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