
Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize - digital55
https://www.quantamagazine.org/karen-uhlenbeck-uniter-of-geometry-and-analysis-wins-abel-prize-20190319/
======
kaitai
Uhlenbeck is delightful, and one of her big ideas discussed in this article --
bubbling -- truly has changed both her field and algebraic geometry. It's a
cool idea. In a previous thread here (How I learned to love algebraic
geometry,
[https://news.ycombinator.com/item?id=19397957](https://news.ycombinator.com/item?id=19397957))
there was some discussion of singularities. Algebraic geometers like
polynomials, while symplectic geometers like Riemann surfaces. These coincide
at times but Riemann surfaces include lots of non-polynomial examples.

Anyhow, as an algebraic geometer, the place I learned about Uhlenbeck's work
was in compactification of moduli of curves. Basically, say I want to look at
all the polynomial curves that can live in a certain 'environment'. (One
'application' is string theory -- what particle interactions can occur given a
certain set of energy constraints? The particle interactions are curves traced
out over time by particles/Riemann surfaces traced out over time by the little
loops that represent closed strings in string theory; the energy constraints
constrain the shapes the particle interaction can take.) If you want to look
at limits of families of these interactions, you're looking at some odd
behavior. An example is the equation xy = t^2 -- this gives you nice
hyperbolas for values of t not equal to zero, but as t -> 0, you get something
singular, two crossed lines. Or x^2 - y^2 - z^2 = t^2: you have a nice smooth
hyperboloid when t neq 0, and a double cone when t=0. However, these examples
are just showing you the idea of how singularities appear in families of
smooth polynomials. Uhlenbeck's older work really worked with the system of
constraints that I mentioned -- figuring out what can appear under those
constraints is a considerably more complicated problem!

This is a very imprecise discussion above and I'm totally blurring together
Uhlenbeck's bubbling and compactness theorems from gauge theory -- but I never
followed her work in a systematic way but instead was plunged into seeing its
aftereffects from another related field that the work affected.

Last quote really resonates: “Along the way I have made great friends and
worked with a number of creative and interesting people. I have been saved
from boredom, dourness, and self-absorption. One cannot ask for more.”

~~~
Cobord
Meanwhile as a mathematical physicist, I learned about her through instantons.

Quite a wide career.

------
otoburb
_" Mathematics research had another feature that appealed to her at the time:
It is something you can work on in solitude, if you wish. In her early life,
she said in 1997, “I regarded anything to do with people as being sort of a
horrible profession.”"_

This sentence struck me as slightly odd if only because Erdõs was renowned for
his social approach to mathematics[1], and my layperson's understanding of
mathematics departments being somewhat collaborative within their sub-fields.

I know there are some notable examples of other solo breakthrough endeavours,
such as Shinichi Mochizuki[2] or Yitang Zhang[3], and but those examples
seemed to be exceptions to the rule.

Perhaps I'm channeling too much Terence Tao[4], or maybe the lone wolf
researcher only applies to literal geniuses who work exclusively in academia.

Is mathematics research still a primarily solitary activity?

[1]
[https://en.wikipedia.org/wiki/Paul_Erd%C5%91s](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s)

[2] [https://www.quantamagazine.org/titans-of-mathematics-
clash-o...](https://www.quantamagazine.org/titans-of-mathematics-clash-over-
epic-proof-of-abc-conjecture-20180920/)

[3] [https://www.cnet.com/news/yitang-zhang-a-prime-number-
proof-...](https://www.cnet.com/news/yitang-zhang-a-prime-number-proof-and-a-
world-of-persistence/)

[4] [https://terrytao.wordpress.com/career-advice/does-one-
have-t...](https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-
genius-to-do-maths/)

~~~
graycat
It is interesting to read what she saw in math, why she liked it. For me, I
just thought it would be useful, especially for physics which I thought was
even more useful!

No one ever explained to me how a career in academic math research would work:
I never suspected that just digging into questions that were fun and curious
and maybe someday somewhat useful, making discoveries, and writing papers
could be a career that could support a family.

In particular, I still don't understand why the US has so much respect for
research universities: My guess was that college was for teaching the
students. But in a research university, nearly all the effort by the
professors is their research, and the teaching is a sideline. E.g., might have
a really good researcher in far out topics in analysis teaching junior level
linear algebra. So, the money to the university, for people directly paying
full tuition often a LOT of money, is only a little for that teaching but much
more for the research. An analogy for restaurants would be that each hamburger
had to come from some three star Michelin place and cost $200.

E.g., when I was a B-school prof, I was really discouraged to see that what
the B-school was teaching had next to nothing to do with business. Instead,
the school was interested in _research_ , e.g., the question P vs NP. If
medicine were taught that way, then no one would go to a hospital no matter
how badly they hurt. B-school tried to be academic research, not professional
training.

