australia. People wear socks in sandals and drink beer directly from the bottle here. I didn't know Brazilians were so proper.
Yes, Australia is a fantastic country too, for many of the same reasons. I spent some time in Sydney when I was a boy, and it had a great impact on me, to see something different from Denmark. It was a nice experience to live in a young and vibrant country full of immigrants.
Joachim Kock (May 24 2020 at 12:57):Math interlude -- Kontsevich's formula for rational plane curves:
An algebraic curve is rational if it can be parametrised by the projective line. Enumerative geometry is the art of counting how many geometric objects (often curves) satisfy certain conditions. The enumerative geometry of rational curves goes back to Antiquity: given 2 points in the plane, there is 1 line (= degree-1 rational curve) that passes through them. Apollonius essentially showed that there is 1 conic (= degree-2 rational curve) that passes through 5 given points. In degree 3: Steiner (1848) computed that 12 rational plane cubics pass through 8 points, and finally, (the great Danish mathematician) Zeuthen (1873) computed the number 620 of rational plane quartics through 11 points.
Then a century passed.
With modern intersection theory à la Fulton (1980s), the results of Zeuthen could finally be verified to modern standards of rigour, and the next number: 87304 rational quintics through 14 points, was computed in 1993 by my thesis advisor Israel Vainsencher, as a byproduct of other investigations.
Then in 1994, Kontsevich found a marvelous recursive formula solving the general case (of rational degree-d curves through 3d-1 points). The formula dropped out 'by accident' from ideas from string theory, and in particular the notion of quantum cohomology. Furthermore, the formula is an instance of the associative law for the product in the quantum cohomology ring! Could it be more beautiful? A century-old problem is given a solution expressed as associativity, arguable one of the most basic notions in mathematics.
Nikolaj Kuntner (May 24 2020 at 13:00):(http://www.math.utah.edu/~yplee/teaching/gw/Koch.pdf) :)
Joachim Kock (May 24 2020 at 13:02):In my thesis I generalised Kontsevich's formula by constructing a certain tangency quantum cohomology ring, whose associativity solves the characteristic-number problem. That is, counting rational curves subject to tangency conditions, such as the famous 3264 conics tangent to 5 given conics. (Recursions for the characteristic numbers had already been found by Pandharipande, Vakil, Ernström, and Kennedy. So my contribution was mainly a conceptual one.)
Joachim Kock (May 24 2020 at 13:11):Nikolaj Kuntner said:
Ha, that's funny. I didn't even know that version existed: it looks like it's an intermediate version between the original Portuguese from 1999 and the final book from 2007.
The lion arms on the cover page is that of Sport Clube do Recife, the glorious football club :heart:
(=_=) (May 24 2020 at 13:57):Joachim Kock said:
Then in 1994, Kontsevich found a marvelous recursive formula solving the general case (of rational degree-d curves through 3d-1 points). The formula dropped out 'by accident' from ideas from string theory, and in particular the notion of quantum cohomology. Furthermore, the formula is an instance of the associative law for the product in the quantum cohomology ring! Could it be more beautiful? A century-old problem is given a solution expressed as associativity, arguable one of the most basic notions in mathematics.
I was going to ask why you made the switch from quantum cohomology to higher category theory, but I think I see a hint here.
Joachim Kock (May 24 2020 at 14:10):Rongmin Lu said:
I was going to ask why you made the switch from quantum cohomology to higher category theory, but I think I see a hint here.
I was already reading the TWFs avidly during my thesis work :-)
Joachim Kock (May 24 2020 at 14:14):I started to drift away from algebraic geometry shortly after my PhD. I came to Stockholm for my first postdoc to work with Lars Ernström, who had been interested in my thesis. But a few weeks after my arrival he quit maths to work for Ericsson! That was in 2000, and Ericsson was one of the biggest companies in the world, and everybody in Stockholm had a mobile phone.
But then I began to talk with Dan Laksov, Michael Shapiro, Carel Faber, Helge Måkestad, and Mats Boij. We started a reading seminar on topological quantum field theories, reading Atiyah, Dubrovin, and Abrams. I also suggested we should read Baez-Dolan, but that was considered too far out!
