Geometry seminar          Fall 2003

Tuesday, September 9
Speaker: David Glickenstein, Department of Mathematics, University of Arizona
Title: An Introduction to Metric Geometry and Gromov-Hausdorff Distance
Abstract:  We will present some elementary definitions of Gromov-Hausdorff distance in order to compare metric spaces and better understand the space of Riemannian manifolds as metric spaces. Some results in the field will be discussed. We will also look at some simple, yet interesting, examples of (Riemannian) metric transformations which arise from these ideas. Our aim is to eventually use metric geometry to describe sequences of solutions of the Ricci flow which collapse, but this is for a future talk. This talk should require very little background knowledge.

Tuesday, September 16
Speaker: David Glickenstein, Department of Mathematics, University of Arizona
Title: Compactness Theorems and the Ricci Flow
Abstract:  The Ricci flow is a partial differential equation on Riemannian metrics which is used by Richard Hamilton and more recently by Grisha Perelman and others to classify manifolds both topologically and metrically. We will survey the role of compactness theorems in the study of the Ricci flow on Riemannian manifolds. Then we shall look at the compactness theorems themselves and their proofs. No background knowledge of the Ricci flow will be assumed.

Tuesday, September 23
Speaker: David Glickenstein, Department of Mathematics, University of Arizona
Title: Compactness Theorems and the Ricci Flow II

Tuesday, September 30
Speaker: Yi Hu, Department of Mathematics, University of Arizona
Title:  Topological Aspects of Chow Quotients. I
Abstract:  Quotients of schemes by reductive algebraic groups arise naturally in many situations. The existence of many moduli spaces, for example, is proved by expressing them as quotients. There are several quotient theories, among them, the Mumford geometric invariant theory is a systematic one. It has become well known now that, for a reductive algebraic group action on a smooth projective variety, Mumford's quotients depend, in a flip-flop fashion, onchoices of linearized line bundles (due to Dolgachev-Hu and Thaddeus). Nevertheless, it is a drawback that none of Mumford's quotients is canonical, in general. Besides this, the closed orbits parameterized by a GIT quotient almost always carry very different topological invariants, and this, among other reasons, oftentimes makes GIT quotients misbehaved compactifications. This is rather unsatisfactory from the viewpoint of a geometric moduli problem, where the moduli space almost always parameterizes geometric objects of same topological type, and awkward to use for purpose of some geometric computations. To overcome these drawbacks, we are led to consider a canonical quotient, the Chow quotient.  In the first out of two talks, I will give an introductory lecture on basics of GIT quotients, Symplectic Reductions, and Chow quotients.

Tuesday, October 7
Speaker: Yi Hu, Department of Mathematics, University of Arizona
Title:  Topological Aspects of Chow Quotients. II
Abstract:  I will talk on some of my recent work on Chow quotients. The main themes are:  over the field of complex numbers, the Chow quotient admits  symplectic and other topological interpretations, namely, symplectically, the moduli spaces of stable orbits with prescribed momentum charges; and topologically, the moduli space of stable action-manifolds. In addtion, we give a computable characterization of the Chow cycles of the Chow quotient, using the so-called perturbing-translating-specializing relation.

Tuesday, October 14
Speaker:  James McKernan, University of California at Santa Babara
Title:  Boundedness of Log Fano pairs of boundex index
Abstract:  It is a classic result of Kollár, Miyaoka and Mori, that the family of all smooth Fano varieties is bounded. Batyrev conjectured that the same holds if one drops the hypothesis on smoothness and adds the hypothesis that some fixed multiple of the canonical divisor is Cartier and that the singularities are log terminal. We prove Batyrev's conjecture.
It suffices to find a bound on the degree of the anticanonical. The classic proof proceeds in two steps. The first is to find an element of the linear system of some high multiple of the anticanonical and exhibit an element of this linear system which is very singular at any given point. This is the method of Fano. The second, the hardest step, is to exhibit a rational curve, through two general points, of low degree. Unfortunately it seems hard to generalise this idea to the singular case, since it is hard to compute intersection multiplicities on singular varieties. Instead we produce covering families of low degree subvarieties, which automatically have large intersection multiplicities with elements of the pluri anti canonical system.

Tuesday, October 21
Speaker: Doug Pickrell, University of Arizona
Title:   Topology and Geometry of Loop Spaces
Abstract:   This is intended to be a relatively elementary and informal introduction to the topoogy and geometry of the space of loops, i.e. closed strings, in a symmetric space, such as a sphere. This talk is definitely not recommended for experts.

Tuesday, October 28
Speaker:  Philip Foth, university of Arizona
Title:   Introduction to symmetric spaces and buildings
Abstract:   This is an introductory talk, where I present some of the background material for the upcoming Colloquium talk by M. Kapovich.  The talk is primarily aimed at graduate students.

Tuesday, November 11  
Veterans Day, no seminar

Tuesday, November 18
Speaker:  Ludmil Katzarkov, University of California at Irvine
Title: HMS for Fano's and beyond
Abstract: In this talk we outline a proof of Homological Mirror Symmetry for some surfaces. We discuss some possible topological applications at the end.

Tuesday, November 25
Thanksgiving break, no seminar