Besides, for me as a college student, I had to conclude that after junior year
level material, the profs really needed to understand what they were teaching
much better than they did. For that, they just needed to do a much better job
learning and presenting what was already in the library and without more
research. E.g., I never had a math or physics prof who had a good grasp of the
theory and applications of Stokes theorem, classic potential theory, multipole
coordinates, or both the theory AND applications of Fourier theory. None of my
undergrad physics profs knew either general relativity or quantum mechanics at
all well. I wanted to learn that stuff, not for research but for applications.

Math research solitary? It always has been for me. For learning the standard
stuff, sure, can have people to talk to. But if they are in the same course
and it is competitive, then maybe they don't want to talk?

The reason my math research was solitary is mostly just because the new work I
was doing was so focused that no one else around knew much about it or had
much interest in it. For a lot of math research, being focused is close to
necessary and, then, the solitary aspect gets to be close to automatic.

Moreover it appears that there can be an effect from outside: Some people are
more socially skilled, polished, talented, interested, motivated, etc. than
others. In some fields, such social skills are important for success. But in
math, if you can learn it, mostly on your own, then you can teach it -- good
social skills can help but can get by with not much. This is even more the
case for math research: Pick a problem, do some research, get some results,
type in a paper, send it to an appropriate journal, maybe all with essentially
no contact from anyone else.

So, since in math can get by with less than the best social skills, people
without great social skills can find math, be successful, and stay, and, then,
the result is that comparatively math is not a very _social_ field.

But, a lot of people can learn to be social: Gee, a lot of grade school girls
(sorry to bring gender into this) have just astounding social skills: E.g., it
seems clear enough that long ago Hollywood discovered that among child actors
the girls were much better -- they paid attention what was going on socially,
reacted to others, had lots of facial expressions on tap, were good at
glancing, averting, head tossing, pose striking, cases of body language,
expressiveness in their voices, etc. Well, what a grade school girl commonly
knows others can pick up, too! Okay performance in the lessons is not
seriously difficult.

Reading some of what Uhlenbeck said about herself, she was doing well in the
E. Fromm recommended social activity "Giving knowledge of oneself" and that
can help a person be social because they are letting others know who they are.
Or little so interests people as other people; if some other person
communicates no better than a stone wall, then there's not much for others to
be interested in; Fromm's _giving knowledge_ can help with that.

For a nutshell description, my view of mathematicians is that they are less
manipulative than most other people. Well, in some activities, can seem to get
progress by manipulation, but in math, can't prove a theorem by a fake,
manipulative proof!

~~~
cadeira
>I never suspected that just digging into questions that were fun and curious
and maybe someday somewhat useful, making discoveries, and writing papers
could be a career that could support a family.

I am interested in pursuing a PhD in Pure Mathematics. What are the career
prospects for it? Based on what I read academic prospects don't seem too
thrilling. It appears you may have to do multiple post-docs at low pay, and
still not be able to be hired at a good school. What is the path to becoming a
professor and making a livable salary?

If that is the case what are the industry prospects? Would whatever software
company want to hire you, if your pure math specialization isn't directly
applicable to what they're focused on? In turn, it seems better to learn how
to code. But will your pure math specialization (as in not applicable to
anything anyone is doing) + coding abilities set you apart?

I would rather study mathematics, pure mathematics and not be unemployable.
But I don't really know what the job prospects are like. Could you speak to
that, say, from a realistic perspective?

~~~
kevinventullo
I did a PhD in pure math and now work for a big tech company. Having the PhD
will get your foot in the door basically anywhere, so you'll at least make it
to the screen, but ultimately you have to pass the same interview as anyone
else. That means doing LeetCode and reading Yegge's stuff etc.

If your goal is to maximize lifetime earnings, a PhD is maybe not the right
choice, but if you want to spend years studying something you love while
financially breaking even then go for it. I definitely don't regret it, even
though I would be worth a lot more if I had gone into tech straight out of
college.

Btw, finance is also a common destination for pure math PhD's looking to leave
academia.

~~~
cadeira
Thank you. This is exactly what I wanted to hear. I’m more interested in
studying math than maximizing my income, but post-graduation opportunities is
a concern, since I’m not well off. Your reply gives me the encouragement I was
looking for to pursue my passion. Thank you.

~~~
gautamdivgi
To add to the discussion. I did a CS PhD with a statistical bent. It's served
me well. However, I would see if an MS provided you with equivalent benefits
from a career standpoint. Very few companies do "research". And if you're in
the US, unless you acquire funding with yourself as the PI its going to be
impossible to find an academic position.

------
dquarks
Enjoyed her IAS lecture on Emmy Noether's laws. She's incredibly fun to watch.
Well deserved. Congratulations, Karen.

------
marai2
The closest I will every get to something like an Erdõs number is that I can
now claim the one and only female Abel prize winner was my Differential
Equations teacher at college! I missed taking a class with Steven Weinberg
when I got to UT because he semi-retired the year before.

~~~
volkadav
I felt very lucky to have had her for an experimental mathematical modeling
course (M375) back around 2000. She was simply delightful as a professor, even
for a mathematical lummox like myself. :) Encouraging, informative, brilliant
... we should all be so lucky as to have teachers and influences like her in
our lives.

