7th HLF – Lecture: Introducing the work of Karen Uhlenbeck, Abel Prize 2019
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Transkript: Englisch(automatisch erzeugt)
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Good morning, welcome back. My name is Gerhard Hüsken from the Mathematical Research Institute
00:29
of Warwulfach and I'm the chair of this session where we do something new, namely introduce the work of new laureates who unfortunately could not make it to this
00:42
meeting. Karen Ullenbeck is the first female mathematician to receive the Abel Prize this year, 2019, and unfortunately she could not make it to Heidelberg this year. Now Elena Mader-Baumdiger will introduce her work and give an overview
01:04
of her research. Professor Mader-Baumdiger is a professor at Darmstadt University working in the area of geometric analysis and she has interacted with Karen Ullenbeck for over a year at Princeton. I'm very much looking forward to her
01:23
presentation. Okay, I hope you can all hear me. It seems to be the case. Okay, I would like to start with reading from the brochure announcing the Abel Prize winner 2019. Karen Cascola Ullenbeck receives the 2019 Abel Prize
01:46
for her pioneering achievements in geometric differential equations, gauge theory and integrable systems and for the fundamental impact of her work on analysis, geometry and mathematical physics. End of quote. So here you see a
02:03
picture of Karen which was made at the Institute for Advanced Study and I like how enthusiastic she is about what she's telling. I would like to tell you a little bit about the life of Karen Ullenbeck. She spent a large portion of her
02:21
childhood in New Jersey. She left Reed and she developed her deep passion for the nature and for outdoor activities there. Here's something very interesting that Karen said about her childhood. I am the eldest of four children and I always considered dealing with my siblings the hardest things I've ever done in my life. That had a great impact on my choosing a
02:45
career. I wanted a career where I didn't have to work with other people. End of quote. So this is quite a statement in my eyes and I will come back to that later. Karen went to the University of Michigan. She got into the Honors
03:02
program at Michigan which gave her a very good education already as a freshman. She got more and more excited about the beauty of mathematics. Originally she wanted to major in physics but she switched to mathematics. After a bachelor's degree she spent one year at the Courant Institute in New York City.
03:20
Then she married the biophysicist Olke Ullenbeck. As he went to Harvard for a PhD she was looking for a place not so far away where she could continue studying mathematics. Karen got an NSF grant for this phase of her career which helped her to be independent. She chose the Brandeis University in the east of Massachusetts. I want to cite Karen about that choice.
03:43
I transferred to the math graduate program at Brandeis University. I did not even apply to MIT and Harvard. I have no idea where this kind of God wisdom came from except that I knew that I wasn't interested in the woman question. I did not want to be an exceptional woman. I can't cite any terrible
04:03
experiences I had. I just knew I would be isolated at those places. Brandeis was a smaller school and I felt that I would be safer there and it was true. I don't think the professors were all that eager to see me. A lot of
04:21
them didn't want or approve of women students but I was a sufficiently good student that whatever their feeling had been about whether women should do math I was able to overcome them without any trauma so to end of quote. Karen's PhD advisor was Richard Palais and the topic was global
04:43
analysis which means very roughly speaking using topology to solve a problem in analysis and using analysis to solve problems in topology. A little bit earlier in the 60s Richard Palais and Stephen Smale had made huge progress in understanding the structure of critical points of functionals on
05:03
infinite dimensional function spaces. So this is a part of the calculus of variations. They define the so-called Palais-Male condition that guarantees the existence of critical points of a functional under this condition. I will mention this later again. After her PhD Karen worked for a
05:24
year at MIT and then for two years at the University of Berkeley. The next stops in her career were the University of Illinois at Urbana-Champaign, the University of Illinois at Chicago and the University of Chicago where she became a full professor in 1983. In 1988 Karen Ullenbeck was called for the
05:46
Sid W. Richardson Foundation Regents Chair in Mathematics at University of Texas at Austin. By that time she had married her second husband Robert F. Williams. Karen stayed at the University of Texas until she retired. Right now
06:03
she's back in New Jersey. She's a visiting associate at the Institute for Advanced Study in Princeton and a visiting senior research scholar at Princeton University. Karen received several awards for her work but let me not mention that long list. Instead I want to say more about her work and
06:24
why it is so fantastic and I also want to say why she's a very good role model. During her time at the University of Illinois at Chicago she wrote two papers with Jonathan Sachs about minimal surfaces. Minimal surfaces are critical points of the area functional. For example surfaces
