The Verlinde formula is a celebrated explicit computation of the
dimension of the space of sections of certain positive line bundles over
the moduli space of semistable vector bundles over an algebraic curve. I
will describe a recent generalization of this formula in which the
moduli of vector bundles is replaced by the moduli of semistable Higgs
bundles, a moduli space of great interest in geometric representation
theory. A key part of the proof is a new ``virtual non-abelian
localization formula" in K-theory, which has broader applications in
enumerative geometry. The localization formula is an application of the
nascent theory of Theta-stratifications, and it serves as a new source
of applications of derived algebraic geometry to more classical questions.

Large amounts of data now stream from daily life; data
analytics has been helping to discover hidden patterns, correlations and
other insights. This talk introduces the mode decomposition problem in
the analysis of oscillatory data. This problem aims at identifying and
separating pre-assumed data patterns from their superposition. It has
motivated new mathematical theory and scientific computing tools in
applied harmonic analysis. These methods are already leading to
interesting and useful results, e.g., electronic health record analysis,
microscopy image analysis in materials science, art and history.

The development of the theory of mirror symmetry in high energy
physics has led to deep conjectures regarding the geometry of a
special class of complex manifolds called Calabi-Yau manifolds. One of
the most intriguing of these conjectures states that various geometric
invariants, some classical and some more homological in nature, agree
for any two Calabi-Yau manifolds which are ``birationally equivalent"
to one another. I will discuss how new methods in equivariant geometry
have shed light on this conjecture over the past few years, leading to
the first substantial progress for compact Calabi-Yau manifolds of
dimension greater than three. The key technique is the new theory of
``Theta-stratifications," which allows one to bring ideas from
equivariant Morse theory into the setting of algebraic geometry.

In this talk I will talk on our recent work on asymptotically hyperbolic Einstein manifolds. I will present a proof for a sharp volume comparison theorem for asymptotically hyperbolic Einstein manifolds, which will imply not only the rigidity theorem for hyperbolic space in general dimension but also curvature estimates for asymptotically hyperbolic Einstein manifolds. In particular, as a consequence of our curvature estimates, one now knows that the asymptotically hyperbolic Einstein metrics with conformal infinities of sufficiently large Yamabe constant have to be negatively curved.

This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the applications area motivates the need for new mathematical approaches and algorithms, and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator for high dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lies near low dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).

The spectra of signed matrices have played a fundamental role in social sciences, graph theory and control theory. They have been key to understanding balance in social networks, to counting perfect matchings in bipartite graphs, and to analyzing robust stability of dynamic systems involving uncertainties. More recently, the results of Marcus et al. have shown that an efficient algorithm to find a signing of a given adjacency matrix that minimizes the largest eigenvalue could immediately lead to efficient construction of Ramanujan expanders.
Motivated by these applications, this talk investigates natural spectral properties of signed matrices and address the computational problems of identifying signings with these spectral properties. There are three main results we will talk about: (a) NP-completeness of three signing related problems with (negative) implications to efficiently constructing expander graphs, (b) a complete characterization of graphs that have all their signed adjacency matrices be singular, which implies a polynomial-time algorithm to verify whether a given matrix has a signing that is invertible, and (c) a polynomial-time algorithm to find a minimum increase in support of a given symmetric matrix so that it has an invertible signing.