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Mathematics Department and Graduate School Colloquium
Archive
2000-2001
- May 29, 2001
Douglas Ravenel (University of Rochester)
Fractals and iterated function systems
- Abstract:
An iterated function system is a simple tool for
producing a large number of well known fractals including the
Cantor set, the Sierpinski triangle and the Jurassic Park dragon.
The main theorem of the subject has an elegant proof that can be
explained to undergraduates and that leads directly to an
effective computer algorithm. It involves applying the
contraction mapping theorem to "the space where fractals live,"
the set of nonempty compact subsets of the plane topologized by
the Hausdorf metric. The talk will be lavishly illustrated.
- May 22, 2001
Igor Dolgachev (University of Michigan)
Algebraic invariants and automorphic functions
- Abstract:
An algebraic invariant is a rational function on a vector space which is
invariant with respect to a linear action of an algebraic group. The field
of such functions is the field of rational functions on the orbit space.
An automorphic function is a meromorphic function on a homogeneous domain
which is invariant with respect to a discrete group of its holomorphic
automorphisms. There are several beautiful examples when a field of
algebraic invariants can be naturally identified with a field of
automorphic functions. The most notorious example is the field of
invariants on the space of binary forms of degree 4 which coincides with
the field of automorphic functions on the upper-half plane invariant with
respect to the modulur group. In this talk we shall explain other examples
of this sort which relate fields of invariants on vector spaces of
homogeneous forms with fields of automorphic forms on a complex ball or
other hermitian symmetric domains.
- May 8, 2001
Hart Smith (University of Washington)
The Wave Equation and Harmonic Analysis
- Abstract:
In the past 30 years, the wave equation has played an
increasingly important role in harmonic analysis, both
as a tool for studying problems in spectral theory,
as well as by motivating the development of estimates
for linear wave equations that are vital for the
study of nonlinear wave equations. I will survey
a few of the more important results, as well as
discuss some recent work on wave equations for
metrics of low regularity.
- April 12, 2001
NOTE UNUSUAL DAY AND TIME
Thursday, 2:30 pm
Paul Goerss (Northwestern University)
Using geometry to parametrize homotopy theory
- Abstract:
It is a remarkable observation of Mike Hopkins that one can use
the algebraic geometry -- and in particular the sheaf theory -- of the
moduli "space" (stack, really) of formal group laws to organize much of
what is known about global phenomena in stable homotopy theory. The
purpose of this talk is to explain this statement; that is, to develop the
geometric ideas, make the connection to homotopy theory, and to
spell out what this gives us. To be specific, the cohomology of
this stack is the input for one of our standard spectral sequences
for computing the homotopy groups of spheres, which says that if
one could somehow lift the sheaf theory to topology we would have
a powerful tool for dissecting these groups. I'll also explain the
progress on this lifting problem.
- April 10, 2001
Gunter Ziegler (Technische Universität Berlin)
Fat 3-Spheres, 4-Polytopes and 5-Lattices
- Abstract:
The following three classes contain very similar objects
- in a topological resp. geometric resp. combinatorial model:
- 3-dimensional CW-spheres with the intersection property,
- 4-dimensional convex polytopes, and
- Eulerian lattices of rank 5.
We introduce and study the parameter of fatness,
for these three classes - which
seems to be a key indicator to show how little we know.
So, it is not clear whether fatness is bounded at all on any
of these classes. Here we construct examples of
- rational 4-dimensional convex polytopes of fatness larger than
5-\varepsilon,
- 4-dimensional convex polytopes of fatness larger than 5.01, and
- 3-dimensional CW-spheres with the intersection property
of fatness larger than 6-\varepsilon.
This implies counter-examples to conjectured f-vector inequalities of
Bayer (1988) and of Billera & Ehrenborg (1999).
Most of our examples are constructed using the "Eppstein construction":
as the convex hull of a 4-polytope with all ridges tangent to S^3,
and its polar.
This construction has a close connection with ball packings in S^3.
Their study should lead to an infinite family of 2-simple 2-simplicial
polytopes.
(Joint work with David Eppstein, UC Irvine)
- March 8, 2001
Thursday, 2:30 pm
Mark Haiman (University of California at San Diego)
Macaulay inverse systems in combinatorics
- Abstract:A Macaulay inverse system is a finite dimensional
space of polynomials closed under differentiation with respect to the
variables. One way to get one is to take the space spanned by all
iterated partial derivatives of a single generating polynomial. A
second way is as the solution space of a system of differential
equations. In this talk I'll introduce examples of each type with
amazing combinatorial properties. The first example is the inverse
system generated by a certain determinant Δµ in
2n
variables, where µ is a partition of n.
Its dimension is n!
and its combinatorics are connected with the Kostka-Macdonald
coefficients in the theory of symmetric functions. The second example is the
space of harmonics for the diagonal action of Sn on the double
of its natural representation. Its dimension is (n+1)n-1
and the combinatorics involves trees, Catalan numbers, and
parking functions, among other things.
