Geometry Seminar
Spring 2007
Fridays at 4 pm in Milner 216

Format: Talks are 50-60 minutes, with the option to continue after a short break.

Spring 2007

Date: Jan 18 [Thursday]
Speaker: A. Rittatore (Uruguay)
Title: The structure of algebraic monoids.
Date: Jan 19
Speaker: J.M. Lansdberg (TAMU)
The postponed working seminar meets.
Date: Jan 26
Speaker: J.M. Lansdberg (TAMU)
The postponed working seminar meets.
Date: Feb 02
No seminar. See the Department Colloquium.
Date: Feb 9
Speaker: J. Weyman (Northeastern)
Title: Jet Schemes of determinental varieties
Date: Feb 16, 1:30 pm
Speaker: G. Farkas (UT Austin)
Title: Koszul divisors on the space of curves
Abstract: Given a moduli space, what is the "best" effective divisor one can construct on this space? (Here, "best" means extremal in the sense of higher dimensional algebraic geometry.) We present a general method of constructing effective divisors on a large class of moduli spaces using the syzygies of the parameterized objects. Applications of this method include: (1) a proof that the moduli space of Prym varieties of dimension g is of general type when g > 13, (2) a proof tha the moduli space of curves of genus 22 is of general type, and (3) shorter (and conceptually different) rederivations of all the Eisenbud-Harris-Mumford calculations on the Picard group of the moduli space of curves.
Date: Feb 16
Speaker: A. Ortega (Morelia)
Title: Dolgachev's conjecture on the moduli space of rank three bundles
Abstract: For a curve of genus 2, the moduli space of rank 3 vector bundles with trivial determinant can be realized as a double cover of the 8-dimensional projective space branched along a sextic hypersurface. On the other hand, Coble proved a century ago that there exists a unique cubic hypersurface in 8-space which is (1) singular along the Jacobian variety of the curve (which is embedded using the system of 3-theta functions) and (2) invariant under the natural action coming from the torsion points of order 3 on the Jacobian. Dolgachev has made the striking conjecture that these two seemingly completely unrelated hypersurfaces are in fact projectively dual to each other. I will discuss the background of the problem and present a proof of Dolgachev's conjecture.
Date: Mar 19-23
Speaker: A. Zelevinsky (Northeastern)
Frontiers Lecture Series
Date: Mar 23-25
Texas Geometry and Topology Conference (TGTC) at TCU.
Date: Mar 26-30
Speaker: Y. Siu (Harvard)
Frontiers Lecture Series
Date: Mar 31--Apr 1
Texas Algebraic Geometry Seminar (TAGS) at UT Austin.
Date: Apr 2-6
Speaker: B. Shiffman (Johns Hopkins)
Frontiers Lecture Series
Date: Apr 6
Speaker: Luke Oeding (TAMU)
Title: On the Holtz-Sturmfels conjecture and homogeneous varieties
Abstract: Given an n x n matrix one can form a vector (of length 2^n) of the principal minors of that matrix. What are relationships between the components of this vector? Given a 2^n vector, can it be realized as the principal minors of some matrix? These, and other questions, arise from matrix theory and probability.

Our approach is to provide geometric information about the set of all principal minors as an algebraic variety. For example, can one determine a complete (minimal) set of defining equations for this variety? As a motivating example, I consider the case of principal minors of skew-symmetric matrices. In this case the variety has already been identified by the work of Landsberg and Manivel. Next, I look at the case of principal minors of symmetric matrices. I describe the variety of principal minors of symmetric matrices, both as a rational map from a projective space, and as the linear projection of a homogeneous variety. Additionally (in the case of symmetric matrices), I identify a group G and prove that it acts invariantly on our variety.

Finally, I will describe the Holtz-Sturmfels conjecture in the case of symmetric matrices.

Date: Apr 13
Speaker: Aaron Bergman (TAMU--Physics)
Title: Moduli spaces and the gauge geometry correspondence.
Date: Apr 20
Speaker: M. Duchin (UC Davis)
Title: Divergence in Teichmuller space and the mapping class group.
Abstract: The rate of divergence of geodesic rays is one of many Gromovian ways to get a handle on curvature. In many settings, there is a gap between linear and exponential rates of divergence. We consider two examples naturally occuring in the study of surface geometry, Teichmuller space (with the Teichmuller metric) and the mapping class group. We show that these spaces have intermediate divergence -- in fact quadratic -- by exploiting the product region structure in each case. This is joint work with Kasra Rafi.
Date: Apr 24 [Tues]
Speaker: I. Zelenko (SISSA)
Title: Differential geometry of curves in Lagrange Grassmannians with given Young diagram
Abstract: Curves in Lagrange Grassmannian appear naturally in the study of geometric structures on a manifold (submanifolds of its tangent bundle). One can consider the time-optimal problem on the set of curves tangent to a geometric structures. Extremals of this optimal problem are integral curves of certain Hamiltonian vector field in the cotangent bundle. The dynamics of the fibers of the cotangent bundle along an extremal w.r.t. to the corresponding Hamiltonian flow is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces an invariant of the original geometric structure.

The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its $k$th column is equal to the rank of the k-th derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical such as sub-Riemannian or sub-Finslerian structures, defined on nonholonomic distributions.

Date: Apr 27
Speaker: E. Carberry (Duke/Sydney)
Title: Bubble, bubble toil and trouble: constant mean curvature surfaces and spectral curves
Abstract: A number of classical integrable systems, for example harmonic maps of the plane to a compact Lie group or symmetric space, can be transformed into a LINEAR flow on a complex torus. This torus is the Jacobian of an algebraic curve, called the spectral curve. I will discuss how this works in a simple example, namely the Gauss map of a constant mean curvature torus (ie a toroidal soap bubble). I will also define a generalisation of a transform of Darboux that is natural in a quaternionic setting, and explain how in this case the spectral curve is essentially (but not quite) the set of such transforms.
Date: May 3 [Thu, Milner 317]
Speaker: D. Knopf (UT Austin)
Title: Local singularities of Ricci flow.
Abstract: In applications of Ricci flow, one evolves a Riemannian metric on a manifold to improve its geometry. This evolution frequently develops singularities, which force changes in topology. The most interesting are local singularities, in which the metric remains regular on an open subset of the manifold. In these cases, an adequate understanding of the geometry in an appropriate space-time neighborhood of the developing singularity reveals how one should modify the manifold by topological-geometric surgeries. I will describe some examples of local singularity formation and show how one can derive precise asymptotic expansions at the most common local singularity, the neckpinch.
Date: May 22
Speaker: Andreas Cap (Vienna)
Title: Parabolic geometries
Date: June 4-8
Speaker: J.M. Landsberg & Jason Morton (TAMU & Berkeley)
Title: Representation Theory Workshop.
Date: June 19
Speaker: J.M. Landsberg (TAMU)
Title: Rigidity, flexibility and Lie algebra cohomology.