### UW Algebra Seminar

Autumn 2006

Tuesday 1:30pm, Padelford C-36

**Speaker:**
Eliot Brenner, Center for Advanced Studies in Mathematics at Ben
Gurion University

**Title:**
Heat Eisenstein Series and Applications

**Date:**
October 3

**Abstract:**
I will define the heat Eisenstein series and sketch some of their
theory, as developed by Jorgenson and Lang in the 90s and early
00s.
Then, I will describe their role in an ongoing project to
generalize
theta relations and spectral zeta functions to arithmetic
quotients of
real Lie groups in higher rank. Finally, I will state some of my
results on exact fundamental domains and explain how they are
used or
are expected to be used in this program.

**Speaker:**
Jon Hanke, Duke University

**Title:**
The 290-Theorem and Representing Numbers by Quadratic Forms

**Date:**
October 10

**Abstract:**
This talk will describe several finiteness theorems for quadratic
forms, and progress on the question: "Which positive definite
integer-valued quadratic forms represent all positive
integers?". The answer to this question depends on settling
the
related question "Which integers are represented by a given
quadratic form?" for finitely many forms. The answer to this
question can involve both arithmetic and analytic techniques,
though only recently has the analytic approach become
practical.
We will describe the theory of quadratic forms as it relates
to
answering these questions, its connections with the theory of
modular forms, and give an idea of how one can obtain explicit
bounds to describe which numbers are represented by a given
quadratic form.

**Speaker:**
Nicole Lemire, University of Western Ontario and MSRI

**Title:**
Galois Module Structure of Galois Cohomology and Applications

**Date:**
October 17

**Abstract:**
For a cyclic p-extension of fields E/F where F contains a
primitive p-th root of unity, we determine the $\F_p[\Gal(E/F)]$
module structure
of $H^m(G_E,\F_p)$ in terms of the field extension E/F. We apply
this to determine
restrictions on the group structure of an absolute Galois group
$G_F$ (with Dave Benson)
and to determine the cohomological dimension of the maximal pro-p
quotient $G_F(p)$ (with John Labute).

**Speaker:**

**Title:**

**Date:**
October 24

**Abstract:**

**Speaker:**
Daniel Chan, University of New South Wales

**Title:**
Clifford algebras and Conic Bundles

**Date:**
October 31

**Abstract:**
Maximal orders on surfaces are examples of noncommutative
surfaces. We consider the question of constructing and studying
such orders in the case where the rank is 4 over the centre. The
approach is via the even Clifford algebra. We also study the
relationship of these orders with conic bundles.

**Speaker:**
Kevin Knudson, University of Mississipi and MSRI

**Title:**
Generating discrete Morse functions from point data

**Date:**
November 7

**Abstract:**
If $K$ is a finite simplicial complex and $h$ is an injective map
from the vertices of $K$ to $\R$ we show how to extend $h$ to a
discrete
Morse function in the sense of Forman in a reasonably efficient
manner so
that the resulting discrete Morse function mirrors the large-scale
behavior of $h$. Examples and an explicit algorithm will be
presented.

**Speaker:**
Julia Hartmann, University of Heidelberg and University of
Pennsylvania

**Title:**
Differential Galois Groups and Patching

**Date:**
November 14

**Abstract:**
The talk gives an introduction to differential Galois theory, and
then addresses
the inverse problem: Which groups can occur as differential Galois
groups over a
given field? Recent results employ patching methods to attack the
problem (joint
work with D. Harbater).

**Speaker:**
Steve Mitchell, University of Washington

**Title:**
Schubert varieties in the affine Grassmannian

**Date:**
November 21

**Abstract:**
The loop space of a compact Lie group is homotopy equivalent to a
certain infinite-dimensional projective algebraic variety, the
"affine Grassmannian". This variety shares many of the beautiful
features of ordinary Grassmannians; for example, it has a
decomposition into Schubert cells whose closures are ordinary
projective varieties. There are two striking differences, however:
The affine Grassmannian is a topological group, and the principal
bundle defining it as a homogeneous space is topologically
trivial.
In these largely expository talks I will present a topologist's
perspective on the affine Grassmannian, and discuss some ongoing
work involving its Schubert varieties: For example, which of these
are nonsingular? This question is closely related to the
combinatorics of the Bruhat order on the affine Weyl group, a
beautiful subject in its own right.

**Speaker:**
Steve Mitchell, University of Washington

**Title:**
Schubert varieties in the affine Grassmannian

**Date:**
November 28

**Abstract:**
The loop space of a compact Lie group is homotopy equivalent to a
certain infinite-dimensional projective algebraic variety, the
"affine Grassmannian". This variety shares many of the beautiful
features of ordinary Grassmannians; for example, it has a
decomposition into Schubert cells whose closures are ordinary
projective varieties. There are two striking differences, however:
The affine Grassmannian is a topological group, and the principal
bundle defining it as a homogeneous space is topologically
trivial.
In these largely expository talks I will present a topologist's
perspective on the affine Grassmannian, and discuss some ongoing
work involving its Schubert varieties: For example, which of these
are nonsingular? This question is closely related to the
combinatorics of the Bruhat order on the affine Weyl group, a
beautiful subject in its own right.

**Speaker:**
Hsian-hua Tseng, UBC

**Title:**
Gromov-Witten theory of twisted projective lines and integrable
hierarchies

**Date:**
December 5

**Abstract:**
It has been expected that the totality of Gromov-Witten invariants
of a Kahler manifold admits certain integrable structures. More
precisely, the
generating function of descendant Gromov-Witten invariants should
be a
tau-function of certain integrable hierarchy. Examples of
manifolds with this
property include a point (Witten's conjecture) and complex
projective line
(Toda conjecture). In this talk we discuss this problem in a class
of new
examples--the twisted projective lines. This is based on the joint
work with
Todor Milanov.

*To request disability accommodations, contact the Office of
the ADA
Coordinator, ten days in advance of the event or as soon as
possible:
543-6450 (voice); 543-6452 (TDD); 685-3885 (FAX); access@u.washington.edu
(E-mail).
*

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