The Abel Prize Laureate 2019 - Karen Keskulla Uhlenbeck www.abelprize.no
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The Abel Prize Laureate 2019 Karen Keskulla Uhlenbeck University of Texas at Austin www.abelprize.no
Karen Keskulla Uhlenbeck receives the 2019 Abel Prize for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.
Citation
The Norwegian Academy of Science and critical points of functionals representing
Letters has decided to award the Abel geometric quantities such as energy and
Prize for 2019 to volume. For example, minimal surfaces
are critical points of the area and harmonic
Karen Keskulla Uhlenbeck maps are critical points of the Dirichlet
University of Texas at Austin energy. Uhlenbeck’s major contributions
include foundational results on minimal
for her pioneering achievements in surfaces and harmonic maps, Yang-Mills
geometric partial differential equations, theory, and integrable systems.
gauge theory and integrable systems, and
for the fundamental impact of her work Minimal surfaces and bubbling
on analysis, geometry and mathematical analysis
physics. An important tool in global analysis,
preceding the work of Uhlenbeck, is the
Palais—Smale compactness condition.
Karen Keskulla Uhlenbeck is a founder This condition, inspired by earlier work
of modern Geometric Analysis. Her of Morse, guarantees existence of
perspective has permeated the field minimisers of geometric functionals and
and led to some of the most dramatic is successful in the case of 1-dimensional
advances in mathematics in the last domains, such as closed geodesics.
40 years. Uhlenbeck realised that the condition
Geometric analysis is a field of of Palais—Smale fails in the case of
mathematics where techniques of surfaces due to topological reasons. The
analysis and differential equations are papers of Uhlenbeck, co-authored with
interwoven with the study of geometrical Sacks, on the energy functional for maps
and topological problems. Specifically, of surfaces into a Riemannian manifold,
one studies objects such as curves, have been extremely influential and
surfaces, connections and fields which are describe in detail what happens when
© In 1987 Karen K. Uhlenbeck moved to the University of Texas at Austin to take up the
Sid W. Richardson Foundation Regents’ Chair in mathematics. She would remain at the
University of Texas until the end of her working career. Currently, Uhlenbeck is a Visiting
Senior Research Scholar at Princeton University as well as a Visiting Associate at the
Institute for Advanced Study (IAS).
photo : Marsha Miller
5
the Palais-Smale condition is violated. of all subsequent research in the area of
A minimising sequence of mappings gauge theory.
converges outside a finite set of singular Gauge theory involves an auxiliary
points and by using rescaling arguments, vector bundle over a Riemannian
they describe the behaviour near the manifold.
singularities as bubbles or instantons, The basic objects of study are
which are the standard solutions of the connections on this vector bundle. After
minimising map from the 2-sphere to the a choice of a trivialisation (gauge), a
target manifold. connection can be described by a matrix
In higher dimensions, Uhlenbeck valued 1-form. Yang-Mills connections
in collaboration with Schoen wrote are critical points of gauge-invariant
two foundational papers on minimising functionals. Uhlenbeck addressed and
harmonic maps. They gave a profound solved the fundamental question of
understanding of singularities of solutions expressing Yang-Mills equations as
of non-linear elliptic partial differential an elliptic system, using the so-called
equations. The singular set, which in the Coulomb gauge. This was the starting
case of surfaces consists only of isolated point for both Uhlenbeck’s celebrated
points, is in higher dimensions replaced by compactness theorem for connections
a set of co-dimension 3. with curvature bounded in Lp, and for her
The methods used in these later results on removable singularities
revolutionary papers are now in the for Yang-Mills equations defined on
standard toolbox of every geometer punctured 4-dimensional balls. The
and analyst. They have been applied removable singularity theory for Yang-
with great success in many other partial Mills equations in higher dimensions was
differential equations and geometric carried out much later by Gang Tian and
contexts. In particular, the bubbling Terence Tao. Uhlenbeck’s compactness
phenomenon appears in many works in theorem was crucial in Non-Abelian
partial differential equations, in the study Hodge Theory and, in particular, in the
of the Yamabe problem, in Gromov’s work proof of the properness of Hitchin’s
on pseudoholomorphic curves, and also map and Corlette’s important result on
in physical applications of instantons, the existence of equivariant harmonic
especially in string theory. mappings.
