In this talk, I will discuss some known results and recent improvements to lower bound on the first eigenvalue of the p-Laplacian as well as introduce a new p-Laplace type operator acting on differential forms.

While classical gauge theoretic invariants for 4-manifolds are usually
defined in the setting that the intersection form has nontrivial positive
part, there are several invariants for a 4-manifold X with the homology S1
cross S3. The first one is the Casson-Seiberg-Witten invariant LSW(X)
defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta
invariant LFO(X). It is conjectured that these two invariants are equal to
each other (This is an analogue of Witten’s conjecture relating Donaldson and
Seiberg-Witten invariants.)

In this talk, I will recall the definition of these two invariants, give
some applications of them (including a new obstruction for metric with
positive scalar curvature), and sketch a prove of this conjecture for
finite-order mapping tori. This is based on a joint work with Danny Ruberman
and Nikolai Saveliev.

A joint seminar with the Geometry & Topology Seminar series.

One of the central topics in mean curvature flow is understanding the singularities. In 1995, Ilmanen conjectured that the first singularity appeared in a smooth mean curvature flow of surfaces must have multiplicity one. Following the theory of generic mean curvature flow developed by Colding-Minicozzi, we prove that a closed singularity with high multiplicity is not generic, in the sense that we may perturb the flow so that this singularity with high multiplicity can never show up. One of the main techniques is the local entropy, which is an extension of the entropy used by Colding-Minicozzi to study the generic mean curvature flow.​

According to the conjecture of Rauch’s from 1974, any eigenfunction corresponding to the principal eigenvalue of the Neumann Laplacian attains its extrema on the boundary of planar domains. After giving an account on the history and validity of the conjecture, we present our own new results for tubular neighbourhoods of curves on surfaces. This is joint work with Matej Tusek.

Location: Skye Hall 347

4:00-4:50pm

Speaker: Yu-Shen Lin (Boston University)

Title: Special Lagrangian fibrations in weak Del Pezzo Surfaces

Abstract: Motivated by the study of mirror symmetry, Strominger-Yau-Zaslow (SYZ) conjectured that Calabi-Yau manifolds admit certain minimal Lagrangian fibrations. These minimal Lagrangians are the special Lagrangian submanifolds studied earlier by Harvey-Lawson. Many of the implication of the SYZ conjecture is proved and it has been the guiding principle for studying mirror symmetry for a long time. However, not many special Lagrangians are known in the literature. In this talk, I will prove the existence of special Lagrangian fibration on the complement of a smooth anti-canonical divisor in a (weak) Del Pezzo surface. If the time allows, I will explain its impact to mirror symmetry. This is joint work with Tristan Collins and Adam Jacob.

5:00-5:50pm

Speaker: Yannis Angelopoulos (UCLA)
Title: Linear and nonlinear waves on extremal Reissner-Nordstrom spacetimes 

Abstract:  I will present several results (that have been obtained jointly with Stefanos Aretakis and Dejan Gajic) from the analysis of solutions of linear and nonlinear wave equations on extremal Reissner-Nordstrom spacetimes, including sharp asymptotics on the horizon and at infinity for linear waves, and instability phenomena for nonlinear waves. These results can be seen as stepping stones to the fully nonlinear problem of stability/instability of extremal black holes.