Geometry Seminar
Fall 2007
Fridays at 4 pm in Milner 216

Format: Talks are 50-60 minutes, with the option to continue after a short break.
* List below includes Department Colloquia and Frontiers Lectures with geometric content.

* This semester's informal meetings with the string theorists in the physics department are morphing into a joint Geometry and String Theory seminar. This semester's erratic schedule will coalesce into a ~once monthly meeting in the Spring.

August 31
M. Becker (TAMU, Physics)
Moduli stabilization in string theory.
Note: K. Becker filled-in for M. Becker.

September 3 (3 o'clock in Milner 317)
H. Derksen (University of Michigan)
Department Colloquium -- Mutations of Quivers
Abstract: Consider a quiver (= directed graph) without loops or oriented 2-cycles. For every vertex, we can define a mutation which transforms the quiver into another one. This combinatorial object appears in various areas of mathematics, for example, the cluster algebras introduced by Fomin and Zelevinsky, triangulations of marked oriented surfaces as in recent work by Fomin-Shapiro-Thurston and even in string theory. In joint work of Jerzy Weyman, Andrei Zelevinsky and the speaker, a generalizion of the classical Bernstein-Gelfand-Ponomarev reflection functors of quiver representations leads to quiver mutations as well. The various areas discussed in this lecture are likely to be connected at an even deeper level.

September 4 (Tuesday)
H. Derksen (University of Michigan)
A counterexample to Okounkov's log-concavity conjecture (joint work with Calin Chindris and Jerzy Weyman)
Abstract: The tensor product multiplicities for representations of the general linear group are given by Littlewood-Richardson coefficients. I will discuss several solved and unsolved conjectures about Littlewood-Richardson coefficients. I will discuss how the theory of quiver representations can be used to generalize, prove or disprove these conjectures. In particular we give a counterexample to Okounkov's conjecture that the Littlewood-Richardson coefficient is a log-concave function on the partitions.

September 7
K. Becker (TAMU, Physics)
Moduli stabilization in string theory, part 2.
Abstract: in this lecture I will discuss how in the context of Calabi-Yau compactifications of string theory to four dimensions moduli stabilization can be implemented in the two dimensional quantum field theory describing the string world-sheet. In particular I will be reviewing how the cohomology classes of the Calabi-Yau manifold are in one to one correspondence with the supersymmetric ground states of the non-linear sigma model.

September 14
B. Hassett (Rice University)
Ample divisors on the moduli space of stable pointed rational curves and its contractions
Abstract: This is a report on thesis work of Matthew Simpson. Consider the moduli space of n-pointed stable curves of genus zero. By the Kleiman criterion, the ample cone of this space is determined by its cone of curves, i.e., by the possible topological invariants of algebraic families of pointed stable rational curves over projective curves. Fulton has conjectured that the cone of curves is generated by one-dimensional boundary strata of the moduli space, which can be enumerated combinatorially.

We show this has beautiful and testable implications for the birational geometry of the moduli space: Log canonical models of the moduli space (taken with respect to multiples of the boundary divisor) are isomorphic to moduli spaces of weighted pointed stable curves. Construction techniques from Geometric Invariant Theory allow us to prove this in a number of specific cases.

September 21
September 28
October 5
October 12
S. Ji (University of Houston)
On proper holomorphic mappings between balls.
Abstract: I will give a survey of the topic.

October 15, 16 & 18 (Mon, Tue & Thu), 4 o'clock in Blocker 120.
N. Mok (University of Hong Kong)
Frontiers Lecture Series
I: From bounded symmetric domains to their compact duals -- rigidity by means of rational curves. (Abstract)
II: Geometric structures on uniruled projective manifolds: varieties of minimal rational tangents. (Abstract)
III: Rigidity of rational homogeneous spaces of Picard number 1 under Kaehler deformation -- from case studies to general principles. (Abstract)

October 19--21
Texas Geometry and Topology Conference at TAMU

October 22, 24 & 25 (Mon, Wed & Thu), 4 o'clock in Blocker 120.
J.-P. Demailly (l'Université de Grenoble I)
Frontiers Lecture Series
Jet bundles, differential equations and hyperbolic algebraic varieties. (Abstract.)