Another major result of Uhlenbeck is
Gauge theory and Yang-Mills her joint work with Yau on the existence
equations of Hermitian Yang-Mills connections on
After hearing a talk by Atiyah in Chicago, stable holomorphic vector bundles over
Uhlenbeck became interested in gauge complex n-manifolds, generalising an
theory. She pioneered the study of Yang- earlier result of Donaldson on complex
Mills equations from a rigorous analytical surfaces. This result of Donaldson-
point of view. Her work formed a base Uhlenbeck-Yau links developments
6
in differential geometry and algebraic families. Based on this observation,
geometry, and is a foundational result for Uhlenbeck described algebraically
applications of heterotic strings to particle harmonic mappings from spheres into
physics. Grasmannians relating them to an infinite
Uhlenbeck’s ideas laid the analytic dimensional integrable system and
foundations for the application of gauge Virasoro actions. This seminal work led to
theory to geometry and topology, to the a series of further foundational papers by
important work of Taubes on the gluing Uhlenbeck and Chuu-Lian Terng on the
of self-dual 4-manifolds, to the ground- subject and the creation of an active and
breaking work of Donaldson on gauge fruitful school.
theory and 4-dimensional topology, and The impact of Uhlenbeck’s pivotal
many other works in this area. The book work goes beyond geometric analysis.
written by Uhlenbeck and Dan Freed on A highly influential early article was
“Instantons and 4-Manifold Topology” devoted to the study of regularity theory
instructed and inspired a generation of of a system of non-linear elliptic
differential geometers. She continued to equations, relevant to the study of the
work in this area, and in particular had an critical map of higher order energy
important result with Lesley Sibner and functionals between Riemannian
Robert Sibner on non self-dual solutions manifolds. This work extends previous
to the Yang-Mills equations. results by Nash, De Giorgi and Moser
on regularity of solutions of single non-
Integrable systems and harmonic linear equations to solutions of systems.
mappings Karen Uhlenbeck’s pioneering
The study of integrable systems has results have had fundamental impact on
its roots in 19th century classical contemporary analysis, geometry and
mechanics. Using the language of mathematical physics, and her ideas
gauge theory, Uhlenbeck and Hitchin and leadership have transformed the
realised that harmonic mappings from mathematical landscape as a whole.
surfaces to homogeneous spaces
come in 1-dimensional parametrised
A biography of
Karen Keskulla Uhlenbeck
By Professor Jim Al-Khalili, Fellow of the Royal Society,
University of Surrey
In 1990, in Kyoto, Japan, Karen Uhlenbeck Ohio in 1942. Her father, Arnold Keskulla,
became only the second woman to give was an engineer and her mother, Carolyn
a Plenary Lecture at the International Windeler Keskulla, an artist and school
Congress of Mathematicians – ICM – the teacher. The family moved to New Jersey
largest and most important gathering of when Karen was in third grade. As a young
mathematicians in the world. It is held girl, she was curious about everything. Her
every four years, and the first woman parents instilled in her a love of art and
to do this was Emmy Noether in 1932. music, and she developed a lifelong love
Such a shocking statistic reflects just of the outdoors, regularly roaming
how hard it is for many women to achieve the local countryside near her home.
the recognition they deserve in a male- Most of all, she loved reading,
dominated field. shutting herself away whenever she
But by that point in her career, could to devour advanced science
Uhlenbeck had already established books, staying up late at night and even
herself as one of the world’s preeminent reading secretly in class. She dreamed of
mathematicians, having overcome becoming a research scientist, particularly
many hurdles, both personally and if it meant avoiding too much interaction
professionally. In 2000, she received the with other people; not that she was a shy
US National Medal of Science. Yet for child, but rather because she enjoyed the
many, the recognition of her achievements peace and solitude of her own company.
should have been far greater, for her work The last thing she wanted to do was
has led to some of the most important to follow in her mother’s footsteps and
advances in mathematics in the last end up teaching – an attitude that would
40 years. change dramatically later in life.
