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Abstracts of 2011-2012 UW-PIMS Mathematics Colloquia
May 18, 2012 at 2:30pm
Mechanical Engineering Building, Room 238 |
Bruce Reznick
University of Illinois at Urbana-Champaign |
The Secret Lives of Polynomial Identities |
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Polynomial identities can reflect deeper mathematical phenomena. In this
talk, I will discuss some of the stories behind four identities (and their
relatives). The stories involve algebra, analysis, number theory, combinatorics,
geometry and numerical analysis. The identities, which don't fit well in plain
text, involve polynomials in two, three and four variables being taken to powers
ranging from the third to the fourteenth. The earliest is due to Viéte, and
dates to the 1590s. Felix Klein makes a special guest appearance.
View slides from this
lecture.
View
video
of this lecture.
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April 6, 2012 at 2:30pm
Chemistry Library Building, Room 015 |
Christopher Hacon
University of Utah |
Geometry of Varieties of General Type |
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In this talk we will discuss issues related to the existence of a moduli
space for varieties of general type. Recall that varieties of general type are
the higher dimensional analog of Riemann surfaces of genus at least 2. We will
explain recent results on the boundedness of these
varieties (once we fix certain invariants, these varieties are expected to be
parametrized by finitely many finite dimensional parameter spaces) and on the
geometry of their possible degenerations.
View
video
of this lecture.
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March 9, 2012 at 2:30pm
Denny Hall, Room 216 |
Jonathan Brundan
University of Oregon |
Representations of General Linear Lie Superalgebras |
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The general linear Lie algebra \(gl_n\)(C) -- endomorphisms of an \(n\)-dimensional
complex vector space \(V\) with operation \([x,y]\) being the commutator \(xy-yx\) -- is the
most basic example of a Lie algebra. If we assume instead that the vector space
\(V\) is equipped with a \(Z/2\)-grading, i.e. \(V = V_0 \oplus V_1\) is the direct sum of
an \(m\)-dimensional "even" and an \(n\)-dimensional "odd" subspace, and replace
commutator with the "supercommutator" (which takes account of parity in the most
natural way), we get the general linear Lie superalgebra \(gl_{m|n}\)(C).
The representation theory of the general linear Lie algebra is unbelievably
rich and has a long history going back to Schur and Weyl, who computed the
characters of the finite dimensional irreducible representations via the theory
of symmetric functions. There is also an important family of (mostly infinite
dimensional) irreducible highest weight representations for which an explicit
character formula was conjectured by Kazhdan and Lusztig in 1979, and
dramatically proved by Beilinson-Bernstein and Brylinski-Kashiwara in 1980,
thereby giving birth to a subject known as "geometric representation theory."
So what happens for the general linear Lie superalgebra? Even the finite
dimensional representations are quite difficult to understand, but now we have a
pretty good picture. There is also a version of the Kazhdan-Lusztig conjecture
which has just been proved (by Cheng, Lam and Wang). In the talk I'll try to
give you the flavor of these results, starting with the base cases \(gl_2\)(C) and
\(gl_{1|1}\)(C), which illustrate the general picture perfectly despite being
trivial from a combinatorial perspective.
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video
of this lecture.
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February 17, 2012 at 2:30pm
Smith Hall, Room 205 |
Edward Witten
Institute for Advanced Study |
Gauge Theory and Khovanov Homology |
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After reviewing ordinary finite-dimensional Morse theory, I will explain how
Morse generalized Morse theory to loop spaces, and how Floer generalized it to
gauge theory on a three-manifold. Then I will describe an analog of Floer
cohomology with the gauge group taken to be a complex Lie group (rather than a
compact group as assumed by Floer), and how this is expected to be related to
the Jones polynomial of knots and Khovanov homology.
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of this lecture.
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January 13, 2012 at 2:30pm
Denny Hall, Room 216 |
Kristin Lauter
Microsoft Research and UW |
Elliptic Curve Cryptography and Applications |
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In the last 25 years, Elliptic Curve Cryptography has become a mainstream
primitive for cryptographic protocols and applications. This talk will give a
survey of elliptic curve cryptography and its applications, including
applications of pairing-based cryptography which are built with elliptic curves.
No prior knowledge about elliptic curves is required for this talk. One of the
information-theoretic applications I will cover is a solution to prevent
pollution attacks in content distribution networks which use network coding to
achieve optimal throughput. One solution is based on a pairing-based signature
scheme using elliptic curves. I will also discuss some applications to privacy
of electronic medical records, and implications for secure and private cloud
storage and cloud computing. View video of this lecture.
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November 8, 2011 at 2:30pm
Mechanical Engineering Building, Room 238 |
Greg Galloway
University of Miami |
On the Topology of Black Holes |
There is a widely held belief in physics that a true astrophysical black hole,
formed from the gravitational collapse of some stellar object, can be described
by a certain exact solution to the Einstein equations discovered by Kerr in the
60s. This belief is based largely on a powerful theorem which shows that the Kerr solution is the unique
solution to the vacuum (source-free) Einstein equations with certain prescribed
properties. A basic step in the proof is Hawking's theorem on the topology of
black holes which asserts that, under physically natural conditions, the surface of a black hole (i.e., cross
section of the event horizon) must be topologically a 2-sphere.
Developments in string theory over the past ten years have generated
considerable interest in gravitation and black holes in higher dimensions. The
remarkable discovery of Emparan and Reall of a 4+1 dimensional vacuum black hole
solution to the Einstein equations with nonspherical horizon topology raised the question as to what horizon topologies
are allowable in higher dimensions. In this talk we review Hawking's theorem on
the topology of black holes in 3+1 dimensions and discuss a generalization of it
to higher dimensions obtained in joint work with Rick Schoen. As we shall
discuss, this latter result, together with a more recent refinement of it, puts
restrictions on the topology of black holes in higher dimensions.View
video
of this lecture.
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October 14, 2011 at 2:30pm
Mechanical Engineering Building, Room 238 |
Joel Spencer
New York University |
Two Needles in Exponential Haystacks |
Erdős Magic, aka The Probabilistic Method, is a powerful tool for
proving the existence of a combinatorial object, such as a coloring.
A probability space is created for which the probability of success
is positive. Hence the desired object must exist. But where is it?
Here we examine instances in which the probability is exponentially
small so that a randomized algorithm would not be in \(P\). Nonetheless,
we give two recent startling successes.
Bansal: A quarter century ago this speaker showed that given \(n\) sets
on \(n\) vertices there is a two-coloring so that all discrepancies are
\(O(\sqrt{n})\). He long conjectured that no polynomial time algorithm
could find the coloring. Wrong! Nikhil Bansal, making ingenious use of
semidefinite programming, finds the coloring and much more.
Moser: Even longer ago, László Lovász, with the Lovász Local
Lemma, showed (roughly!) that when bad events are mostly independent
there is a positive probability that the random object has no bad events. Robin Moser
gives a simple "fix-it" randomized algorithm to find the object. The proof that the algorithm
works, however, is most original. It gives a new and seemingly quite different proof
of the Local Lemma itself.
When the probabilistic method sieves an event with exponentially
small probability the usual randomized algorithms will not find an actualization.
We discuss two recent startling successes: Moser et.al. on the Lovász Local
Lemma and Bansal on the speaker's "Six Standard Deviations Suffice."
View
video
of this lecture.
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