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Home , Abel Prize, Dan Freed, Jacques Tits, Karen Uhlenbeck, Lennart Carleson, Louis Nirenberg

Karen Keskulla Uhlenbeck, Winner

Elizabeth Denne

Washington & Lee University

October 16, 2019 Karen Keskulla Uhlenbeck

Elizabeth Denne (W&L) Abel Prize October 16, 2019 2 / 24 Abel Prize citation

Karen Uhlenbeck receives the 2019 Abel Prize

for her pioneering achievements in geometric partial differential equations, and integrable systems, and for the fundamental impact of her work on analysis, and mathematical .

Elizabeth Denne (W&L) Abel Prize October 16, 2019 3 / 24 About the Abel Prize

Named after: Neils Henrik Abel (1802–1829) Awarded by: the Norwegian Academy of Science and Letters. Dates: Annually since 2003. Prize: cash award of about US$730,000. Awarded for: outstanding scientific work in the field of , including mathematical aspects of computer science, , probability, numerical analysis, scientific computing, statistics, and also applications of mathematics in the sciences.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 4 / 24 Previous winners

2018 Robert P. Langlands 2017 2016 Sir Andrew J. Wiles 2015 John Forbes Nash Jr. and 2014 Yakov G. Sinai 2013 2012 Endre Szemerédi 2011 John Milor 2010 John Torrence Tate 2009 Mikhail Leonidovich Gromov 2008 John Griggs Thompson and 2007 Srinivasa S.R. Varadhan 2006 2005 Peter D. Lax 2004 Sir Michael Francis Atiyah and Isadore M. Singer 2003 Jean-Pierre Serre

Elizabeth Denne (W&L) Abel Prize October 16, 2019 5 / 24 At the Abel Prize ceremony

Karen Uhlenbeck is the first woman to be awarded the Abel Prize.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 6 / 24 Biography

Born: August 24, 1942, in Cleveland ; eldest of four children. Parents: Arnold Keskulla (engineer), Carolyn Windeler Keskulla (artist and school teacher) Moved to when Karen was in 3rd grade. Undergraduate education: (1964) Love of mathematics started here. Had been planning on physics, but hated labs!

Elizabeth Denne (W&L) Abel Prize October 16, 2019 7 / 24 Biography

1964 married Olke Ulhenbeck (biophysicist). 1968 PhD Brandeis with NSF Fellowship. (Husband at Harvard.) PhD advisor: . Thesis: The and Global Analysis. 1968–69 Instructor MIT. 1969–71 Lecturer University of California, Berkeley. Studied general relativity and geometry of space-time.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 8 / 24 Biography

1971–76 Assistant Professor, University of Illinois Urbana-Champaign. 1977–83 Associate Professor, University of Illinois, Chicago. (Divorced husband.) 1983–88 Professor at . Mathematical interests: nonlinear partial differential equations, , gauge theory, topological quantum field theory, integrable systems. 1984 co-authored Instantons and 4-Manifolds with . Instantons describe the behavior of surfaces in 4-dimensional space-time. This book inspired a whole generations of . 1987-88 Sid W. Richardson Foundation Regents’ Chair in Mathematics at University of Texas at Austin. Worked with winning physicist .

Elizabeth Denne (W&L) Abel Prize October 16, 2019 9 / 24 Biography

1988 Married Robert R. Williams (). Currently Professor Emerita University of Texas at Austin. Distinguished Visiting Professor at Institute for Advanced Study. Visiting Senior Research Scholar at .

Elizabeth Denne (W&L) Abel Prize October 16, 2019 10 / 24 The mathematics

Karen Uhlenbeck is a founder of modern .

This is a field of mathematics where: techniques of analysis and differential equations are interwoven with the study of geometrical and topological problems.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 11 / 24 Calculus of Variations

Brachistochrone problem: Find the curve connecting two points, on which a bead slides frictionlessly in the shortest time. Link

https://scipython.com/blog/the-brachistochrone-problem/

Elizabeth Denne (W&L) Abel Prize October 16, 2019 12 / 24 Calculus of Variations: 1-dimensional

General Problem: Let J be a functional that assigns a value to any function in some function space. Find the function that minimizes J. Question: What is the shortest distance between two points?

