In this talk, we discuss the Diederich–Fornæss index in several complex variables. A domain Ω ⊂ Cn is said to be pseudoconvex if −log(−δ(z)) is plurisubharmonic in Ω, where δ is a signed distance function of Ω. The Diederich–Fornæss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich–Fornæss index for many types of domains. 

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) for discrete random Schrödinger operators with finite-range potentials with respect to the probability measure. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the technique are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime for dimension one. The talk is based on joint work with Peter Hislop (Univ. of Kentucky) 

We discuss localization and local eigenvalue statistics for Schr\"odinger operators with random point interactions on $R^d$, for $d=1,2,3$. The results rely on probabilistic estimates, such as the Wegner and Minami estimate, for the eigenvalues of the Schr\"odinger operator restricted to cubes. The special structure of the point interactions facilitates the proofs of these eigenvalue correlation estimates.
One of the main results is that the local eigenvalue statistics is given by a Poisson point process in the localization regime, one of the first examples of Poisson eigenvalue statistics for multi-dimensional random Schr\"odinger operators in the continuum.  This is joint work with M.\ Krishna and W.\ Kirsch.

On a complex manifold, a Higgs bundle is a pair containing a holomorphic vector bundle E and a holomorphic End(E)-valued 1-form. In this talk, we focus on nilpotent Higgs bundles, for example, the ones arising from variations of Hodge structures for a deformation family of Kaehler manifolds. We first give an optimal upper bound of the curvature of Hodge metric of the deformation space of Calabi-Yau manifolds. Secondly, we prove a rigidity theorem of the holonomy of polystable nilpotent Higgs bundles via the non-abelian Hodge theory when the base manifold is a Riemann surface. This is joint work with Song Dai.

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifold-structured data by combing with recent advances in deep learning.