GEOMETRY SEMINAR     SPRING 2005

Tuesday, January 18

Paul Bressler, Department of Mathematics, The University of Arizona, will speak on “A gentle introduction to the Borel-Weil theory” at 4:00 PM in Math 402.

In this (and the subsequent, should it take place) talk I will give a gentle introduction to some of the basic ideas of geometric representation theory. The talk should be accessible to anyone possessing rudimentary notions of (linear) algebra, calculus and geometry. I will focus on the geometric approach to the finite dimensional representation theory of the group SL(2,C) of invertible two-by-two matrices with complex entries having determinant equal to one. The goal is to illustrate the Borel-Weil theory (a method of constuctig representations) by explicit calculations in this example. Time permitting, I will go on to treat in analogous fashion a class of (infinite dimensional) representations of the Lie algebra sl(2,C) of two-by-two matrices of trace zero.


Tuesday, January 25

Paul Bressler, Department of Mathematics, The University of Arizona, will speak on “A gentle introduction to the Borel-Weil theory, part II” at 4:00 PM in Math 402.


Tuesday, February 1

Paul Bressler, Department of Mathematics, The University of Arizona, will speak on “Highest Weight Modules” at 4:00 PM in Math 402.

I will, using mainly the example of sl(2), give a rapid introduction to the highest weight (infinite dimensional) representation theory (of a reductive Lie algebra) and the corresponding geometric representation theory (which describes this representation theory in terms of D-modules [= systems of linear PDE with rational coefficients] on the flag variety). In the case of sl(2) this quickly boils down to the classical theory of linear ODE (D-modules on the projective line discussed in the previous talks). The talk will make use (after a very brief recollection) of the universal enveloping algebra of a Lie algebra and the ring of differential operators.


Tuesday, February 8

Doug Pickrell, Department of Mathematics, The University of Arizona, will speak on “Introduction to SL(2,R) and its unitary representations” at 4:00 PM in Math 402.

I intend this talk to be an elementary (as possible) introduction to the structure of (the universal covering of) SL(2,R) and its unitary representations. In contrast to the case of a compact Lie group, a noncompact Lie group such as SL(2,R) has different types of elements (elliptic, parabolic, and hyperbolic), and different types of representations (highest and lowest weight type, principal type and so on). These were classified by Bargmann (a physicist) half a century ago (because of their relevance to quantum mechanics), but (at least for me) puzzles remain. In a possible sequel to this talk, I may pursue some structural analogies which exist between SL(2,R) and the symmetry group of a string, Diff(S^1).


Thursday, February 10

Chenchang Zhu, Department of Mathematics, ETH Zurich, will speak on “Symplectic (stacky) groupoids of Poisson manifolds” at 2:00 PM in Math 401N. (Please note unusual place and time.)

A symplectic groupoid is a symplectic manifold with a submersive Poisson morphism onto a Poisson manifold. It was designed with the purpose to relate quantization of symplectic manifolds and Poisson manifolds. But not every Poisson manifold can be associated with such a symplectic groupoid. In this talk, we will first explain Weinstein's plan to associate Poisson manifolds symplectic groupoids, and then how to put a symplectic structure on a stack and how to solve the above problem after we enter the world of stacks (joint with H. Tseng). We will attack stacks from the viewpoint of groupoids in this talk. All the necessary definitions will be given and introductory examples explained.



Tuesday, February 15

Tara Holm, Department of Mathematics, UC Berkeley, will speak on “Morse Theory in Real Symplectic Geometry” at 4pm in Math 402.

Much as real algebraic geometry is the study of the real points of complex varieties, real symplectic geometry examines the real loci of symplectic manifolds. While arbitrary submanifolds of a manifold can be fiendishly complicated, real loci display a surprising and beautiful structure. The relationship between the topology of the real locus and the topology of the ambient symplectic manifold can be best understood using Morse theory. In this lecture, I will first review the basics of Morse theory in the concrete setting of the height function on the torus. I will then explain several facts from symplectic geometry and examine their analogues in the context of real loci, giving examples along the way.


Tuesday, February 22

Doug Pickrell, Department of Mathematics, The University of Arizona, will speak on “More about SL(2,R)” at 4:00 PM in Math 402.

I will first back up to recall some things about hyperbolic geometry. I will then continue discussing the geometry of the unitary representations of SL(2,R). In whatever time remains, I will describe how this is applied to harmonic analysis in the hyperbolic plane.




Tuesday, March 1

Xiaofeng Sun, Department of Mathematics, Harvard University, will speak on “Good Metrics on the Moduli Space of Riemann Surfaces” at 4:00 PM in Math 402.

