In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa the genus one string partition function in the B-model is identified with certain analytic torsion of the Hodge Laplacian on a K\"ahler manifold. In a joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the analogous torsion in Landau--Ginzburg models. I will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the Schr\"odinger operator. I will also explain the rigorous definition of the BCOV torsion for homogeneous polynomials on ${\mathbb C}^N$. Lastly I will explain the conjecture stating that in the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly equation for marginal deformations.

A free boundary minimal hypersurface in the unit Euclidean ball is a critical point of the area functional among all hypersurfaces with boundaries in the unit sphere, the boundary of the ball. While regularity and existence aspects of this subjecct have been extensively investigated, little is known about uniqueness. That motivates the study of the Morse index, which quantitatively measures the number of deformations decreasing the area to second order. Henceforth, A. Fraser and R. Schoen proposed a fundamental conjecture concerning surfaces with low indices. In this talk, we discuss recent developments including a joint work with Ari Stern, Detang Zhou, and Graham Smith.

By a gluing construction, we produce steady Kahler-Ricci solitons on equivariant crepant resolutions of C^n/G, where G is a finite subgroup of SU(n), generalizing Cao's construction of such a soliton on a resolution of C^n/Z_n.  This is joint work with Olivier Biquard.

Lecture 1

Speaker: Otis Chodosh

Time/place: Surge 284 3:40~4:30

Title:Properties of Allen--Cahn min-max constructions on 3-manifolds

Abstract:

I will describe recent joint work with C. Mantoulidis in which we study the properties of bounded Morse index solutions to the Allen--Cahn equation on 3-manifolds. One consequence of our work is that a generic Riemannian 3-manifold contains an embedded minimal surface with Morse index p, for each positive integer p.

Lecture 2

Speaker:  Ved Datar

Time/place: Surge 284 4:40~5:30

Title: Hermitian-Yang-Mills connections on collapsing K3 surfaces

Abstract:

Let $X$ be an elliptically fibered K3 surface with a fixed $SU(n)$ bundle $\mathcal{E}$. I will talk about degenerations of connections on $\mathcal{E}$ that are Hermitian-Yang-Mills with respect to a collapsing family of Ricci flat metrics. This can be thought of as a vector bundle analog of the degeneration of Ricci flat metrics studied by Gross-Wilson and Gross-Tosatti-Zhang. I will show that under some mild conditions on the bundle, the restriction of the connections to a generic elliptic fiber converges to a flat connection. I will also talk about some ongoing work on strengthening this result. This is based on joint work with Adam Jacob and Yuguang Zhang.

Speaker #1: Renato Bettiol (University of Pennsylvania) 

Title:  A Weitzenbock viewpoint on sectional curvature and application

Abstract:  In this talk, I will describe a new algebraic characterization of sectional curvature bounds that only involves curvature terms in the Weitzenboeck formulae for symmetric tensors. This characterization is further clarified by means of a symmetric analogue of the Kulkarni-Nomizu product, which renders it computationally amenable. Furthermore, a related application of the Bochner technique to closed 4-manifolds with indefinite intersection form and positive or nonnegative sectional curvature will be discussed, yielding some new nsight about the Hopf Conjecture. This is based on joint work with R. Mendes (Univ. Koln, Germany).

Speaker #2: Or Hershkovitzs (Stanford University)

Title: The topology of self-shrinkers and sharp entropy bounds

Abstract: The Gaussian entropy, introduced by Colding and Minicozzi, is a rigid motion and scaling invariant functional which measures the complexity of hypersurfaces of the Euclidean space. It is defined to be the supremal Gaussian area of all dilations and translations of the hypeprsurface, and as such, is well adapted to be studied by mean curvature flow. In the case of the n-th sphere in Rn+1, the entropy can be computed explicitly, and is decreasing as a function of the dimension n. A few years ago, Colding Ilmanen Minicozzi and White proved that all closed, smooth self-shrinking solutions of the MCF have larger entropy than the entropy of the n-th sphere. In this talk, I will describe a generalization of this result, which derives better (sharp) entropy bounds under topological constraints. More precisely, we show that if M is any closed self-shrinker in Rn+1 with a non-vanishing k-th homotopy group (with k less than or equal to n), then its entropy is higher than the entropy of the k-th sphere in Rk+1. This is a joint work with Brian White.