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Abstract: In the early 1960s, as a math-loving teen-ager, I was swept up in an intoxicating binge of mathematical exploration, centered on the discovery (or invention?) of integer sequences. The discovery that launched this binge came out of an empirical exploration of how triangular numbers intermingle with squares. The strange hidden order in the sequence that reflected their intermingling was extremely unexpected and exciting to me.
This burst of joy gave rise to the desire to repeat the experience, which meant to recreate essentially the same phenomenon again, but in a new “place”, which meant to generalize outwards, and I carried out this generalization by exploring many “nearby” analogues, where the terms “place” and “nearby” suggest a metaphorical space of ideas, and in it, some kind of metaphorical metric.
Over the next few years, analogies and sequences came to me in wondrous flurries, giving rise to all sorts of discoveries, some very rich and inspirational, some less so, but in any case, these coevolving discoveries pushed the envelope of interconnected ideas outwards, revealing unsuspected new caverns in the idea-space I had stumbled upon. Some explorations gave rise to patterns I could fully understand and prove; others gave rise to mysterious, chaotic behaviors too deep for me to fathom, let alone prove. After a while, alas, I started hitting up against the limits of my own imagination, and I gradually ran out of steam, but that several-year voyage was and remains a high point of my life.
My talk will thus be all about the very human, intuition-driven, analogy-permeated nature of mathematical discovery, invention, and exploration — not at the highly abstract level of the greatest of mathematicians, to be sure — but hopefully, the essence of the mental processes mediating the modest meanderings of a middling, minor mathematician is more or less the same as that of those that mediate the marvelous and majestic masterstrokes of a major, mature mathematician.

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Abstract: TBA

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Abstract: The proof of a quadratic estimate used in the literature to give the convergence rate of Glimm approximations to conservation laws requires the development of Lagrangian type representation of solutions, similar to linear transport equations.

In this talk I will show:
- an overview of hyperbolic systems of conservation laws in one space dimension
- why this kind of problems arises naturally and how has been addressed in the literature
- compactness results which can be deduced from this kind of analysis

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Abstract: The Cherednik algebra B(c,n), generated by symmetric polynomials and the quantum Calogero-Moser Hamiltonian, appears in many areas of mathematics. It depends on two parameters - the coupling constant c and number of variables n. I will talk about representations of this algebra, and in particular about a mysterious isomorphism between the representations of B(m/n,n) and B(n/m,m) of minimal functional dimension. This symmetry between m and n is made manifest by the fact that the characters of these representations can be expressed in terms of the colored HOMFLY polynomial of the torus knot T(m/d,n/d), where d=GCD(m,n). The talk is based on my joint work with E. Gorsky and I. Losev.

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Abstract: If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian is a a family of operators acting on the total space X of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian and the geodesic flow. Up to lower order terms, the hypoelliptic Laplacian is the weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow.
In the talk, I will explain the construction of the hypoelliptic Laplacian in de Rham theory as a natural interpolation between the Hodge Laplacian and a symplectic Laplacian. In the case of locally symmetric spaces, the spectrum of the elliptic Laplacian is essentially preserved by the deformation.
The stochastic process corresponding to the hypoelliptic Laplacian interpo-lates naturally between Brownian motion and the geodesic flow. Many proper-ties of the hypoelliptic Laplacian can be properly understood from this point of view.

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Abstract: The central object of consideration in the study of the birational geometry of a proper variety is an important and very often elusive invariant called its nef cone. In my talk I will describe the F-Conjecture, which gives a very simple combinatorial description of the nef cone of the moduli space of curves. I will also talk about elements of the nef cone that come from geometry. Namely, vector spaces of conformal blocks which are associated to a simple complex Lie algebra, n representations, and a non-negative integer $\ell$, fit together to form vector bundles on $\overline{M}_{g,n}$. These give rise to nef line bundles in genus 0. After a brief introduction to these bundles, I will discuss their basic properties, and how a growing understanding of them relates to the F-Conjecture.

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Abstract: Quantum mechanics (QM) provides a theory of matter at the atomic scale, and numerical solutions to Schrödinger's equation allow calculation of many properties of molecules and materials. A major challenge is the computational cost of accurate solutions, which scales as a polynomial of high degree in system size, severely limiting applicability. Machine learning (ML), in particular non-linear regression, has been successfully used to interpolate between QM reference calculations, leading to computational savings of several orders of magnitude. In this talk, I will introduce the concepts underlying hybrid QM/ML models, with a focus on kernel-based learning, an elegant, systematically non-linear form of ML that has seen increased interest and a variety of creative applications in QM over the last years.