Karen Keskulla Uhlenbeck, the eldest Uhlenbeck’s love affair with mathe
of four children, was born in Cleveland, matics developed only after she had
8
© Karen Uhlenbeck giving a talk at the Institute for Advanced Study. photo: Andrea Kane 9
started at university. Having been inspired where she studied general relativity and
in high school by the writings of great the geometry of space-time – topics
physicists such as Fred Hoyle and George that would shape her future research
Gamow, she enrolled at the University work. Although a pure mathematician,
of Michigan, initially planning to major in Uhlenbeck has drawn inspiration for her
physics. However, she soon discovered work from theoretical physics and, in
that the intellectual challenge of pure return, she has had a major influence in
mathematics was what really excited her. shaping it by developing ideas with
It also meant she didn’t have to do any lab a wide range of different applications.
work, which she disliked. For example, physicists had predicted
Graduating in 1964, she married her the existence of mathematical objects
biophysicist boyfriend Olke Uhlenbeck called instantons, which describe the
a year later and decided to embark on behaviour of surfaces in four-dimensional
postgraduate study. Already well aware space-time. Uhlenbeck became one
of the predominantly male and often of the world’s leading experts in this
misogynistic culture in academia, she field. The classic textbook, Instantons
avoided applying to prestigious schools and 4-Manifolds, which she co-wrote in
such as Harvard, where Olke was heading 1984 with Dan Freed, inspired a whole
for his PhD and where competition to generation of mathematicians.
succeed was likely to be fierce. Instead, In 1971, she became an assistant
she enrolled at Brandeis University professor at the University of Illinois at
where she received a generous graduate Urbana-Champaign where she felt isolated
fellowship from the National Science and undervalued. So, five years later she
Foundation. There, she completed her left for the University of Illinois at Chicago.
PhD in mathematics, working on the Here, there were other female professors,
calculus of variations; a technique that who offered advice and support, as well
involves the study of how small changes as other mathematicians who took her
in one quantity can help us find the work more seriously. In 1983, she took
maximum or minimum value of another up a full professorship at the University
quantity – like finding the shortest of Chicago, establishing herself as one
distance between two points. You might of the preeminent mathematicians of
think this would be a straight line, but her generation. Her interests included
it is not always so straightforward. For nonlinear partial differential equations,
example, if you have to drive through differential geometry, gauge theory,
a busy city, the quickest route is not topological quantum field theory and
necessarily the shortest. Needless, to say, integrable systems. In 1987, she moved
Uhlenbeck’s contribution to the field was to the University of Texas at Austin to take
somewhat more complicated than this! up the Sid W. Richardson Foundation
After a brief teaching period at Regents’ Chair in mathematics. There, she
MIT, she moved to Berkeley, California, broadened her understanding of physics
10by studying with Nobel Prize winning science. She has come a long way from
physicist Steven Weinberg. She would the young girl who wished to be alone.
remain at the University of Texas until the For a while, she struggled to come to
end of her working career. terms with her own success, but now
Uhlenbeck’s most noted work says she appreciates it as a privilege.
focused on gauge theories. Her papers She has stated that she is aware of
analysed the Yang-Mills equations in four being a role model, for young female
dimensions, laying some of the analytical mathematicians in particular, but that
groundwork for many of the most exciting “it’s hard, because what you really need
ideas in modern physics, from the to do is show students how imperfect
Standard Model of particle physics to the people can be and still succeed. Everyone
search for a theory of quantum gravity. knows that if people are smart, funny,
Her papers also inspired mathematicians pretty, or well-dressed they will succeed.
