Abstracts and Materials

Yamabe Symposium: October 5 - 7, 2012

Geometry of Gradient Ricci Solitons, Huai-dong Cao (Lehigh University)

Abstract: The concept of Ricci solitons was introduced by Richard Hamilton in mid 1980's to study singularity formations in the Ricci flow. Ricci solitons are natural extensions of Einstein manifolds. They are also self-similar solutions to the Ricic flow and often arise as singularity models. In this talk, I shall describe some recent progress on the geometry and classifications of Ricci solitons, especially my joint work with Qiang Chen on Bach-flat shrinking Ricci solitons.

Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Jean-Pierre Demailly (University of Grenoble)

Abstract: The goal of the talk will be to investigate the existence and properties of entire curves drawn in a complex projective variety. More specifically, we show that all such curves must satisfy certain global differential equations as soon as the variety is of general type. This is strongly related to the Green-Griffiths-Lang conjecture which asserts the algebraic degeneracy of entire curves, and also to some fundamental questions of arithmetic geometry (higher dimensional generalaizations of the Mordell-Faltings theorem). Our strategy relies on a use of holomorphic Morse inequalities and on a probabilistic estimate of the cohomology of jet bundles.

Minimal volume metrics, entropy, and degree, Benson Farb (University of Chicago)

Abstract: In this talk I will explain some major open problems, and some progress, on the following topics:

  • Finding (suitably normalized) metrics that minimize volume and/or entropy on a fixed manifold.
  • What is the relationship between closed manifolds M,N admitting a degree d>0 map f:M--->N, and the volumes Vol(M), Vol(N)?
Some Homology and Cohomology Theories for a Metric Space, Robert Hardt (Rice University)

Abstract: Various classes of chains and cochains may reveal geometric as well as topological properties of metric spaces. In 1957, Whitney introduced a geometric "flat norm" on polyhedral chains in Euclidean space, completed to get flat chains, and defined flat cochains as the dual space. Federer and Fleming also considered these in the sixties and seventies, for homology and cohomology of Euclidean Lipschitz neighborhood retracts. These include smooth manifolds and polyhedra, but not algebraic varieties or subspaces of some Banach spaces. In work with Thierry De Pauw and Washek Pfeffer, we find generalizations and alternate topologies for general metric spaces. With these, we homologically characterize Lipschitz path connectedness and obtain several facts about spaces that satisfy local linear isoperimetric inequalities.

Polyhedral complexes and topology of complex-projective varieties, Misha Kapovic (University of California, Davis)

Abstract: I will explain how to use polyhedral complexes of constant curvature and hyperbolic orbifolds in order to construct some interesting complex-projective varieties. Among other things, I will show how to construct projective varieties with normal crossing singularities and with prescribed finitely-presented fundamental groups.
One of the key tools of the construction is the geometry of 3-dimensional hyperbolic orbifolds related to the Jorgensen-Marden conjecture on cut-locus of such orbifolds.

Instantons in G2 geometry, Conan Leung (Chinese University of Hong Kong)

Abstract: M-theory on G2 manifolds is an analog of string theory on symplectic manifolds. The role of holomorphic curves with Lagrangian boundary conditions is replaced by associative submanifolds with coassociative boundary conditions. The work of Fukaya-Oh related holomorphic disks in cotangent bundles with Morse flow lines in Lagrangian submanifolds. Wang, Zhu and I generalized this to the G2 setting, namely thin associative submanifolds can be constructed from regular holomorphic curves in coassiciative submanifolds. This can be used to construct new examples of associative submanifolds.

Ancient solutions and singular solutions to the Yamabe flow, Natasa Sesum (Rutgers University)

Abstract: In the first part of the talk I will discuss a way of constructing new solutions to the Yamabe flow that look like towers of bubbles, that is, spheres joined by short necks. This is a joint work with P. Daskalopoulos. In the second part of the talk I will mention the results on the Ricci flow (same as the Yamabe flow in two dimensions) on conic surfaces. In the joint work with Mazzeo and Rubenstein we construct a solution to the Ricci flow o conic surfaces that preserves the conic angles. Long time behavior of the flow is related to the Uniformization theorem.

Parabolic flows in complex geometry, Ben Weinkove (UCSD/Northwestern)

Abstract: Parabolic flows are powerful tools in the study of geometric structures on manifolds. In this talk I will discuss some work (joint with Jian Song) on the behavior of the Ricci flow on Kahler manifolds. In particular, we analyze the singularities that form in complex dimension two and show how the flow can be continued through the singularities. I will also talk about some joint work with Valentino Tosatti on another parabolic flow, called the Chern-Ricci flow. This flow was first introduced by Matt Gill, and is a natural flow to consider on more general complex manifolds. I will discuss the behavior of this flow on complex surfaces, and give a number of examples.