Math
Colloquium
Fall 2008
Math colloquium
meets on Mondays at 12:15 in Schreiber 006,
Previous talks: 2002-2003 , 2003-2004 , 2004-2005 , 2005-2006 , 2006-fall, 2007-spring
3.11.2008
Boris Feigin, Landau Institute for Theoretical Physics, Russian Academy
Sackler
lecture
Part of a
lecture series titled: Quantum group construction of maximal commutative
subalgebra inside
universal
enveloping algebra of an affine Kac-Moody algebra.
Title of the
colloquium talk:
Review of
quantum group approach - why quantum groups are so efficient.
ABSTRACT: I will try to explain the role of quantum groups in
mathematics. Sometimes quantum groups appear naturally - sometimes not, but
they give us understanding and technical tools to solve problems or to create
the new ones. Surely the idea of symmetry is so natural for human beings
because of evident reason. The conceptions of beauty and symmetry are very
close. So the theory of quantum group studies the new type of symmetry and as a
theory it is beautiful by itself. Well, but the real question is why the
quantum groups are so efficient (and also so popular)? The idea of symmetry is
universal and deep but in the same time trivial. For example one of the most
well-known applications of the quantum groups - invariants of knots and
3-dimensional manifolds. But when you study this topological subject you get a
mixed feeling. On one side you definitely see the manifestation of idea of
quantum symmetry. Especially it is clear from the topological field theory
point of view. But at the same time you feel that the real essence lies in some
combination – quantum group and something – and this
”something” is rather elusive. You can find many interesting
technical ideas; find deep connections with very different parts of
mathematics. So you see the complicated net of details, you definitely feel
something important behind it, but you can not catch it. Historically quantum
groups appear as a part of the theory of quantum integrable system. And still
the main applications are there. Quantum groups are used as a tool for
constructing "big" commutative subalgebras inside some associative
algebras and after that diagonalizing the action of these commutative algebras
in representations of this associative algebra.
I plan to review the known construction of
the "big commutative algebras” and present several new results. I
want to remark that here the situation here is similar in the one in the theory
of invariants of knots and 3-manifolds. Again the huge amount of ideas,
technical details, connections with other theories – and one has a
feeling that something mysterious is hidden behind. More concretely I will talk
about the following subjects.
1. R -matrix construction of commutative
subalgebras
2. Construction of commutative subalgebras
using the representation theory of affine Kac-Moody algebras "on the
critical level". Connections with Langlands duality.
3. "Center on a critical level"
for quantum affine Kac-Moody algebras.
4. Commutative subalgebras inside
universal enveloping algebra of an affine Kac-Moody Lie algebra.
5. Quantization of classical integrable
systems..
10.11.2008, 12:15
Rabin Memorial
day – No Colloquium
17.11.2008
Bill Casselman, University
of British Columbia
The role of
pictures in mathematics.
ABSTRACT: Professional mathematicians are ambivalent about the use
of illustrations in mathematics. I claim that in fact a more widespread and
clever use of illustrations in elementary classes as well as professional exposition
is the surest way to get audiences to understand what mathematics is all about.
One of the problems is that mathematicians are reluctant to teach
themselves how to produce good illustrations, and I'll talk also about tools
available, and how much work is involved in learning how to use them.
24.11.2008, 12:15
Bo'az Klartag, Tel Aviv University
High-dimensional distributions with convexity properties.
ABSTRACT: We review recent advances in the understanding of probability
measures with geometric characteristics on Rn, for large n. These
advances include the central limit theorem for convex sets, according to which
the uniform measure on a high-dimensional convex body has marginals that are
approximately gaussian.
1.12.2008, 12:15
Mikhail
Gromov, IHES and NYU
Homological Isoperimetry and Morse Theory.
8.12.2008, 12:15
Shahar Mendelson, Technion and ANU.
Reconstruction and random operators.
ABSTRACT: In a reconstruction problem one is interested in the
ability to approximate an unknown vector in a given subset T of Rd
using linear (usually random) measurements. This problem has been studied in
recent years by Donoho, Candes and Tao, and Rudelson and Vershynin in very
specific cases (i.e. for specific choices of the set T and of the
measurements), and was solved in a very general case by Pajor,
Tomczak-Jaegermann and myself. I will show that this problem has
strong connections to the way random operators act on subsets of the sphere, or
more generally, to the behavior of certain empirical processes, which
leads to the solution of the general problem. I will also present a
simple argument that solves the specific cases mentioned above.
15.12.2008, 12:15
Manor Mendel, Open University.
BiLipschitz embeddings and dichotomies.
ABSTRACT: BiLipschitz embedding of
a metric spaces is a mapping that preserves distances up to some
constant multiplicative factor.
We will discuss
some basic facts of biLipschitz embeddings of finite metric spaces, and their
applications to Computer Science. We will then discuss the following
dichotomy: For every host space H, either H contains all finite metric spaces
almost isometrically; or there exists a sequence of finite metric spaces that
does not embed in H biLipschitzly.
Proofs of this
and related facts are obtained by transferring analogues ideas from the
geometry of Banach spaces.
22.12.2008, 12:15
NO COLLOQUIUM THIS WEEK
29.12.2008, 12:15
Sergei Kuksin,
Ecole Politechnique.
Mathematics of 2d turbulence.
ABSTRACT: In my talk I will review
recent progress in the study of the qualitative theory of stochastic 2D
Navier-Stokes equations and discuss its relevance for the theory of turbulence.
5.1.2000, 12:15
NO COLLOQUIUM THIS WEEK
12.1.2009, 12:15
Sergei Treil, Brown University
Haplitz
operators, systems and functions..
ABSTRACT: In my talk I will consider several topics in the theory of
Hankel and Toeplitz (Haplitz) operators, related to the work of Israel Gohberg.
I will try to explain why these operators are interesting and important object
of investigation and discuss their relation to complex analysis and to the
system theory.
19.1.2009, 12:15
Eitan Sayag, Ben Gurion University.
Periods of modular forms and representation theory.
ABSTRACT:
Periods of modular forms are objects of central importance to number theorists
and have been a source of inspiration for workers in representation theory and
arithmetic algebraic geometry.
I will give an overview on the role of representation theory in the modern
study of periods of automorphic functions starting with the example of GL(2).
I will then explain the role of periods in detecting poles of L-functions and
in their role in detecting the image of a Langlands
functoriality.
Finally, I will describe some new results concerning periods of automorphic
forms on GL(n) and some new problems in Harmonic
Analysis and representation theory arising from this study.
26.1.2009, 12:15
Charles
Fefferman, Princeton University
CANCELLED.