06:45
that have the least area bounding a certain curve. So if you have a wire and you dip it into soap water what you get out is a minimal surface. I want to cite Karen about that time. At the time I had not had contact with a large
07:02
number of mathematicians. I had gotten my PhD at Brandeis not from Harvard or MIT or Berkeley and when I was at Berkeley and MIT I didn't really work with anyone else. That was valuable in some sense because I didn't know what people were thinking about minimal surfaces. So when Jonathan Sachs told me
07:22
the problem I didn't have any built-in machinery to think about it so I was able to think about it on my own. End of quote. So this is a very good example of how Karen works. Of course she's very interested in reading papers of other mathematicians and she's able to understand them but she's also not a
07:44
right to start her own very individual approach to solve questions she's interested in and she's absolutely driven by curiosity. So I would like to tell you a little bit more about what Jonathan Sachs and Karen Wombeck actually proved. So they were looking for a method to prove the existence of
08:04
minimal surfaces so he keep this piece of soap bubble in mind. Again those minimal surfaces are critical points of a geometric functional, see the area functional. But unfortunately for analytical methods it's hard to prove
08:22
existence of critical points because it's functional has a lot of invariances. So one invariance is instead of looking at a domain with valuable X you can define lambda dot X where lambda is just a parameter or a real number and the functional will not change okay. And this invariance cause a
08:43
lot of problems. And in particular the mentioned Pauli-Smale condition is not satisfied okay. So it's much harder to find critical points of that functional. So first step which people already recognized earlier was that instead of looking at the area functional you could look at the
09:03
Dirichlet energy which is now another functional but you can somehow come back so if you have a critical point for the Dirichlet energy and you have so-called conformal parameterization then it's also a critical point for the area functional okay. So why do we do that? Because the Dirichlet integral
09:20
has a much better invariance property. Still some invariance group so still the Pauli-Smale condition is not satisfied unfortunately but there's at least some hope and this is also shown by Karen's work. So what Sachs and Ullenbeck did is
09:40
that they defined a perturbation of the Dirichlet functional okay. It consists of a family of functionals depending on a parameter. Let's call it alpha okay. Then they showed that all these new functional nulls do actually satisfy the Pauli-Smale condition. So one gets a by the Pauli-Smale theory so you get a
10:02
sequence of critical points for all these alpha okay. Now you want to let alpha go to zero because in the end you want to find a harmonic map what's called a critical point for the Dirichlet energy and you would hope that yeah by going to zero you would find such a map but what actually can happen is that around certain points the certain amount of energy concentrates
10:25
and this is called bubbling and what's actually what they actually were able to prove is that if you let this alpha go to zero so do you converge to a harmonic map but only away from finally many points and they understood
10:44
also what's going on around these exceptional points okay and for this they used the scaling. You know they have these exceptional points you have a little disk around it and what they did is use a magnifying glass somehow zoom in and try to see if you find something. This is the scaling okay they
11:04
rescaled it with an appropriate factor such that you can actually see something and what you see is a harmonic map on all of R2 because you can scale it to all of R2 and then I mean R2 is actually not so much
11:23
different from the surface of a sphere you only have to add one point okay and then they proved also and that these how many maps that they found on R2 can be extended across the singularity the singularity is now infinity okay and then they developed this they found this harmonic maps of
11:43
the type of a sphere okay and they're called bubbles okay it's very nice pictures we see swimming in you see these bubbles and in a final step they've also showed that you can parametrize the bubbles conformally so actually that these are actually minimal bubbles okay this bubble
12:02
bubbling analysis is not absolutely standard in geometric analysis but it is very new territory back then. Another fundamental result of CERN is the study of regularity for a class of nonlinear elliptic systems. CERN herself said several times that this is the hardest paper she ever did and that
12:21
she had to work very hard for it. It was worth it. I could continue to tell you about CERN's fundamental work in gauge theory. It became incredibly important for all the following tremendous success of application of gauge theory in topology and geometry especially for the work of Simon
12:41
Donaldson but it is even more technical to explain than the bubbling analysis so instead of going into detail here I decided to cite CERN. I started out my mathematics career by working on Pauli's modern formulation of a very useful classical theory the calculus of variations. I decided
13:02
