Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Speaker: 

Yi Wang

Institution: 

Johns Hopkins University

Time: 

Tuesday, October 9, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

306 Rowland Hall

We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when $2k<n+1$. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen’s example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.

Bershadsky--Cecotti--Ooguri--Vafa torsion in Landau--Ginzburg models

Speaker: 

Guangbo Xu

Institution: 

SUNY Stony Brook

Time: 

Tuesday, October 9, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

306 Rowland Hall

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.

The Morse index of Free Boundary Minimal Hypersurfaces

Speaker: 

Hung Tran

Institution: 

Texas Tech University

Time: 

Tuesday, May 22, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

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.

ALE Kahler Manifolds

Speaker: 

Rares Rasdeaconu

Institution: 

Vanderbilt

Time: 

Tuesday, June 5, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

In recent years the scalar flat asymptotically locally Euclidean (ALE) Kahler manifolds attracted a lot of attention, and many examples were constructed. However, their classification is not understood, except for the case of ALE Ricci flat Kahler surfaces. In this talk, I will present a first step in this direction: the underlying complex structure of ALE Kahler manifolds is exposed to be a resolution of a deformation of an isolated quotient singularity. The talk is based on a joint work with Hans-Joachim Hein and Ioana Suvaina.

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