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Mathematics Department and Graduate School Colloquium
Archive
2002-2003
- Tuesday May 27, 2003
Victor Guillemin
(MIT)
Title: Moment Polytopes
- Abstract:
It was observed in the early 1980's by Sternberg and myself and by Atiyah
and Kirwan that if a compact Lie group acts as a group of symmetries on a
compact symplectic manifold one can associate with this action a convex
polytope: the moment polytope. At the time this result seemed interesting
mainly because it fitted a grab-bag of previous convexity theorems (by
Horn, Kostant, Weyl and others) into a general framework. However, it has
turned out to have a number of other unanticipated applications over the
course of the last two decades. In this talk I'll report on some of these
applications.
- Tuesday May 20, 2003
Robert McIntosh
(Office of the Washington State Superintendent of Public Instruction)
Title: The K-12 Mathematics Scene
-
- Abstract:
Before students arrive on the UW campus, they have spent
twelve or thirteen years being taught mathematics in K-12
schools. How that mathematics is taught has a considerable
impact on how and what can be taught at UW. It is also the
subject of debate, controversy, legislation and a lot of hard
work at local, state and national levels. This talk will focus
particularly on the perspective from Olympia: the WASL and the
EALR's (translations will be supplied), the response to Bush's
"No Child Left Behind" legislation and the areas where change
is occurring or hoped for.
- Tuesday April 29, 2003
Mircea Mustata
(Clay Institute/Harvard University)
Title: Integration and singularities
- Abstract:
I will discuss connections between several integration
theories and the study of singularities. The story started
with the work of Gelfand, Bernstein and Atiyah on complex powers and
continued with Igusa's study of the p-adic case.
I will explain the more recent approach based on the theory
of motivic integration due to Kontsevich, Batyrev, and
Denef and Loeser, and the applications to the study of singularities.
- Tuesday April 22, 2003
Herb Clemens
(Ohio-State University)
Title: Complex structures and their deformations.
- Abstract:
This talk will be a personal journey through my (hopefully still
developing) understanding of what it means to have a complex structure on a
differentiable manifold and how one can view changes in that structure. I
will begin with the example of a 2-real-dimensional torus, as it might have
been thought about 100 years ago (and as I had to first think about it
before being able to approach more modern formulations). The endpoint of
the talk will be a brief encounter with Kuranishi's beautiful formulation of
the deformation theory of complex structures on an arbitrary (compact)
differentiable manifold.
- Tuesday April 15, 2003
János Kollár
(Princeton University)
Milliman Lecture I: What is the biggest multiplicity of a root of a
degree d polynomaial?
- Abstract:
TBA
- Tuesday April 8, 2003
Jürgen Herzog
(Universität Essen/MSRI
Title: The impact of Groebner basis theory on algebra and combinatorics.
- Abstract:
Commutative algebra since Dedekind, Hilbert and Emmy
Noether favors non-constructive arguments over
concrete algorithms, because often they are shorter
and more elegant. The arrival of computers and their
increasing power changed the picture to some extend,
as now known algorithms which were beyond the scope of
what can be done with pencil and and paper have become
effective tools in computations. The actual
implementations of these algorithms to provide
solutions to concrete problems substantially use
Groebner bases. The algorithm to compute Groebner
bases was introduced by Bruno Buchberger in 1965.
However Groebner basis theory is also an instrument to
attack and solve theoretical problems in commutativwe
algebra and combinatorics. In my lecture this point of
view will be stressed. One of the earliest and most
impressive examples of a theoretical application of
Groebner bases is Macaulay's theorem on Hilbert
functions published in 1927. This theorem is strongly
related to algebraic shifting theory which was
invented by Kalai 1984 in the study of simplicial
complexes.
Many applications of Groebner basis theory rely on the
fact that the initial ideal of a graded ideal has the
same Hilbert function as the graded ideal itself and
that the graded ideal shares all nice properties (for
example being Cohen-Macaulay, Gorenstein or Koszul)
with its initial ideal. Thus problems about ideals in
polynomial rings can be reduced to questions about
monomial ideals which are far easier and are
accessible to combinatorial techniques. This strategy
has been applied very successfully to the study of
determinantal and toric rings with fundamental
contributions by Sturmfels. Some of these developments
will be surveyed in this lecture.