Cliff Taubes and Simon Donaldson, paving But it’s also possible to succeed with
the way for the work that won Donaldson all of your imperfections. I may be a
the Fields Medal in 1986. wonderful mathematician and famous
Uhlenbeck, now back in New Jersey, because of it, but I’m also very human.”
remains a staunch advocate for greater Karen Uhlenbeck is certainly a remarkable
gender diversity in mathematics and in human.
Karen K. Uhlenbeck will meet schoolchildren for mathematical games in Archimedes’ Labyrinth in the
Botanical Garden at the University of Bergen during the 2019 Abel Week. The labyrinth is designed by
Hans Munthe-Kaas, chair of the Abel Committee. Photo: Avinor/Daniel Volle - Pandora FilmCC BY-SA 3.0 12
A glimpse of the Laureate’s work
“I’m Forever
Blowing Bubbles”
Arne B. Sletsjøe, ass. professor, Department of Mathematics,
University of Oslo
I’m forever blowing bubbles, Soap bubbles are beautiful objects,
Pretty bubbles in the air. perfectly shaped and with a marvellous
They fly so high, play of colours, due to interference of
Nearly reach the sky, light reflecting off the front and back
Then like my dreams, surfaces of the soap film. Soap bubbles
They fade and die. are beautiful objects in a mathematical
setting as well, as they constitute
Jaan Kenbrovin examples of minimal surfaces. When the
enclosed volume of air inside the bubble
is fixed, the soap film will minimize the
wall tension, pulling the bubble into the
shape of the least surface enclosing a
fixed volume, known for centuries to be a
perfect sphere.
If we instead of blowing the bubble,
dip a heavily deformed wire loop into a
soap bubble solution, the soap film will
form a disc with its boundary given by
the wire loop and of minimal area. Unlike
the sphere-shaped bubble, this film has
equal pressure on each side, hence it is
a surface with zero mean curvature, i.e.
the average curvature along all directions
is zero. Even if the soap film almost
instantly is able to form a minimal surface,
computing the shape of the surface
analytically is a rather complicated task.
13Among curves connecting two points of minimal surfaces, Uhlenbeck wanted to
in space, we can always find a shortest understand what happens when Condition
path. The analogous statement is not true C is violated. In a paper co-authored
for surfaces when considering their area. with Jonathan Sacks, they describe in
The problem is that in order to reduce the detail the situation where you cannot
area of a surface, a consequence could rely on the conclusion of Condition C.
be that the surface is shrunk to a curve, They construct a sequence of mappings
which of course does not count as a of a sphere into the target space which
minimal surface. An example of this is the satisfies Condition C, but in such a way
minimal tubular surface connecting two that their limit does not. Outside of a finite
parallel circles. If the distance between the set of singular points everything works
circles is small compared to their radius, well, but near the singularities the so-
the minimal surface looks like a slightly called bubbling phenomenon appears.
concave cylinder. When pulling the circles The area of the limit surface is strictly
apart the cylinder will shrink in the region less than the limit of the areas of the
between the circles, forming a surface surfaces in the sequence. The difference
known as a catenoid. At a certain point, is concentrated in a finite set of isolated
the middle part of the curved cylinder will points, being the limit of “bubbles” in
collapse along the line connecting the the sequence of surfaces. The idea and
centres of the two parallel circles. When the methods of this revolutionary paper
pulling the circles further apart there is no has since it was published become a
tubular minimal surface connecting them. successful mathematical tool. In particular,
the bubbling phenomenon has had
Mapping spaces great influence as a method for solving
In 1968 Karen Uhlenbeck received her problems in various parts of mathematics.