October 23 (Tuesday)
M. Eastwood (University of Adelaide)
The geodesics of a metric connection.
Abstract: How can one tell what are the geodesics of a Riemannian metric? More precisely, suppose a torsion-free connection is given and we ask whether there is a Riemannian metric whose geodesics, regarded as unparameterised curves, coincide with those of the given connection. This problem gives rise to a certain closed system of partial differential equations and hence to obstructions to finding such a metric. In two dimensions, the primary obstruction is an invariantly defined scalar. This is recent and current joint work with Robert Bryant, Maciej Dunajski, and Vladimir Matveev.

November 1 (Thursday in Milner 317)
S. Kumar (University of North Carolina)
Deparment colloquium -- Hermitian eigenvalue problem and its generalization to any semisimple group: A survey.
Abstract: The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A+B of two Hermitian matrices A and B, provided we fix the eigenvalues of A and B. A systematic study of this problem was initiated by H. Weyl (1912). By virtue of contributions from a long list of mathematicians, notably Weyl (1912), Horn (1962), Klyachko (1998) and Knutson-Tao (1999), the problem is finally settled. The solution asserts that the eigenvalues of A+B are given in terms of certain system of linear inequalities in the eigenvalues of A and B. These inequalities are given explicitly in terms of certain triples of Schubert classes in the singular cohomology of Grassmannians and the standard cup product. The Hermitian eigenvalue problem has been extended by Berenstein-Sjamaar (2000) and Kapovich-Leeb-Millson (2005) for any semisimple complex algebraic group G. Their solution is again in terms of a system of linear inequalities obtained from certain triples of Schubert classes in the singular cohomology of the partial flag varieties G/P (P being a maximal parabolic subgroup) and the standard cup product. However, their solution is far from being optimal. In a joint work with P. Belkale, we have given an optimal solution of the problem for any G. We define a deformation of the cup product in the cohomology of G/P and use this new product to generate a certain system of inequalities which solves the problem for any G optimally. The talk should be accessible to general mathematical audience.

November 2
S. Kumar (University of North Carolina)
Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups.
Abstract: This is a joint work with P. Belkale. We consider the eigenvalue problem, intersection theory of homogenous spaces (in particular, the Horn problem) and the saturation problem for the symplectic and odd orthogonal groups. The classical embeddings of these groups in the special linear groups play an important role. We deduce properties for these classical groups from the known properties for the special linear groups. The tangent space techniques play a crucial role. Another crucial ingredient is the relationship between the intersection theory of the homogeneous spaces for Sp(2n) and SO(2n+1). We solve the modified Horn and saturation problems for these classical groups.

November 6 (Tuesday) at 3 o'clock in Milner 317
J.-Y. Cai (U. Wisconsin & Radcliffe Inst., Harvard U., Comp. Sci.)
Developments in Holographic Algorithms
Abstract: Valiant's theory of holographic algorithms is a new design method to produce polynomial time algorithms. Information is represented in a superposition of linear vectors in a holographic mix. This mixture creates the possibility for exponential sized cancellations of fragments of local computations. The underlying computation is done by invoking the Fisher-Kasteleyn-Temperley method for counting perfect matchings for planar graphs, which uses Pfaffians and runs in polynomial time. In this way some seemingly exponential time computations can be done in polynomial time, and some minor variations of the problems are known to be NP-hard or #P-hard. Holographic algorithms challenge our conception of what polynomial time computations can do, in view of the P vs. NP question.

In this talk we will survey some new developments in holographic algorithms.

November 9
D. Fox (University of California, Irving)
Cayley cones, pseudoholomorphic curves, and minimal surfaces in the six sphere.

November 16

November 19 (Monday) at 10 o'clock in ENPH 501
Geometry and String Theory seminar (Joint w/ Physics Dept.)
Michael Douglas (Rutgers)
Overview of theory of boundary states
Abstract: Boundary states in two-dimensional conformal field theory have important applications in string theory (Dirichlet branes) and in condensed matter physics (quantum junctions, edge states in the quantum hall effect). We survey recent developments in this area and discuss the prospects for developing a general theory of boundary states in c>1 theories.

November 23
November 30
Geometry and String Theory seminar (Joint w/ Physics Dept.)
E. Sharpe (Virginia Tech.)
Recent results in heterotic compactifications.