# On the eigenfunctions of the flat torus

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I will present some recent results on equidistributive properties of toral eigenfunctions. Only a minimal knowledge of Fourier analysis is required to follow all the details of this talk.

# TBA

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# An introduction to ABP estimate

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The celebrated *Alexandrov-Bakelman-Pucci Maximum Principle* (often abbreviated as *ABP estimate*) is a pointwise estimate for solutions of elliptic equations, which was introduced in the 1960s. It was motivated by beautiful geometric ideas and has been a fundamental tool in the study of non-divergent PDEs. More recently, this PDE technique also pays back to geometry - the ABP estimate and its extensions can be used to prove some optimal classical geometric inequalities such as the Isoperimetric and Minkowski inequalites.

# Localization in the random XXZ quantum spin chain

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# Random Integer Matrices and Random Finite Abelian Groups

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How do you choose a random finite abelian group?

A d x d integer matrix M gives a linear map from **Z**^d to** Z**^d. The cokernel of M is **Z**^d/Im(M). If **det**(M) is nonzero, then the cokernel is a finite abelian group of order **det**(M) and rank at most d.

What do these groups ‘look like’? How often are they cyclic? What can we say about their p-Sylow subgroups? What happens if instead of looking at all matrices, we only consider symmetric ones? We will discuss distributions on finite abelian p-groups, focusing on ones that come from cokernels of families of random matrices. We will explain how these distributions are related to questions from number theory about ideal class groups, elliptic curves, and sublattices of **Z**^d.

# Neoclassical Theory of Electromagnetic Interactions

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The theory of electromagnetic (EM) phenomena known as electrodynamics is one of the major theories in science. At macroscopic scales the interaction of the EM field with matter is described by the classical electrodynamics based on the Maxwell-Lorentz theory. Many of electromagnetic phenomena at microscopic scales are covered by the so-called semiclassical theory that treats the matter according to the quantum mechanics, whereas the EM field is treated classically. The subject of this presentation is a recently advanced by us neoclassical electromagnetic theory that describes EM phenomena at all spatial scales –microscopic and macroscopic. This theory modifies the classical electrodynamics into a theory that applies to all spatial scales including atomic and nanoscales. The neoclassical theory is conceived as one theory for all spatial scales in which the classical and quantum aspects are naturally unified and emerge as approximations. It is a classical Lagrangian field theory, and consequently it is a local and deterministic theory. Probabilistic aspects of the theory may arise in it effectively through complex nonlinear dynamical evolution. This presentation is to provide an introduction to our theory including a concise historical review.

(Joint work with Anatoli Babin)

# A logician's/set theorist's point of view

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I will discuss the important phenomenon in mathematics that the solution to a question may reach out to concepts of complexity significantly higher than concepts needed for the formulation of the question.

# Characterizations of the unit ball in Euclidean spaces.

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In this talk, I will give you many ways to characterize the unit ball

in $R^n$ or in $C^n$. It involves, differential equations, first eigenvalue of Laplace-Beltrami

operator, etc.

# Mathematical and computational modeling in the applied sciences

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Mathematical and computational modeling have become an indispensible component of research across the sciences. Nevertheless, there are still many examples of research across the sciences where decision making processes are strongly influenced by empirical approaches rather than theory. One of the primary challenges in developing rigorous models of complex processes is capturing the nonlinear interactions of processes across multiple scales in space and time. At the same time, because such models may contain many parameters and can describe wide ranges of behaviors, new methods for parameter estimation and inference are needed as well. In this talk, I will give several examples of new multiscale models and novel applications of parameter inference methodologies in applications ranging from tumor biology to engineering. I will discuss some open problems where there are significant opportunities for future research.