Ph.D. from Brandies with the thesis
“The Calculus of Variations and Global Footprints of gauge
Analysis”. Her supervisor was Richard Karen Uhlenbeck also left her footprints
Palais, who a few years earlier and in the field called gauge theory. Gauge
together with Stephen Smale, had theory is a mathematical theory
introduced the so-called Palais-Smale introduced by Hermann Weyl in 1918,
Condition C. This condition gives a which originated in theoretical physics and
criterion for the existence of minimizers Einstein´s theory of general relativity. A
for functionals on mapping spaces. key idea in Einstein´s work is that laws of
“Minimizers for functionals on mapping physics should be the same in all frames
spaces” is a more general phrase than of reference. This is also the general idea
“find the surface of least area”, but of a gauge theory, to find connections that
Condition C can also be applied to the compare measurements taken at different
minimal surface problem, but then it fails. points in a space and look for quantities
Motivated by the general non-existence that do not change. The physical
14interpretation was brought further by Yang which both originate from a wish to
and Mills in the fifties, in what is now understand nature. When mathematicians
called the Yang-Mills equations. To reveal get interested in such problems, the theory
the secrets of theoretical physics you have propagates into theoretical constructions
to work in a (at least) four-dimensional far beyond the tangible objects of nature.
space, three spatial coordinates and one But even if the mathematical theory seems
time-coordinate. A physical law should to be soaring, scientists often benefit from
be the same wherever you are located in the generalized theory. Karen Uhlenbecks
space-time, i.e. independent of choice of mathematical achievements constitute
frame of reference. important examples of such processes.
Minimal surfaces (I’m Forever Blowing Bubbles is an
Karen Uhlenbeck attacked this problem American song from the Broadway
from the mathematical point of view. musical The Passing Show of 1918. It
She pioneered the study of Yang-Mills was released in 1918, the same year as
equations in a rigorous analytical way. Hermann Weyl introduced the notion
Her work formed a base of all subsequent of a gauge. In addition to the obvious
research in the area of gauge theory. Her connection to minimal surfaces, the
analysis of the Yang-Mills equations in four lyrics may have some associations to
dimensions together with C. H. Taubes, mathematical research in general. The
also laid the ground for the theories of song is also adapted as the club anthem
Simon Donaldson, who later was awarded of West Ham United, a London-based
the Fields Medal in 1986 for his work on football club.)
the topology of four-manifolds.
Minimal surfaces and gauge theory
are two separate fields of mathematics
15Karl Johans gate, the main street of Oslo during the Abel week. photo: Thomas Brun/NTB 16
About the
Abel Prize
The Abel Prize is an international award Developing Countries, and the Bernt
for outstanding scientific work in the field Michael Holmboe Memorial Prize for
of mathematics, including mathematical excellence in teaching mathematics in
aspects of computer science, mathe Norway. In addition, national mathematical
matical physics, probability, numerical contests, and various other projects and
analysis, scientific computing, statistics, activities are supported in order to
and also applications of mathematics in stimulate interest in mathematics
the sciences. among children and youth.
The Abel Prize has been awarded At the Heidelberg Laureate Forum
since 2003 by the Norwegian Academy in Germany young mathematicians get
of Science and Letters. The choice of the opportunity to meet winners of the
laureates is based on the recommen Abel Prize.
dations from the Abel Committee. The
prize carries a cash award of 6 million Call for nominations 2020
NOK (about 650,000 Euro or about The Norwegian Academy of Science and
730,000 USD). Letters hereby calls for nominations for
The prize is named after the the Abel Prize 2020, and invites you (or
exceptional Norwegian mathematician your society or institution) to nominate
Niels Henrik Abel (1802–1829). According candidate(s). Nominations are confidential
to the statutes of the Abel Prize the and a nomination should not be made
objective is both to award the annual Abel known to the nominee.
Prize, and to contribute towards raising
the status of mathematics in society and Deadline for nominations for the Abel Prize
stimulating the interest of children and 2020 is September 15, 2019.
young people in mathematics.
Among initiatives supported are the Please consult www.abelprize.no for
Abel Symposium, the International more information.
Mathematical Union’s Commission for
18© DNVA / Snøhetta 19
The laureate wall in The Abel-room at the Norwegian Academy of Science and Letters. photo: Eirik Furu Baardsen
2018 Robert P. Langlands
“for his visionary program connecting representation theory to number
theory”
2017 Yves Meyer
“for his pivotal role in the development of the mathematical theory of
wavelets.”