- Special Colloquium
Thursday April 3, 2003
Michael Roeckner
(University of Bielefeld)
Title: Heat equation in infinitely many variables and applications to
stochastic partial differential equations
- Abstract:
I will first review the connection between ordinary stochastic differential
equations (OSDE) on ${\bf R}^d$ and parabolic partial
differential equations (such as the classical heat equation). Subsequently,
a generalization to OSDE's in infinite dimensions will be given. A purely
analytic approach to solve the corresponding (generalized) heat equation in
infinitely many variables will be presented. Infinite
dimensional OSDE's related to parabolic stochastic partial differential
equation
will be discussed. Examples will include
the stochastic Ginzburg-Landau equation, the generalized stochastic Burgers
equation,
and the stochastic Navier-Stokes equation.
- Tuesday February 25, 2003
Kristin Lauter
(Microsoft Research)
Title: Curves in Cryptography
- Abstract:
Elliptic Curve Cryptography (ECC) was proposed in 1985 as an alternative
to the RSA and traditional Diffie-Hellman cryptosystems. ECC has now been
encorporated into many of the standard protocols for cryptography which
serve to secure internet transactions and encrypt communications. This
talk will give a quick overview of how standard elliptic curve
cryptography works, and also point to a couple of non-standard uses of
elliptic curves for cryptographic purposes, such as a recently proposed
system for doing identity-based-encryption (IBE). The talk will also
indicate the wealth of interesting mathematical questions in number theory
and algebraic geometry which arise from these applications.
- Tuesday February 11, 2003
László Lovász
(Microsoft Research)
Discrete Analytic Functions and Global Information from Local Observation
-
- Abstract:
We observe a certain random process on a graph "locally", i.e., in the
neighborhood of a node, and would like to derive information about
"global" properties of the graph. For example, what can we know about a
graph based on observing the returns of a random walk to a given node?
This can be considered as a discrete version of "Can you hear the shape of
a drum?"
Our main result concerns a graph embedded in an orientable surface with
genus g, and a process, consisting of random excitations of edges and
random balancing around nodes and faces. It is shown that by observing the
process locally in a "small" neighborhood of any node sufficiently (but
only polynomially) long, we can determine the genus of the surface. The
result depends on the notion of "discrete analytic functions" on graphs
embedded in a surface, and extensions of basic results on analytic
functions to such discrete objects; one of these is the fact that such
functions are determined by their values in a "small" neighborhood of any
node.
This is joint work with Itai Benjamini.
- MICROCENTURY TALK
Tuesday January 28, 2003
Cathleen Morawetz
(Courant Institute, NYU)
Mixed equations, transonic flows and boundary value problems that show that
the road to supersonic flow is difficult.
- Abstract:
The study of equations that change from elliptic to hyperbolic began with
a paper by Tricomi in 1923. The subject became important twenty-five years
later in developing subsonic airplanes flying at maximum speed without
shocks. The mathematical theory of well-posed boundary value problems was
used to explain why shocks almost always occur near the airplane wing as
soon as the speed of the plane "approaches" sonic. Proofs for similar
problems using Noether's conservation law will be sketched and some
special unsolved singular problems discussed.
-
Special Colloquium
Tuesday January 14, 2003
10:30am (note unusually early time)
Jeffrey Brock
(Columbia University)
The classification problem for hyperbolic 3-manifolds.
- Abstract:
After proving his revolutionary hyperbolization theorem for
Haken 3-manifolds in the late 1970s, W. Thurston proposed a conjectural
classification and parameterization for infinite volume hyperbolic
3-manifolds with finitely generated fundamental group. His conjectural
picture, which has topological and geometric components, predicts that
such a hyperbolic 3-manifold is determined by simple topological,
combinatorial, and conformal data. In this talk I will discuss recent
progress on the classification problem, including a full solution for
hyperbolic 3-manifolds with freely-indecomposable fundamental group.