Answer: Minimize arc-length:

Z b q J(f ) = 1 + f 0(x)2 dx a b

y = g(x) 0 y = f (x) f (x) 1 a

Elizabeth Denne (W&L) Abel Prize October 16, 2019 13 / 24 Shortest paths: straight lines and great circles

a b

a R2 b S2

Elizabeth Denne (W&L) Abel Prize October 16, 2019 14 / 24 Minimal Surfaces

Minimize surface area subject to a boundary constraint.

Let f : S → M. Here S is a surface, and M is a manifold (think Euclidean 3-space).

wikepedia.org and soapbubble.dk

Elizabeth Denne (W&L) Abel Prize October 16, 2019 15 / 24 Minimal Surfaces: catenoid

Again, f : S → M, minimize area. Here S is a cylinder, with boundary two circles.

Link

soapbubble.dk

Elizabeth Denne (W&L) Abel Prize October 16, 2019 16 / 24 Generalize to higher dimensions

Richard Palais and :

Palais-Smale Condition C gave a criterion for the existence of minimizers for functionals on mapping spaces.

Apply Condition C to problem: it fails!

Uhlenbeck studied what happens when Condition C is violated.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 17 / 24 Condition C and Uhlenbeck

Paper with Jonathan Sacks developed “bubbling phenomenon” method of solving problems. Sequence of mappings of a sphere into target space, each of which satisfies Condition C. Limit of sequences does not satisfy Condition C. Outside of finite set of singular points everything works. Near singularities, so-called “bubbling phenomenon" appears. Area of limit is strictly less than limit of areas of sequence. Difference is concentrated in finite set of isolated points, being the limit of “bubbles” in the sequence of surfaces.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 18 / 24 Uhlenbeck’s impact on mathematics

Uhlenbeck’s major contributions include foundational results on:

Minimal surfaces and harmonic maps. Gauge theory and Yang-Mills theory. Integrable systems and harmonic mappings.

Elizabeth Denne (W&L) Abel Prize October 16, 2019 19 / 24 Awards and Honors

1983-88 MacArthur Fellow 1985 Fellow of the American Academy of Arts and Sciences 1986 Member of National Academy of Sciences 1990 Plenary speaker at International Congress of Mathematicians (second woman after 1932) 2000 National Medal of Science 2001 Guggenheim Fellow 2007 Leroy P. Stelle Prize (AMS) 2019 Abel Prize Honorary Doctorates: University of Illinois at Urbana-Champaign (2000), (2001), University of Michigan (2004), (2007), Princeton University (2012).

Elizabeth Denne (W&L) Abel Prize October 16, 2019 20 / 24 Outreach

“I remain quite disappointed at the 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 pressures from outside. Changing the culture is a momentous task in comparison to the other minor accomplishments I have mentioned.”

Elizabeth Denne (W&L) Abel Prize October 16, 2019 21 / 24 Outreach

1991 Co-founder (with and Dan Freed) of the IAS/Park City Mathematics Institute. 3 week program bringing together mathematicians of all levels (students, researchers, university faculty, K-12 teachers, etc). 1994 Co-founder (with Chuu-Lian Terng) of the Women and Mathematics Program at Institute for Advanced Study in Princeton. >1500 participants to date, and 2019 Winner of AMS “Programs that Makes a Difference" Award. Saturday Morning Math at UT Austin. Mentored many mathematicians (>17 PhD students).

Elizabeth Denne (W&L) Abel Prize October 16, 2019 22 / 24 On being a role model

“. . . it’s hard, because what you really need to do is show students how imperfect people can be and still succeed. Everyone knows that if people are smart, funny, pretty, or well-dressed they will succeed. But it’s also possible to succeed with all of your imperfections. In this respect, being a role model is a very unglamorous position to be in, showing people all your bad sides. I may be a wonderful mathematician and famous because of it, but I’m also very human.”

Elizabeth Denne (W&L) Abel Prize October 16, 2019 23 / 24 Thank you Karen for inspiring so many of us!

Elizabeth Denne (W&L) Abel Prize October 16, 2019 24 / 24