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Abstract: TBA

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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.)