# Seminars & Events for Special Lecture

##### On fractional analogues of k-Hessian operators

**Geometric Analysis Learning Seminar:** We consider k-Hessian operators(and when k=n, it is the Monge Amp\’{e}re operator) as convex envelopes of linear operators. And then we will do the fractional analogue of k-Hessian operators, and show the existence and semiconcavity of the solutions of the fractional k-Hessian equations, under a relatively simple framework of global solutions prescribing data at infinity and global barriers. In addition, the uniformly ellipticity of the fractional k-Hessian operators can be proved, thus the regularity results of nonlocal equations developed by L.Caffarelli and L.Silvestre can be applied to prove C^{2s+\alpha} regularity.

##### Moduli of Riemann surface and Bers conjecture

**THIS IS A SPECIAL ANALYSIS / GEOMETRY SEMINAR. **It was Keobe who first proved that closed Riemann surface can be uniformized by Schottky groups. However Marden (1974) showed that not every Schottky group is generated by geometric circle reflections in the complex plane, which is called "classical"(original definition by Schottky himself) Schottky group. Bers (1975) and Hejhal (1975) and Ahlfors made detailed studies on Schottky space of moduli space of Riemann surface. And Bers made the following conjecture: "Every closed Riemann surface can be uniformized by classical Schottky group." In this talk I will describe and present resolution of this conjecture based on two recent works. In fact, I will present the solution which actually answer a lot more to the original problem.

##### Moduli of Riemann surface and Bers conjecture

This is a continuation of the October 24 talk. It was Koebe who first proved that closed Riemann surface can be uniformized by Schottky groups. However Marden (1974) showed that not every Schottky group is generated by geometric circle reflections in the complex plane, which is called "classical"(original definition by Schottky himself) Schottky group. Bers (1975) and Hejhal (1975) and Ahlfors made detailed studies on Schottky space of moduli space of Riemann surface. And Bers made the following conjecture: "Every closed Riemann surface can be uniformized by classical Schottky group." In this talk I will describe and present resolution of this conjecture based on two recent works. In fact, I will present the solution which actually answer a lot more to the original problem.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Monotone Lagrangian tori in vector spaces and projective spaces

A basic open problem in symplectic topology is to classify Lagrangian submanifolds (up to, say, Hamiltonian isotopy) in a given symplectic manifold. In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (eg. vector spaces or complex projective spaces) contain many more Lagrangian tori than previously thought. We will present some of the recent developments on this problem, and discuss some of the connections (established and conjectural) between Lagrangian tori, cluster mutations, and toric degenerations, that arise out of this story.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Khovanov homology in characteristic two and involutive monopole Floer homology

There are various spectral sequences starting from the Khovanov homology of a link in S^3 (which is an algebraically defined quantum invariant) to various versions of Floer homology (which are invariants with a more analytical nature). In this talk, we discuss what are the implications in this context of the conjugation symmetry in Seiberg-Witten theory.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Kahler-Einstein metrics and normalized multiplicity

Recently Fujita showed that projective spaces have maximal volume among all K\"ahler-Einstein Fano varieties. I will discuss a refinement of Fujita's result in singular cases, and its connection to minimizing normalized multiplicity of local ideals.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Cornered Asymptotically Hyperbolic Metrics

The talk will introduce asymptotically hyperbolic metrics that have a finite boundary in additional to the usual infinite one (and thus a corner at infinity), will describe some of the analytic and geometric complications that this creates, and will then briefly describe some results that have been obtained and work in progress, as well as future directions.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Invariants for mapping classes of surfaces

Suppose we are given a mapping class on a surface with boundary, s.t. the boundary is fixed. We will show how to assign to such object two invariants. The first one is an A-infinity bimodule, which is defined using intersections of curves and their images on a surface. The second is fixed point Floer homology, which counts fixed points of a map representing the mapping class. We will conclude with stating a conjectural connection between these two invariants.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Min-Max constructions for Ginzburg-Landau functionals and applications to geometric measure theory

The Ginzburg-Landau functionals (with no magnetic field) are a one-parameter family of functionals defined on complex-valued maps, whose variational theory is connected to that of the Dirichlet energy for S^1-valued maps and the area functional for submanifolds of codimension two. We describe a natural min-max method for producing critical points of these functionals on a given manifold, and discuss applications to the existence of critical points of the codimension-two area functional.

##### GEOMETRY & TOPOLOGY AT PRINCETON: Hermitian flows on complex manifolds and weak Capana-Peternell conjecture

Campana-Peternell conjecture states that a Fano manifold with nef tangent bundle is rational homogeneous. We formulate weak differential-geometrical version of Campana-Peternell conjecture and discuss a possible approach \ to it based on the study of metric flows on Hermitian manifolds.