I actually worked full time on the seminar, and really made an effort to iron out everything. I was very much inspired by This Week's Finds in Mathematical Physics. A year later I had the opportunity to give a not-quite-mini course on this material at a summer school back in Recife, and I wrote it all up into a book. I think at this point I could not longer hide my categorical bias :-)
Valeria de Paiva (May 24 2020 at 15:25):Joachim Kock said:
Math interlude -- Kontsevich's formula for rational plane curves:
Then in 1994, Kontsevich found a marvelous recursive formula solving the general case (of rational degree-d curves through 3d-1 points). The formula dropped out 'by accident' from ideas from string theory, and in particular the notion of quantum cohomology. Furthermore, the formula is an instance of the associative law for the product in the quantum cohomology ring! Could it be more beautiful? A century-old problem is given a solution expressed as associativity, arguable one of the most basic notions in mathematics.
WOW!!! Beautiful! and very surprising.
but shouldn't you perhaps mention your dad?
Joachim Kock (May 24 2020 at 16:04):Valeria de Paiva said:
but shouldn't you perhaps mention your dad?
Of course! But at this point in life I was more concerned with not becoming a category theorist! (Actually previously I had also tried to avoid becoming a mathematician, but that did not work out.)
But of course my father had a big influence on my maths, even in my thesis work. When I learned algebraic geometry, he told me a lot about representable functors and the big Zariski site, and the importance of nilpotent elements, and Grothendieck fibrations, and so on. And when I studied Riemannian geometry I also got a good dose of synthetic differential geometry. And countless small things and viewpoints I was maybe not even aware of at the time. Probably even just the taste for abstract mathematics. All that only really sunk in much later.
Valeria de Paiva (May 24 2020 at 16:29):Joachim Kock said:
Valeria de Paiva said:
but shouldn't you perhaps mention your dad?
Of course! But at this point in life I was more concerned with not becoming a category theorist! (Actually previously I had also tried to avoid becoming a mathematician, but that did not work out.)
[...]
Probably even just the taste for abstract mathematics. All that only really sunk in much later.
oh well, I never managed to get my kids interested in mathematical ideas.
and yes, I do count your dad (Anders Kock, if people don't know) as a friend. did you know that he once spend the weekend at my house in Cambridge? for a PSSL, we messed up the reservations or ran out of rooms or something and Anders stayed at my place. I hope this was at the very nice house (in 4 years I had to move 4 times, so the houses get a bit jumbled up in my memory). and yes someone should write up the story of the PSSL which together with TAC is one of things category theorists should be proud of.
Nikolaj Kuntner (May 24 2020 at 16:31):Must be nice to have someone close to you in the field you study, is it?
(=_=) (May 25 2020 at 01:06):Valeria de Paiva said:
for a PSSL, we messed up the reservations or ran out of rooms or something and Anders stayed at my place.
For the uninitiated: PSSL is the Peripatetic Seminar on Sheaves and Logic, which "grew out of Dana Scott's seminars for his research students at Oxford."
For historical interest: Gavin Wraith maintains a list of PSSLs up until the 70th (1976-99), while Valeria has a list covering PSSL 71-94. The most recent PSSL would have been the 106th at Leeds if not for COVID-19.
(=_=) (May 25 2020 at 05:21):I've just created a topic on PSSL: #learning: history of ideas > History of PSSL. If you have stories, please contribute there. Thank you!
(=_=) (May 25 2020 at 13:06):Joachim Kock said:
I was already reading the TWFs avidly during my thesis work :-)
How did you find out about the TWFs by John Baez? Probably a bit unfair, since I can't remember myself, but maybe you do.
Joachim Kock (May 25 2020 at 20:59):Rongmin Lu said:
How did you find out about the TWFs by John Baez? Probably a bit unfair, since I can't remember myself, but maybe you do.
Actually, my father told me about it :-)
Joachim Kock (May 25 2020 at 21:04):Nikolaj Kuntner said:
Must be nice to have someone close to you in the field you study, is it?
Yes, I do realise it is a privilege. I ought to exploit it more. But I have been living quite far from my father for 30 years, and when we do see each other in family contexts, there is very little time for math. We do correspond by email, of course.
We did write one small paper together: Local fibred right adjoints are polynomial. That was a very nice experience. Fun to work out the math, and very easy to agree on notation and style, for some reason :-)
John Baez (May 25 2020 at 21:15):Joachim Kock said:
Rongmin Lu said:
How did you find out about the TWFs by John Baez? Probably a bit unfair, since I can't remember myself, but maybe you do.
Actually, my father told me about it :-)
Go ahead, make me feel old. :upside_down: These days I get people saying "I really liked This Week's Finds back when I was in high school".
(=_=) (May 26 2020 at 04:30):John Baez said:
Joachim Kock said:
Rongmin Lu said:
How did you find out about the TWFs by John Baez? Probably a bit unfair, since I can't remember myself, but maybe you do.