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Abstract: Dynamics of polynomial automorphisms of C^2 is a fairly new area of research that has been intensely explored since mid 1980s. It combines ideas from holomorphic dynamics, ergodic theory, and pluri-potential theory, displaying a number of phenomena that are not observable in dimension one. We will give a survey of this field including recent developments on classification of periodic Fatou components and a complex version of the Palis Conjecture on the structural stability, homoclinic tangencies, and Newhouse phenomenon.

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Abstract: Deformation theory is a classical subject, going back to the
foundational work of Kodaira-Spencer on complex manifolds. Nowadays, a
simple yet powerful general principle from the mid 1980's due to
Deligne, Goldman-Millson, and others, states that: in characteristic
zero, every deformation problem is governed by a differential graded
Lie algebra (DGLA). We propose and illustrate an enhanced principle:
every deformation problem with cohomology constraints is governed by a
module over a DGLA. Joint work with Botong Wang.

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Abstract: In this talk I will present some recent developments in the study of dispersive differential equations on compact manifolds that can also be viewed as infinite dimensional Hamiltonian systems. I will talk about Strichartz estimates, weak turbulence, Gibbs measures, symplectic structures and non-squeezing theorems. A list of open problems will conclude the talk.

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Abstract: The goal of the talk is to survey the emerging field of integrable probability,
whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

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Abstract: I will describe connections between the minimal surface theory,
as develop from the point of view of sets with minimal perimeter and
different approaches to free boundary regularity. Will also describe some
problems in phase transition that combine both.

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Abstract: Computer-based simulation of partial differential equations involves approximation of the unknown fields and a description of geometrical entities as, for example, the computational domain and the properties of the media.
There are a variety of situations: in some cases this description is very complex, in some other the governing equations are very difficult to discretize. Starting with an historical perspective, I will describe the recent efforts to improve the interplay between the mathematical approaches characterizing these two aspects of the problem.

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Abstract: We consider the problem whether small perturbations of integrable mechanical systems
can have very large effects.

It is known that in many cases, the effects of the perturbations average out, but there
are exceptional cases (resonances) where the perturbations do accumulate. It is a complicated
problem whether this can keep on happening because once the instability accumulates, the system
moves out of resonance.

V. Arnold discovered in 1964 some geometric structures that lead to accumulation
in carefully constructed examples. We will present some other geometric structures
that lead to the same effect in more general systems and that can be verified in
concrete systems. In particular, we will present an application to
the restricted 3 body problem. We show that, given some conditions, for all
sufficiently small (but non-zero) values of the eccentricity, there are orbits
near a Lagrange point that gain a fixed amount of energy. These conditions
(amount to the non-vanishing of an integral) are verified numerically.

Joint work with M. Capinski, M. Gidea, T. M-Seara

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Abstract: The theory of motives was envisioned by Grothendieck in the
sixties. In contrast, noncommutative motives were only recently introduced
by Kontsevich. The central tenet of noncommutative motives is that one
should not work with algebraic varieties but rather with their (bounded)
derived category of coherent sheaves: the later carries much more
symmetries than the former. In this talk I will survey the foundations of
noncommutative motives and describe some applications. I will end the talk
with some speculative remarks regarding where the research on
noncommutative motives might take us.

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Abstract: The cited work of both Avila and Mirzakhani includes
contributions to the study of area-preserving flows on surfaces,
(and related systems, such as Interval Exchange Transformations
and Billiards in Rational Polygons) and/or to the study of the
corresponding ''renormalization dynamics'', that is, the so-called
Teichmueller geodesic flow on the moduli space of Riemann surfaces.
In this talk we will survey their main results on these topics and discuss
their significance mainly from the point of view of dynamical systems.

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Abstract: The purpose of this talk is to present an elementary introduction to the stochastic calculus of variations or Malliavin calculus.
This is a differential calculus on a Gaussian space introduced by Paul Malliavin in the 70s to provide a probabilistic proof of Hormander's hypoellipticity theorem. We will also discuss a recent application of Malliavin calculus, combined with Stein's method, to normal approximations.

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Abstract: We construct a small time strong solution of a nonlocal Hamilton–Jacobi equation
introduced by Lions, the so-called master equation, which finds its origins in the theory of Mean Field Games. As a consequence we prove the existence of a Nash equilibrium for a game with a continuum of players, called non-atomic game by R. J. Aumann and L. S. Shapley. (This is a joint work with A. Swiech).

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Abstract: It is a mystery which groups can occur as fundamental groups of smooth complex projective varieties. It is conceivable
that whenever the fundamental group is infinite, the variety has some "negative curvature" properties. We discuss a result
in this direction, in terms of "symmetric differentials". There are interesting open questions even about the special case
of compact quotients of the unit ball in C^n. (Joint work with Yohan Brunebarbe and Bruno Klingler.)