# Spring Term 2011

Gabriele La Nave (Yeshiva and UIUC). Finite time singularities of the Kahler-Ricci flow: surgeries and symplectic quotients. Friday January 14th. Huxley 130, 1.30-2.30pm.

Just as in Perelman’s proof of the Geometrization conjecture, one of the main stumbling blocks in the study of the Kahler-Ricci flow on Kahler manifolds is the formation of finite time singularities. In a recent paper with Gang Tian, we suggested an approach for the study of such singularities which consists in interpreting the Kahler-Ricci flow as arising via a symplectic quotient of an elliptic equation of soliton type –which we call the V-soliton equation– on a higher dimensional manifold (constructed from the original one) endowed  with a complex Hamiltonian C^*-action. The original Kahler manifold with initial metric is but one of the symplectic quotients and the flow is described (after reparametrization and up to suitable diffeomorphisms) in terms of the variation of the moment map parameter. The necessary surgeries are then described in terms of variations of symplectic quotients. We will briefly summarize this construction, draw some connection with the Minimal Model Program and discuss the Analyses involved in solving the V-soliton equation and its associated equation of Monge-Ampere type.

Daniel Huybrechts (Bonn). Global Torelli theorem for hyperkaehler manifolds (after Verbitsky). Friday January 21st. Huxley 130, 1.30-2.30pm.

Compact hyperkaehler manifolds are the natural generalization of K3 surfaces. But it has been known for quite some time that the celebrated Global Torelli theorem for K3 surfaces (due to Piatetski-Shapiro/Shafarevich and Burns/Rapoport) does not hold in higher dimensions. In this talk I explain the approach of Verbitsky to prove one half of the Global Torelli theorem.

Yuguang Zhang (IHES & Beijing). Continuity of  Extremal Transitions  and Flops for Calabi-Yau manifolds. Friday January 28th. Huxley 130, 1.30-2.30pm.

In this talk, we present a proof of a weaker version of the Candelas-de la Ossa conjecture, i.e.,  extremal transitions and flops for Ricci-flat Calabi-Yau  manifolds are  continuous in the Gromov-Hausdorff sense.

Rafe Mazzeo (Stanford). Ricci flow on conic surfaces. Friday February 4th. Huxley 130, 1.30-2.30pm.
I will discuss some new results concerning existence, nonuniqueness and convergence for the Ricci flow on Riemann surfaces with conic singularities. The main technical result is a refined regularity theorem for the linearized heat equation. This is joint work with Yanir Rubinstein and Natasa Sesum.

Dario Martelli (King’s College). Non-Kahler geometries in String Theory. Friday February 11th. Huxley 130, 1.30-2.30pm.

After briefly reviewing the notion of geometries characterised by SU(3) structures, I will discuss a particular class known (in String Theory) as “non-Kahler”. I will explain how this arises both in Type II and Heterotic Supergravities. Two explicit constructions of these geometries will be discussed. One is a one-parameter solution corresponding to fivebranes wrapped on the S^2 of the resolved conifold, that can be thought of as a non-Kahler analog of the conifold. The other is a general construction of one-parameter non-Kahler deformations of Calabi-Yau manifolds with a U(1) isometry, where the non-Abelian Yang-Mills field of the Heterotic is non-trivial.

Angelo Vistoli (SNS Pisa)Essential dimension of homogeneous forms. Friday February 18th. Huxley 130, 1.30-2.30pm.

Essential dimension is a concept that has emerged in the last 15 years and has attracted a lot of attention since then. The essential dimension of an algebraic or algebro-geometric object (e.g., of an algebra, a quadratic form, or an algebraic curve) is the minimal  number of independent parameters required to define the underlying  structure. In many cases computing the essential dimension is a delicate question, linked with long-standing open problems. I will survey the basic concepts, give some examples, and present recent results due to Reichstein and myself on essential dimension of homogeneous forms.

Costante Bellettini (ETH Zurich). Calibrated Integral Currents in Geometry. Friday February 25th. Huxley 130, 1.20-2.20pm.

Calibrated integral currents are a particular class of mass-minimizers. In addition to providing interesting concrete examples of solutions to Plateau’s problem, they naturally appear when dealing with several geometric questions. After a brief introduction to rectifiable integral currents and calibrations, we will overview some geometric problems where the regularity of calibrated currents plays a key role.

Ian McIntosh (York). Representations of surface groups in SU(2,1) via minimal Lagrangian surfaces in the complex hyperbolic plane. Friday March 4th. Huxley 130, 1.20-2.20pm.

I will describe how an “open ball” worth of representations of a surface group (fundamental group of a compact surface, genus at least 2) into the isometry group of the complex hyperbolic plane can be found by the solving the equations governing minimal Lagrangian surfaces. To be precise, each representation is obtained from a minimal Lagrangian embedding of the Poincare disc, as universal cover, which is equivariant with respect to that representation. The geometric picture is that all these representations are deformations of the representation into SO(2,1) corresponding to a choice of hyperbolic metric on the surface. The construction is based on proving the existing of solutions to the Gauss-Codazzi equations for such equivariant embeddings. These equations are the complex hyperbolic cousins of the Tzitzeica equations (elliptic and quasi-linear). This is joint work with John Loftin at Rutgers-Newark.

Duco van Straten (Mainz). Local systems of Calabi-Yau type, monodromy, and Frobenius. Friday March 11th. Huxley 130, 1.20-2.20pm.

We give an overview of the ongoing hunt of differential operators of Calabi-Yau type and their remarkable properties.

Valery Alexeev (Georgia). On the three compactifications of Siegel space. Friday March 18th. Huxley 130. 1.20-2.20pm.

The moduli space $A_g$ of abelian varieties has three classical toroidal compactifications: (1) perfect, (2) 2nd Voronoi, and (3) Igusa blowup, each with its own distinct geometric meaning. It is an interesting problem to understand exactly how these compactifications are related. I will show that (1) and (2) are isomorphic in a neighborhood of the image of a regular map from the Deligne-Mumford’s moduli space $\overline{M}_g$, and that the rational map $\overline{M}_g \dashrightarrow \overline{A}_g$ for (3) is not regular for $g>8$. This is a joint work with Adrian Brunyate.

Tamas Hausel (Oxford). Arithmetic and physics of Higgs moduli spaces. Friday March 25th. Huxley 130, 1.20-2.20pm, and
Kevin Costello (Northwestern).
Some remarks on the elliptic genus. Friday March 25th. Huxley **343**, 3-4pm.

Hausel’s abstract: In this talk we discuss the connection between conjectures with Villegas on mixed Hodge polynomials of character varieties of Riemann surfaces achieved by arithmetic means and conjectures on the cohomology of Higgs moduli spaces derived by physicists Chuang-Diaconescu-Pan.

Costello’s abstract: I’ll explain how one can view the (Witten) elliptic genus of a complex manifold X from a rigorous analysis of a 2d quantum field theory of maps from an elliptic E to X.