Einstein's general relativity was too hard but managed to learn a lot about geometry of space-time. I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three-manifold topology, learned
13:22
gauge theory and then some about applications to four manifolds and have recently been working on equations with algebraic infinite symmetries. I find that I am bored with anything I understand. My excuse is that I am too
13:40
poor an expositor to want to spend time on formal matters. End of quote. From this statement you see how broad CERN's interest is. In several of the fields you mentioned she made absolutely fundamental contributions. Most of the techniques that CERN developed are now standard machinery in geometric
14:00
analysis. I found it absolutely amazing in almost every paper I read I find something that I can trace back to CERN's work. She deserves to be named one of the founders of the modern geometric analysis. I want to mention another part of her life that became very important for her. At a certain time in
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her career she recognized that the number of women in mathematics didn't increase although she had expected that. I think CERN has a lot of interesting things to say about the phenomenon that there are not so many women in mathematics. The first quote is about her time as a graduate student at the Brandeis University. There were a handful of women in my graduate
14:46
program although I was not close friends with any of them. It was self-evident that you wouldn't get ahead in mathematics if you hung around with women. We were told that we couldn't do math because we were women.
15:00
So if anything there was a tendency to not be friendly with other women. Another statement refers to a later time of her career. There certainly was legal discrimination against women when I first found a job but then the law was changed. Most mathematicians believed that once the
15:22
legal barriers were removed women and minorities would walk through the gates in great numbers and whatever problem there was would be solved. However in early 80s and early 90s it just wasn't happening. I mean women were actually having a difficult time. I and some other female
15:42
mathematicians noticed that the numbers of women had stopped increasing. We hadn't had anything to do with programs since we hadn't thought it would be necessary. When we realized it was necessary many of us became involved. There was a time when Karen co-founded the Women in Mathematics program at the
16:03
IES, the Institute for Advanced Study in Princeton. It is an annual two weeks program with a specific topic. It helps women of all mathematical levels to connect, to speak about recent developments in mathematics and to build research collaborations. At another occasion Karen said the
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following, starting from my days in Berkeley the issue of women has never been far from my thoughts. I have undergone wide swings of feelings and opinion on the matter. I remain quite disappointed at the
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numbers of women doing mathematics and in leadership positions. This is to my mind primarily due to the culture of the mathematical community as well as harsh societal pressure from outside. Changing the culture is a momentous task
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in comparison to other minor accomplishments. End of quote. I met Karen in Princeton in person when I was there for Research Day. She's a Here's one advice for younger mathematicians that I found written down.
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It shows that she somehow changed her mind about working alone. I tell my students that the most important thing if you want to keep doing mathematics is that you establish mathematical contacts. Even if you don't need to work with them
17:41
you're going to get depressed sooner or later and you're going to need some sort of input. Whether people stay as research mathematicians or not I think the big item is that they have some contact in the mathematics community of a personal nature. That sounds weird because mathematicians are crazy. They
18:02
work by themselves and you sort of think of them as sitting in the room working by themselves. But every mathematician hits bad points and how do you get over it? Somebody has to has got to come along and say cut it out kit or somebody has to come in with a new idea and hit you on the head with
18:23
it. End of quote. Karen has her own way of doing mathematics. She follows her own interest and intuition for which topic to choose. Then she finds new and creative ways to explore the unknown. At the same time she's very
18:41
kind, strong and generous. Karen Ulmbeck is without a doubt a role model for every young mathematician female or not. Thank you Karen Ulmbeck. Thank you
19:05
very much for this presentation. Other questions or additions? If not let me add one story how I met Karen the first time. I was a young postdoc at UC
19:21
San Diego and Karen arrived as a visiting professor. The very first thing she did is she took her travel grant and donated it to all the postdocs including me at UC San Diego at the time. So in addition to her inspiring
19:41
mathematics I can confirm that Karen was an extremely generous and kind person all throughout her career and we hope that maybe next year or so she can be here in person. Thank you very much again.
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