I will discuss my recent joint works with Prof. Kefeng Liu and Prof. Shing-Tung Yau.
We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces, and study their curvatures and boundary behaviors by using the singular perturbation techniques from partial differential equations. These new metrics have Poincare growth near the boundary of the moduli space and have bounded geometry. Furthermore, we showed the Weil–Petersson metric is good in the sense of Mumford.
Based on the detailed analysis of these new metrics, we obtain good understanding of all of the known classical complete Kahler metrics, in particular the Kahler–Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable in the sense of Mumford.
By studying the Monge–Ampere equation together with the Kahler–Ricci flow on complete non-compact manifolds, we derive Ck estimates directly without using the C0 estimate. Based on these analysis, we prove that the Kahler–Einstein metric has strongly bounded geometry.
Another corollary is a proof of the equivalences of all of the known classical complete metrics to these new metrics, in particular Yau's conjectures on the equivalences of the Kahler–Einstein metric to the Teichmuller and the Bergman metric.




Friday, March 11 [Cancelled]

Ravi Vakil, Department of Mathematics, Stanford University, will speak on “A geometric Littlewood–Richardson rule” at 1:30 PM in Math 402. (Please note unusual place and time.)

I will describe an explicit geometric Littlewood–Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood–Richardson rules, such as tableaux and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the Littlewood–Richardson rule. It has a host of geometric consequences, which I may describe, time permitting. The rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory, and in fact a Littlewood–Richardson rule in equivariant K-theory (ongoing work with Knutson). The rule suggests a natural approach to the open question of finding a Littlewood–Richardson rule for the flag variety, leading to a conjecture, shown to be true up to dimension 5. Finally, the rule suggests approaches to similar open problems, such as Littlewood–Richardson rules for the symplectic Grassmannian and two-step flag varieties.



Tuesday, March  29 [Cancelled]

John Millson, Department of Mathematics, The University of Maryland, will speak on “The toric geometry and algebra of ordered points on the projective line” at 4:00 PM in Math 402.

I will present joint work with Ben Howard and Andrew Snowden on the structure of the ring of projective invariants for n ordered points on the projective line. In 1894 A. Kempe proved that this ring is generated by the ring of lowest degree invariants. We will give a new proof of this theorem using a toric degeneration. We will extend this theorem to the case in which the points are weighted (this involves a simple trick — the side-splitting map). Our main theorem describes the relations in this ring. In case the number of points is odd then the ideal of relations is generated by the quadratic relations. In case the number of points is even then the lowest (even) degree invariants generate the ring of even degree invariants and the ideal of relations among these generators is generated by the quadratic relations. Again these results extend to the weighted case and again the proof depends on the study of a toric degeneration.




Tuesday, April 5

Arlo Caine, Department of Mathematics, The University of Arizona, will speak on “Determinants of Cauchy Riemann Operators over Riemann Surfaces” at 4:00 PM in Math 402.

The determinant of an operator acting in an infinite dimensional space is difficult to define. Even if the operator has only eigenvalues in its spectrum, the infinite product of these values need not converge. In this talk I will present Quillen's construction of the determinant function for Cauchy-Riemann operators acting in complex vector bundles over Riemann surfaces. We will carefully motivate and then state the problem which Quillen solved, outline his solution, and finish with some of the amazing details of the proof. I will give a preparatory talk on some of the background material in complex geometry and functional analysis in the Graduate geometry seminar, Monday 4/4 at 1:00pm in 402.



Tuesday, April 12

Arlo Caine, Department of Mathematics, The University of Arizona, will speak on “Determinants of Cauchy Riemann Operators over Riemann Surfaces Part II” at 4:00 PM in Math 402.

Last time I presented Quillen's construction of the Determinant line bundle over Fredholm operators and outlined how Quillen used this to give a construction of a determinant function for Cauchy Riemann operators. This time I will sketch some of the details of Quillen's solution and show how Quillen's contstruction gives geometric meaning to the zeta function determinant of Laplacians. I will give another preparatory talk on some of the background material in complex geometry and functional analysis in the Graduate geometry seminar, Monday 4/11 at 1:00pm in 402.



Tuesday, April 26

Michael Otto, Department of Mathematics, The University of Arizona, will speak on “Semisimple Lie groups and Poisson geometry” at 4:00 PM in Math 402. 

The Killing form of a complex semisimple Lie group G gives rise to a variety of interesting Poisson structures on manifolds related to G, e.g., on certain orbits in the Riemannian space G/K. One might then use results from symplectic geometry to analyze the Lie theoretic structure of G. We will discuss such a symplectic approach to van den Ban's convexity theorem for semisimple symmetric spaces. [Joint work with P. Foth]