2005-06 Geometry seminar
Fridays, Milner 216 at 4pm
Generally the speaker will give a 50-60 minute
talk, followed by a break, followed by more for
those who are interested.
Spring 2006
Jan. 20 A. Yampolsky (A&M)
Title: On
totally geodesic unit vector fields.
Also possibly of interest : the 1/27 algebra
and
combinatorics seminar is on secant varieties of
Segre varieties
Jan 27 L Matusevich (A&M)
Title: Horn Hypergeometric D-modules
Abstract: I will give an introduction to the theory of D-modules, whose goal is an
algebraic treatment of linear PDEs. In this theory, hypergeometric systems
play the role of toric varieties: they provide examples with enough
combinatorial underpinning to be tractable by our current technology.
This is particularly interesting, since hypergeometric functions are
useful in all sorts of different areas. In this context I will discuss
joint work with Alicia Dickenstein and Ezra Miller on Horn systems.
Note the colloquim Feb. 2 might be of interest (Tevelev, UT Austin)
Feb. 3 Dan
Freed (UT Austin)
Title: Loop groups and twisted K-theory
Abstract: I will describe joint work with Mike Hopkins and Constantin
Teleman. We construct an isomorphism between the Verlinde algebra in the
theory of loop groups and a certain twisted K-theory group. I will mostly
explain a finite dimensional analog which relates to the Kirillov orbit
method and the Borel-Weil-Bott theorem in the theory of compact Lie groups.
Feb. 10 Joe Harris (Harvard)
Title: Brill-Noether theory
and the slope conjecture
Abstract:
Brill-Noether theory describes the family of linear systems on a
general curve of genus $g$. The slope conjecture is concerned with the
cone of effective divisors in the moduli space $\overline M_g$ of
stable curves, and essentially asserts that divisors comprised of
curves admitting Brill-Noether exceptional linear series are extremal
rays in this cone.
Recent
work of Farkas and Khosla has provided strong evidence that the slope
conjecture is fundamentally false. At the same time, however, it has in
a way strengthened the connection between Brill-Noether theory and the
divisor class theory of moduli.
In
this talk I hope to describe the development of Brill-Noether theory,
leading up to current conjectures; then the divisor class theory of
$\overline M_g$ and its relation to Brill-Noether theory; and finally
the work of Farkas and Khosla.
note the colloquim
Tues Feb 14 may be of interest (A. Savage, U. Toronto)
as well as the colloquium
Wed. Feb 15 (K. Knudson, Missisipi State)
Thursday Feb 16 (NOTE SPECIAL DAY and time) Ruth
Gornet (UT
Arlington) 3pm
Title: Laplace and Length Spectra and the
Wave Invariants on Riemannian Two-Step Nilmanifolds
Abstract: We compare the behavior
of the length spectrum and the wave invariants on certain
families of isospectral
nilmanifolds. The clean intersection hypothesis is discussed in detail.
And don't miss the Texas
Geometry and Topology Conference
at the University of Houston Feb. 17-19!
Feb 20 Robert Bryant
(Duke) (NOTE special day)
Title:
Real hypersurfaces in unimodular complex surfaces
Abstract: A
unimodular
complex surface is a complex 2-manifold X endowed with a holomorphic
volume form. A strictly pseudoconvex real hypersurface M in X inherits
not only a CR-structure but a canonical coframing as well.
In this article, this canonical coframing on M is defined, its
invariants are discussed and interpreted geometrically, and its basic
properties are studied. A natural evolution equation for strictly
pseudoconvex real
hypersurfaces in unimodular complex surfaces is defined, some of its
properties are discussed, and several examples are computed. The
locally homogeneous examples are determined and used to illustrate
various features of the geometry of the induced structure on the
hypersurface.
the colloquia Feb.
22(Radko, UCLA) and Feb
23 (Robles, Rochester) may be of interest
March 3 No seminar - recovering from the last few weeks.
March 10 No seminar- spring break starts early
March 17 No seminar: spring break
DO
NOT MISS!:
March 20,21,22. P.
Griffiths
(IAS)
Frontiers Lectures
March 31 D. Allcock (UT Austin)
Title: Moduli of Cubic Threefolds and Complex Hyperbolic Geometry
Abstract: The geometric-invariant-theory moduli space of cubic threefolds is a
quotient of the complex 10-ball (=complex hyperbolic 10-space) by a
specific discrete group. The meaning of "is" is that there is a map
from the moduli space to the Bailey-Borel compactification of the ball
quotient, which blows up one specific point and then blows down one
specific curve, and is otherwise an isomorphism. The main tools are
Hodge theory and an analysis of the fundamental group of the space of
smooth cubic threefolds. This is joint work with Jim Carlson and
Domingo Toledo.
April 7 James Lewis (U. Alberta)
Title: Arithmetic Invariants on
Algebraic Cycles
Abstract: Let X be a
projective algebraic manifold and
let CH^r(X; Q ) be the Chow group of algebraic cycles
of
codimension r on X, modulo rational equivalence.
We explain some motivation for working with a candidate ``motivic''
filtration F^{\nu} on CH^r(X; Q ),
and introduce a space of arithmetical Hodge theoretic
invariants \nabla J^{r,\nu}(X) which captures
information about the graded piece
Gr_{F}^{\nu}CH^r(X; Q).
In particular we discuss the image and
kernel of a corresponding
map \phi_{X}^{r,\nu} : Gr_{F}^{\nu}CH^r(X; Q )
-> \nabla J^{r,\nu}(X).
This talk is based on joint work with S. Saito,
and is aimed at nonexperts.
April 14 S. Keel (UT Austin)
Title: A functorial normal crossing
compactification of
moduli of smooth cubic surfaces.
Abstract: I'll explain recent
work with Tevelev and Hacking,
where we obtain a nice compactification, with a nice universal
family, via tropicalisation.
WED April 19: J. Dilles
(UPenn) NOTE SPECIAL DAY AND ROOM
Milner 313
Title: Construction of Calabi-Yau
threefolds.
Abstract: String theory has
stimulated the study of Calabi-Yau threefolds. Many questions remain
without answersand we still do not know 'enough' examples, i.e.
varieties which could
be usable to construct a sound physical theory. We will exhibit
and analyse two analoguous constructions. The first one is
related to group theory while the second requires the study of the
geometry of K3 surfaces.
April 21 D. Khosla (UT Austin)
Title: Divisors on Moduli Spaces of Curves
Abstract: For which g is it possible to write down the general curve of genus g?
-- that is, to give an expression with indeterminate parameters that
represents almost every curve? To answer this question it turned out
to be important to know which divisor classes in the moduli space of
curves represent actual subvarieties of codimension 1, the "effective
cone" of divisors. A well-known conjecture of Harris and Morrison
reflects a lot of what is known. However, it is false; I will explain
constructions using the space of curves with a map of given degree to
a given projective space that give many counterexamples.
May 20-21 Texas Algebraic
geometry
conference here at A&M
Previous semesters
fall 05
spring 05