Modular forms are ubiquitous mathematical objects. Roughly speaking, they are functions in the complex upper half plane satisfying certain symmetry properties. More importantly, they occur naturally in connection with many problems in different areas of mathematics, e.g., number theory, geometry, mathematical physics, etc.

In this talk, I will give a brief introduction to the theory of classical modular forms while providing examples of various applications.

Wave maps are natural hyperbolic analogue of harmonic maps. The study of the wave maps over the last twenty plus years has led to many beautiful and deep ideas, culminating in the proof of the ``ground state conjecture'' by Sterbenz and Tataru (with independent proof by Tao, and by Krieger&Schlag when the map has hyperbolic plane as target). The understanding of wave maps is now quite satisfactory. There are however still some remaining interesting problems, involving more detailed dynamics of wave maps. In this talk, we shall look at the problem of ruling out the so called ``null concentration of energy'' for wave maps. We will briefly review the history of wave maps, and explain why the null concentration of energy is relevant, and why the channel of energy inequality seems to be uniquely good in ruling out such possible energy concentration in the presence of solitons.

In this talk, I will describe my results on superconvergence estimates of mixed methods using Raviart--Thomas finite elements. First I prove the canonical interpolant and finite element solution approximating the vector variable are superclose in $L^2$ norm. The main tool is a triangular integral identity in Bank and Xu SIAM J. Numer. Anal 41 (2003) 2294-2312, and a discrete Helmholtz decomposition. Comparing to previous supercloseness results (eg. Brandts Numer. Math. 68 (1994) 311--324), my proof is constructive and works on irregular triangular meshes. Even on a special uniform grid, my result shows that the previous supercloseness result is suboptimal. Next I will describe several postprocessing operators based on simple edge averaging, $L^2$ projection or superconvergence patch recovery. Then I will show the postprocessed finite element solution superconverges to the true solution. If time permits, I will also briefly describe applications to Maxwell's equations an
d generalizations to fourth order elliptic equations.

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection C of all of them, and |C| is smaller than each of the sets. There has been a lot of recent progress on determining the maximum size of a sunflower-free family of subsets of [n]. We consider the problems of determining the maximum sum and product of k families of subsets of [n] that contain no sunflower of size k with one set from each family. We solve the sum problem exactly and make partial progress towards the product problem.

\noindent{Joint work with Lujia Wang.}

This talk introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The techniques and theory touch on spectral theory of Laplacians and heat kernels, optimization, and linear algebra.
Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Entanglement is a core concept in quantum mechanics and quantum information theory. Put simply: a tensor is entangled if is not a product state. Measuring precisely how much entanglement a given tensor has is a big question with competing answers in the physics community. One natural measure is the {\bf geometric measure of entanglement}, which is a version of the Hilbert--Schmidt distance of the given tensor from the set of product states. It can also be described as the log of the spectral norm.

In 2009, Gross, Flammia, and Eisert showed that, as the mode of the tensor grows, the geometric measure of entanglement of a random tensor is, with high probability, very close to the theoretical maximum. In this talk, I will describe my joint work with Shmuel Friedland on the analogous situation for symmetric tensors. While symmetric tensors are inherently entangled, it turns out their maximum geometric measure of entanglement is exponentially smaller than for generic tensors. Using tools from representation theory and random matrix theory, we prove that, nevertheless, random symmetric tensors are, with high probability, very close to maximally entangled.

Equidistribution results play an important role in dynamical
systems and their applications in number theory. Often in such
applications it is necessary for equidistribution to be effective (i.e.
the rate of convergence is known) so that number-theoretic methods such
as sieving can be applied. In this talk, we will give a brief history
of effective equidistribution results in homogeneous dynamics and their
applications to number theory. We will then present an effective
equidistribution result for certain flows on the space of lattices and
discuss a number-theoretic application regarding almost-prime times for
these flows.

We explore the implications of molecular charge fluctuations on the kinetics of electrostatically dominated diffusion-controlled association, employing diffusion theory and Brownian computer simulations. In general, stochastically fluctuating degrees of freedom that dynamically couple to the diffusional transport can yield additional dissipative forces, i.e., friction, resonant response or feedback mechanisms. Using a minimalistc model we present how charge fluctuations due to randomly fluctuating protonation of a charged ligand influence the association kinetics to a charged membrane. Here, the exploration of the limiting and intermediate frequency domains of the fluctuations serve to clarify the use of concepts of an average charge, an average potential, and an associated capacitance for quantitative rate prediction.

In quantum mechanics, at an energy level
E, there arises an allowed region A(E), a forbidden
region F(E) and an interface C(E) between them.
Most quantities of interest, ranging from sizes of
quantum states to their nodal sets, exhibit a transition
across the interface between allowed and forbidden behavior. I will illustrate these interfaces with two
different types of problems: nodal sets of eigenfunctions
of Schrodinger equations and ``partial Bergman kernels''
for ample line bundles over Kahler manifolds. The two
settings seem quite different at first sight but they are just two types of geometry in phase space. No prior knowledge of quantum mechanics is assumed.

We consider the problem of detecting and localizing an submatrix with larger-than-usual
entries inside a large, noisy matrix. This problem arises from analysis of data in
genetics, bioinformatics, and social sciences. We consider that entries of the data matrix are
independently following distributions from a natural exponential family, which generalizes
the common Gaussian assumptions in the literature. Distribution-free methods of detection and
size-adaptive methods of both detection and localization problems are studied with their
asymptotic behaviors illustrated.

I will give a brief introduction to the Hodge conjectures and
describe some examples where the conjectures are nontrivial.

Rigidity phenomena in homogeneous dynamics have been extensively
studied over the past few decades with several striking results and
applications. In this talk we will give an overview of these results
and also the more recent activities which aim at presenting
qualitative versions of them.

The goal of the meeting is to bring together world class mathematicians with expertise in complex analysis/geometry and CR geometry, as well a provide a forum for the exchange of ideas with graduates.

The goal of the meeting is to bring together world class mathematicians with expertise in complex analysis/geometry and CR geometry, as well a provide a forum for the exchange of ideas with graduates.