The eta form of Bismut–Cheeger is the higher degree version of the Atiyah-Patodi-Singer eta invariant, i.e. it is exactly the boundary correction term in the family index theorem for manifolds with boundary. In this talk, I'll study the properties of eta forms and extend them to the equivariant version for compact Lie group action. Moreover, the applications of eta forms in differential K theory will be discussed.

Kontsevich-Witten tau-function and the Hodge tau-function
are generating functions for two types of intersection numbers on
moduli spaces of stable curves. Both of them are tau functions for the
KP hierarchy. In this talk, I will describe how to connect these two
tau-functions by differential operators belonging to the
$\widehat{GL(\infty)}$ group. Indeed, these two tau-functions can be
connected using Virasoro operators. This proves a conjecture posted by
Alexandrov. This is a joint work with Gehao Wang.

In the middle of 1980s, Andreas Floer invented a new theory,
which is nowadays called Floer (co)homology.  I would like to describe a general story of Floer theory for Lagrangian submanifolds and explain some of its applications.

Program:

3:10 - 4:00 PM    Pengzi Miao (Univ. of Miami)

4:10 - 5:00 PM    Jonathan Luk (Stanford Univ.)

Title/Abstract:

Pengzi Miao (University of Miami)

Title:  Minimal hypersurfaces and boundary behavior of compact manifolds with
nonnegative scalar curvature

Abstract:
On a compact Riemannian manifold with boundary having positive mean
curvature, a fundamental result of Shi and Tam states that, if the
manifold has nonnegative scalar curvature and if the boundary is
isometric to a strictly convex hypersurface in the Euclidean space,
then the total mean curvature of the boundary is no greater than the
total mean curvature of the corresponding Euclidean hypersurface. In
3-dimension, Shi-Tam's result is known to be equivalent to the
Riemannian positive mass theorem.

In this talk, we will discuss a supplement to Shi-Tam's theorem
by including the effect of minimal hypersurfaces on a chosen boundary
component. More precisely, we consider a compact manifold with
nonnegative scalar curvature, whose boundary consists of two parts,
the outer boundary and the horizon boundary. Here the horizon
boundary is the union of all closed minimal hypersurfaces in the
manifold and the outer boundary is assumed to be a topological
sphere. In a relativistic context, such a manifold represents a body
surrounding apparent horizon of black holes in a time symmetric
initial data set. By assuming the outer boundary is isometric to a
suitable 2-convex hypersurface in a Schwarzschild manifold of
positive mass m, we establish an inequality relating m, the area of
the horizon boundary, and two weighted total mean curvatures of the
outer boundary and the hypersurface in the Schwarzschild manifold. In
3-dimension, our result is equivalent to the Riemannian Penrose
inequality. This is joint work with Siyuan Lu.

Jonathan Luk (Stanford University)

Title: Strong cosmic censorship in spherical symmetry for two-ended
asymptotically flat data

Abstract:
I will present a recent work (joint with Sung-Jin Oh) on the strong
cosmic censorship conjecture for the
Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for
two-ended asymptotically flat data. For this model, it was previously
proved (by M. Dafermos and I. Rodnianski) that a certain formulation
of the strong cosmic censorship conjecture is false, namely, the
maximal globally hyperbolic development of a data set in this class
is extendible as a Lorentzian manifold with a C0 metric. Our main
result is that, nevertheless, a weaker formulation of the conjecture
is true for this model, i.e., for a generic (possibly large) data set
in this class, the maximal globally hyperbolic development is
inextendible as a Lorentzian manifold with a C2 metric.

In this talk, we introduce the Laplacian eigenvalue problem and briefly go over its history.  Then we will present a recent result which gives a sharp lower bound of the fundamental gap for convex domain of spheres motivated by the modulus of continuity approach introduced by Andrews-Clutterbuck.  This is joint work with Lili Wang and Guofang Wei.