- February 27, 2001
Tony Pantev (University of Pennsylvania)
K-theory and strings
- Abstract:
This will be a survey talk on some recent discoveries in
geometry and topology which were provoked by advances in the string
theory of D-branes and the D-brane charges. In particular I will
explain a
construction of a new K-theoretic intermediate Jacobian of Riemannian
manifolds (due to Witten) and the construction of new generalized
cohomology theories of differential type (due to Freed, Hopkins and
Singer. The talk is aimed at a
general mathematical audience
and will not assume any familiarity with string theory and K-theory.
- February 23, 2001
Friday, 4:00
David Morrison (Duke University)
Is There a Pot of Gold at the End of Every Rainbow? The Search for New
Dualities in String Theory
- Abstract:
There has been a major revolution in string theory
during the past six years, in which new "duality" relations have
been observed which link the different types of string theory together
in a common framework. These dualities have provided an important
tool for studying strongly-coupled physical systems to which traditional
perturbative methods do not apply. They have also revealed surprising
connections between different parts of mathematics which were not
previously recognized as being related.
- February 20, 2001
Bob Moody (University of Alberta, Edmonton, Canada)
Aperiodic Order and Diffraction
- Abstract:
The advent of physical quasicrystals, combined with the earlier work on
aperiodic tilings, launched a flurry of activity and a new, intriguing
area of mathematics. One of the many interesting things (both
for physicists and mathematicians) was the unexpected occurrence of
pure point diffraction in the context of non-crystallographic symmetry.
In this talk we will give a lot of interesting mathematical examples
of aperiodic structures that show how common this phenomenon really is
and go on to describe the relevant mathematics -- principally a mixture
of discrete geometry and harmonic analysis -- that lie behind it.
This leads to the difficult question of trying to determine exactly
which point sets diffract.
The talk will be addressed to a mathematics audience. Fortunately
there are lots of pictures!
- February 5, 2001
Leonid Kunyansky (Caltech)
Fast, high-order solution of surface scattering problems
- Leonid Kunyansky is a candidate for Assistant Professor in our department.
- Abstract: We present a new algorithm for the numerical solution
of
problems of acoustic scattering by surfaces in three-dimensional space.
This algorithm evaluates scattered fields through fast, high-order
solution of the corresponding boundary integral equation. The high-order
accuracy of our solver is achieved through use of PARTITIONS OF UNITY
together with ANALYTICAL resolution of kernel singularities. The
acceleration, in turn, results from use of a novel approach which, based
on high-order "TWO-FACE" EQUIVALENT SOURCE approximations, reduces the
computation of long-range interactions to evaluation of 3-D FFTs. The
presentation will include details of our numerical analysis of the method,
including a convergence analysis for its basic integrator, a proof of
validity of the two-face approximations we use, and a discussion of the
techniques used for resolution of geometric singularities in the Lipschitz
case.
The overall algorithm exhibits super-algebraic convergence, it can be
applied to smooth and non-smooth scatterers, and it does not suffer
from accuracy breakdowns of any kind. Our comparative numerical studies
show that the present algorithm can evaluate accurately in a personal
computer scattering from bodies of acoustical sizes of several hundreds
- a goal, otherwise achievable today only by supercomputing.
Joint work with O. Bruno
- January 30, 2001
Joseph Zaks (Haifa University, Israel and currently visiting Cornell
University)
The Beckman-Quarles Theorem for rational spaces
- Abstract: The Beckman-Quarles Theorem states that every
unit-preserving mapping
of to itself is an isometry, for all
; i.e. if for all
and ,
implies that , then
for all and .
The analogue of this theorem concerning unit-preserving mappings from
the rational -space to itself is quite
challenging. Based on results
of Perles and Benda (unpublished), one can easily show that the analogues
for the (rational) cases , and do not hold.
Recently A. Tyszka proved that the rational analogue for
holds.
I have extended Tyszka's result for infinitely many even
(odd) dimensions.
R. Connelly and I have recently extended these results for all even
dimensions , .
- January 29, 2001
Lisa Korf (University of Washington)
Variational Analysis and Sampling in Stochastic Optimization
- Lisa Korf is a candidate for Assistant Professor in our department.
- Abstract: Stochastic optimization is concerned with the
minimization of a
function over a set, with the additional feature that some of the
problem's parameters which are not known with certainty are modeled as
random variables, with a discrete or more general distribution. Many
challenging theoretical questions arise out of the consideration of the
numerical and computational issues associated with these inherently very
large (possibly infinite-dimensional) models. In particular, algorithmic
procedures for solving these problems must rely on approximation schemes.
Sampling from the underlying distribution provides an important means by
which to accomplish this.
But in order to justify this approach, one must ensure that the
approximate solutions from the sampled problems converge to the solution
of the original model. In the past, such justification could generally
only be obtained on a problem by problem basis, or for specific classes
of problems using classical variational methods. However, through the
techniques of variational analysis, the justification for sampling in
stochastic optimization may now be obtained as a consequence of the
consistency of the stochastic optimization problems themselves. This
results in the ability to encompass all of the problem classes in one
broad-reaching theory. To do this, it is shown that one can appeal to
the i.i.d. or ergodic properties of random lower semicontinuous functions.