Actually, my father told me about it :-)
Go ahead, make me feel old. :upside_down: These days I get people saying "I really liked This Week's Finds back when I was in high school".
TWF started when I was in middle school, so yeah...
But I think it was David Michael Roberts and the others at Adelaide who told me about it.
David Michael Roberts (May 26 2020 at 05:18):Maybe. I stumbled on it in first year of uni, after finding John's material on general relativity, which I was trying to teach myself.
(=_=) (May 26 2020 at 09:37):Joachim Kock said:
We did write one small paper together: Local fibred right adjoints are polynomial.
You've done a lot of work on polynomial functors, and in your notes on them, you acknowledged the influence of Joyal. How did this work begin?
(=_=) (May 26 2020 at 09:49):You've also done a lot of work on the algebraic structure underlying renormalisation. Can you take us through the history of that work?
Joachim Kock (May 28 2020 at 08:23):Rongmin Lu said:
You've done a lot of work on polynomial functors, and in your notes on them, you acknowledged the influence of Joyal. How did this work begin?
I was a postdoc with André Joyal in Montreal in 2003-2004. Up to the summer 2004 we worked on weak units and the Simpson conjecture.
Then in July 2004, we both participated in the conference in Minneapolis on n-categories, organised by John Baez and Peter May. A fantastic conference, very inspiring. Actually it was more than a conference -- it was a work camp: talks started at 8 o'clock in the morning; dinner was just a sandwich during the Russian-style evening talks! Very intense -- it was hard to find time to go to Dinky Town and drink pitchers.
Joachim Kock (May 28 2020 at 08:31):There were several talks on opetopes, very fascinating objects invented by Baez and Dolan. The opetope speakers Eugenia Cheng, Mihail Makkai, and Tom Leinster were very helpful to explain details. At some point André Joyal, Michael Batanin, Gian-Franco Mascari and myself were discussing at a blackboard, trying to see if we could give a purely combinatorial definition of opetopes, without relying on higher-dimensional polytope-style lego drawings (and without relying on syntax such as
-- that's a real-life 3-opetope from the Hermida-Makkai-Power papers). Especially Gian-Franco had many crazy ideas we tried to formalise.
After the conference, Michael spent two weeks in Montreal, and we could not stop thinking about opetopes. At some point, André pulled polynomial functors out of his hat: a combinatorial framework for certain nice operads -- including all the operads resulting from the iterated Baez-Dolan construction. Polynomial functors are sums of representables. The representing data is a configuration of sets , and everything becomes very combinatorial. (In the 1-variable case, this reduces to just as in David Spivak's talk later today in the MIT seminar.)
Joachim Kock (May 28 2020 at 08:40):Between August and November, André guided me through the basic theory of polynomial functors. I was completely absorbed. It is a beautiful theory. At first it is a cute categorical version of some elementary algebra. Soon it becomes a general toolbox for substitution and induction, intimately related with dependent type theory. Very conveniently, Nicola Gambino came visiting in October, and he was already working with polynomial functors in connection with type theory. Then in November 2004 I moved to Barcelona.
I think André was happy someone could take up the theory, and he invested some time in setting it in the right direction. He did not have time to develop it himself, as he was busy with the grand project of quasi-categories. I feel very privileged for having been there just at the right time. For the past 16 years, polynomial functors have been a main theme in my work.
Joachim Kock (May 28 2020 at 08:45):It took three more years before we finished the opetope project, Polynomial functors and opetopes https://arxiv.org/abs/0706.1033. Meanwhile, Nicola and I began to write what we intended to be a kind of foundational paper on Polynomial functors and polynomial monads https://arxiv.org/abs/0906.4931, incorporating also a lot of stuff from parallel developments in computer science (where polynomial functors had been studied under the name 'containers').
(=_=) (May 28 2020 at 09:37):Thank you. That's a very fascinating story! I didn't know John had so much to do with it.
Joachim Kock said:
Then in July 2004, we both participated in the conference in Minneapolis on n-categories, organised by John Baez and Peter May. A fantastic conference, very inspiring.
For those who'd like to know more:
John wrote about the conference here, which also has some nice photos. Sadly, the conference website itself is down. Towards Higher Categories is a book associated with that conference, but it is not a proceedings.
(=_=) (May 29 2020 at 08:23):You noted in Polynomial functors and combinatorial Dyson–Schwinger equations that
It was Kurusch Ebrahimi-Fard who first got me interested in quantum field theory and renormalisation [ten years ago,] and his continuing guidance and help since then has been essential for this work, and is gratefully acknowledged.
Can you tell us more about that?