- Special Colloquium
Monday January 13, 2003
4:00pm
Charles Doran
(Columbia University)
Integral Structures and Mirror Symmetry.
- Abstract:
We begin by introducing the notion of a Hodge Structure on an
algebraic variety and of its Variation in families (VHS). The
complex projective threefolds whose VHS we study are "Calabi-Yau"
--- a natural generalization of curves of genus 1 --- and are of
great interest to geometers and string theorists alike. The best
known example of such a family is the Fermat quintic pencil, whose
investigation over a decade ago by physicists marked the birth of
Mirror Symmetry as a mathematical subject. Under certain
hypotheses motivated by this example, we explicitly determine the
corresponding abstract integral VHS on the thrice punctured sphere
by describing the underlying complex monodromy representations
and compatible skew forms (given by 4x4 matrices) and the integral
lattices. We then compare the integral structures which arise from
known Calabi-Yau families to those which are predicted by a very
general formulation of the Mirror Symmetry Conjecture due to
Kontsevich, providing a nontrivial partial verification of the
conjecture. This is joint work with John Morgan.
- Special Colloquium
Friday January 10, 2003
2:30pm
Romyar Sharifi
(Harvard University)
Title: Galois groups with restricted ramification
- Abstract:
A prime ideal in the ring of integers of a number field is said to ramify
in an extension of number fields if it does not factor as a product of
distinct prime ideals in the extension ring. We may consider the Galois
group of the maximal extension of a number field unramified outside a
given set of prime ideals. Such Galois groups play an important role in
algebraic number theory, yet their structure is generally far from
understood.
Our focus will be the Galois group of the maximal pro-p unramified outside
p extension of the cyclotomic field of pth roots of unity, for a prime p.
I will explain what information can be gained from a pairing on cyclotomic
p-units defined by McCallum and myself. I will then describe my results
on the structure of an interesting Lie algebra associated to the outer
action of this group on the fundamental group of the projective line minus
three points.
- Special Colloquium
Thursday January 9, 2003
4:00pm
Antonella Grassi
(University of Pennsylvania)
Geometry of open-closed string duality.
- Abstract:
The open-closed dualities in physics involve transformations among
(complex) Calabi-Yau threefolds, holomorphic curves with and without
boundaries, Chern Simons theory on (real) 3 manifolds and knot invariants.
We will discuss mathematical implication of these dualities and relations
among these various invariants.
- Special Colloquium
Tuesday January 7, 2003
1:30pm
John Garnett
(UCLA)
Analytic Capacity, Bilipschitz Maps, Cantor Sets and Menger Curvature
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Abstract
- MICROCENTURY TALK
Tuesday November 19, 2002
Neil Trudinger
(Australian National University)
Analytic methods in affine geometry.
- Abstract:
Following Klein's famous Erlangen lecture in 1872, the study of affine
differential geometry flourished in the early 20th century under the
guidance of Blashke and his associates. Recently fundamental problems
for locally convex surfaces have been solved using the modern theory
of nonlinear elliptic partial differential equations. In this talk we
shall introduce the analytic framework and describe recent advances
concerning affine completeness and the affine Bernstein and Plateau
problems.
- Tuesday November 12, 2002
George Papanicolaou
(Stanford University)
Remote sensing and communications in scattering
environments.
- Abstract:
I will describe what remote sensing in a randomly
scattering environment is and how it can be used
in assessing the performance of some wireless
communications systems. The mathematics
of these problems involves some interesting questions
in stochastic analysis.
- MICROCENTURY TALK
Tuesday October 22, 2002
Karen Smith
(University of Michigan)
Uniform Results: A recent trend in geometry and algebra.
- Abstract:
In this talk, a recent research trend in algebraic geometry and
commutative algebra towards proving "uniform results" will be illustrated
through five fairly concrete problems on different topics. These problems
are unified by the approaches to their solutions: all use a machine called
"multiplier ideals" which has arisen independently over the last 15 years
in analysis, algebraic geometry and commutative algebra.
The talk is geared at non-specialists, including graduate students.
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