The analysis also relies on the development and use of probability
theory on the space of lower semicontinuous functions, and a special
scalarization of such extended-real-valued functions.
- January 23, 2001
Van Ha Vu (Microsoft Research)
On an economical versio of Waring's problem
- Van Ha Vu is a candidate for Assistant Professor in our department.
- Abstract: In 1770 Waring asserted, without proof, that for every
natural
number k,
there exists a number s such that every natural number can be
represented as sum of s non-negative kth powers.
For instance, every
natural number is sum of four squares, 9 cubes and so on. Waring's
conjecture was first proved by Hilbert in the beginning of the last
century. A different and more efficient proof was found by Hardy and
Littlewood in 1920.
In 1980 Nathanson conjectured that one could use only a small subset of
the set of all kth to represent the natural numbers.
Small here means
that the subset in question has nearly best possible density. Partial
results were obtained by Erdos, Choi, Nathanson, Zollner and Wirsing. In
this talk, we shall sketch the proof of Nathanson's conjecture. The proof
uses tools from probability but no special knowledge in number theory and
probability is required.
- January 18, 2001
Thursday, 2:30
Ilia Binder (Harvard University)
Harmonic measure and polynomial Julia sets
- Ilia Binder is a candidate for Assistant Professor in our department.
- Abstract: We discuss the multifractal analysis of planar harmonic
measure. We are especially concerned with sharp bounds for the local dimension
of harmonic measure --- so
called universal spectra. The theory is well established for simply
connected domains, since in this case the technique of conformal mappings
can be applied.
It is conjectured that the universal spectra for the non simply connected
domains are the same as for the simply connected ones. We will discuss the
proof of the conjecture for a particular class of planar domains, the
basins of attraction to infinity of polynomials. The proof uses holomorphic
motion in the dynamical Teichmüller space.
- January 16, 2001
Yu Yuan (University of Texas at Austin)
A Priori Estimates for Fully Nonlinear Elliptic Equations
- Yu Yuan is a candidate for Assistant Professor in our department.
- Abstract: The theory of a priori estimates for fully nonlinear
equations with a convexity condition is well understood.
We survey some recent development on a priori estimates for
fully nonlinear elliptic equations without convexity. These equations
arise in differential geometry, and stochastic control theory.
- January 11, 2001
Thursday, 2:30
Ravi Vakil (MIT)
Branched covers of the sphere and the moduli space of curves:
Geometry, physics, representation theory, combinatorics
- Ravi Vakil is a candidate for Assistant Professor in our department.
- Abstract: The moduli space of curves has been a central tool and
object of study in
algebraic geometry for many decades, and yet its topology (and
intersection theory) has proved remarkably difficult to understand,
despite the belief that much of it is in some sense combinatorial.
Recent developments in Gromov-Witten theory have established a connection
between intersection theory on the moduli space of curves and branched
covers of the sphere, bringing to bear ideas from the fields of
mathematics mentioned in the title.
I'll describe the moduli space of pointed curves, some of the major
conjectures (Witten's, Faber's, and Virasoro), and discuss some of the
developments in the area (both old and new), and future prospects.
- November 21, 2000
This colloquium will be given in 120 Communications
David Damanik (UC Irvine)
Spectral theory of one-dimensional quasicrystals
- Abstract: We discuss recent progress on the spectral properties
of typical
models for one-dimensional quasicrystals. The primary example is a
one-dimensional Schrödinger operator whose potential is given by the
Fibonacci sequence. It was conjectured in the early eighties that such
operators should have purely singular continuous spectrum. We review the
history of this problem and its recent solution which has motivated
several connections with fields outside spectral theory.
- November 14, 2000
Craig Tracy (UC Davis)
On recent applications of random matrix theory
- Abstract: Methods first developed in random matrix theory are
finding application to combinatorics, number theory, growth processes,
statistics and queueing theory. This talk will survey some of these most
recent developments.
- October 26, 2000 (Note: Thursday, 2:30 in C-36 Padelford Hall)
Alexander Schrijver (CWI and University of Amsterdam, temporarily at
Microsoft Research)
Permanents and edge-colouring
Alexander Schrijver is head of the "Cluster: named Probability, Networks and
Algorithms for the Centrum voor Wiskunde en Informatica in Amsterdam. He is
one of the world's foremost experts on combinatorial optimization and
integer programming. He has written several books in these fields.
- October 24, 2000
Robert Coleman (UC Berkeley)
P-adic Spectral Curves
- Abstract: Sometimes operators on Banach spaces come in families.
If each operator is nuclear their spectra trace out a curve. I will explain
how conjectures of Gouvea and Mazur on p-adic properties of the
coefficients of modular forms can be re-interpreted in terms of properties
of such a curve.
- October 17, 2000
Persi Diaconis (Stanford University)
What do we know about the metropolis algorithm?
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