2016 Sir Andrew J. Wiles
“for his stunning proof of Fermat’s Last Theorem by way of the
modularity conjecture for semistable elliptic curves, opening a new era
in number theory.”
2015 John Forbes Nash, Jr.
and Louis Nirenberg
“for striking and seminal contributions to the theory of nonlinear partial
differential equations and its applications to geometric analysis.”
2014 Yakov G. Sinai
“for his fundamental contributions to dynamical systems,
ergodic theory, and mathematical physics.”
2013 Pierre Deligne
“for seminal contributions to algebraic geometry and for their
transformative impact on number theory, representation theory,
and related fields.”
2012 Endre Szemerédi
“for his fundamental contributions to discrete mathematics and
theoretical computer science, and in recognition of the profound
and lasting impact of these contributions on additive number theory
and ergodic theory.”
2011 John Milnor
“for pioneering discoveries in topology, geometry and algebra.”2010 John Torrence Tate
“for his vast and lasting impact on the theory of numbers.”
2009 Mikhail Leonidovich Gromov
“for his revolutionary contributions to geometry.”
2008 John Griggs Thompson
and Jacques Tits
“for their profound achievements in algebra and in particular
for shaping modern group theory.”
2007 Srinivasa S. R. Varadhan
“for his fundamental contributions to probability theory and
in particular for creating a unified theory of large deviations.”
2006 Lennart Carleson
“for his profound and seminal contributions to harmonic
analysis and the theory of
smooth dynamical systems.”
2005 Peter D. Lax
“for his groundbreaking contributions to the theory and application of
partial differential equations and to the computation of their solutions.”
2004 Sir Michael Francis Atiyah
and Isadore M. Singer
“for their discovery and proof of the index theorem, bringing together
topology, geometry and analysis, and their outstanding role in building
new bridges between mathematics and theoretical physics.”
2003 Jean-Pierre Serre
“for playing a key role in shaping the modern form of many
parts of mathematics, including topology, algebraic geometry
and number theory.”
22After hearing a talk by 2004 Abel laureate, the late Sir Michael Atiyah, Karen Uhlenbeck became interested in gauge theory. photo: Didier Vandenbosch 23
Programme Abel Banquet at Akershus Castle
Iselin Nybø, Minister of Research and Higher
Abel Week 2019
Education, hosts the Norwegian government’s
banquet at Akershus Castle in honour of the
Abel Laureate (by invitation only)
May 20
front page photo :
—
Holmboe Prize Award Ceremony
May 22
—
Jan Tore Sanner, Minister of Education
The Abel Lectures
and Integration, presents the Bernt Michael
The Laureate will give the Abel Prize lecture
Holmboe Memorial Prize for teachers
© Marsha Miller
at the University of Oslo, Georg Sverdrups Hus,
of mathematics at Oslo Cathedral School
Aud. 1. This will be followed by other lectures
with topics related to the prize winner’s work
Wreath-laying at the Abel Monument
by the Abel Prize Laureate in the Palace Park
The Abel Party
at the Norwegian Academy of Science
Dinner in honour of the Abel Laureate
and Letters (by invitation only)
at the Norwegian Academy of Science and
Letters (by invitation only)
May 23
May 21 —
Abel Prize Lecture and mathematical games
—
The Abel Laureate gives a lecture at the
Abel Prize Award Ceremony
University of Bergen. Schoolchildren will invited
His Majesty King Harald V presents the
to play mathematical games with the prize
Abel Prize to the Laureate in the University
winner in Archimedes Labyrinth in Bergen
Aula, Oslo, Norway
Botanical Garden
Reception and interview with
Register online at: www.abelprize.no from
the Abel Laureate
mid-April or contact abelprisen@dnva.no,
at Det Norske Teatret, Oslo, Norway
facebook.com/Abelprize
Press contact: For other information:
Marina Tofting Håkon Sandbakken
marina.tofting@dnva.no hakon.sandbakken@dnva.no
+ 47 938 66 312 + 47 454 03 077
Anne-Marie Astad
anne.marie.astad@dnva.no
+ 47 415 67 406You can also read
