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Abstract: Given a Borel A is R^n of positive measure, one can consider its
semisum S=(A+A)/2. It is clear that S contains A, and it is not difficult
to prove that they have the same measure if and only if A is equal to his
convex hull minus a set of measure zero.
We now wonder whether this statement is stable: if the measure of S is
close to the one of A, is A close to his convex hull?
More in general, one may consider the semisum of two different sets A and
B, in which case our question corresponds to proving a stability result
for the Brunn-Minkowski inequality.
When n=1, one can approximate a set with finite unions of intervals to
translate the problem onto Z, and in the discrete setting this question
becomes a well studied problem in additive combinatorics, usually known as
Freiman's Theorem.
In this talk I'll review some results in the one-dimensional discrete
setting, and show how to answer to this problem in arbitrary dimension.

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Abstract: Although the theoretical foundations of Gabor analysis have been established more or less by the end of the last century there is still a lot to be done in Gabor analysis, and various important questions have been settled in the meantime. It is clear that the Banach Gelfand Trip consisting of the Segal Algebra (So,L2,So')(G) is the most approprate setting for many questions in time-frequency analysis.
We will walk a panorama, from the classical setting, the basic facts derived from the specific properties of the Schroedinger representation of the Heisenberg group (resp. phase space) to recent results concernig the robustness of Gabor expansions, the properties of Gabor multipliers, the computation of approximate duals, or the localization of dual Gabor families derived from the Wiener property of certain Banach algebras of infinite matrices.

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Abstract: I will present a general overview of the theory of stochastic homogenization for Hamilton-Jacobi, "viscous"-Hamilton-Jacobi and nonlinear elliptic equations. I will describe the general setting, describe the basic differences from the "standard" periodic theory, state the main results and discuss some applications to front propagation in random media.

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Abstract: At MIT we are halfway through a two year project to renovate our introductory probability and statistics class as a technology enhanced active learning (TEAL) course. The course has on online component using the edX platform and an in class component in a room built especially for TEAL. This term is the first time we've taught the revised course. Most of the talk will be focused on the lessons we've learned about preparing materials, using technology, the layout of the room and working with students in an active learning environment. A smaller part of the talk will look at the decisions we made to modernize the statistics syllabus for today's life sciences majors.

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Abstract: The usual heat equation is not suitable to preserve the topology of divergence-free
vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topology-preserving
diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of ”dissipative solutions”, which shares common features with both P.-L. Lions’ dissipative solutions to the Euler equations and the concept of ”curves of maximal slopes”, a la De Giorgi, recently used by Gigli and collaborators to study the scalar heat equation in very general metric spaces. We show that the initial value problem admits global "dissipative" solutions (at least for two space dimensions) and they are unique whenever they are smooth.

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Abstract: ``Quantum ergodicity'' usually deals with the study of eigenfunctions of
the Laplacian on Riemannian manifolds, in the high-frequency asymptotics.
The rough idea is that, under certain geometric assumptions (like negative
curvature), the eigenfunctions should become spatially uniformly
distributed, in the high-frequency limit. There are a many conjectures,
some of which have been turned into theorems recently. Physicists like Uzy
Smilansky or John Keating have suggested looking for similar questions and
results on large (finite) discrete graphs. Take a large graph $G=(V, E)$
and an eigenfunction $\psi$ of the discrete Laplacian -- normalized in
$L^2(V)$. What can we say about the probability measure $|\psi(x)|^2$
($x\in V$)? Is it close to uniform, or can it, on the contrary, be
concentrated in small sets? I will talk about recent work with Etienne Le
Masson, in the case of large regular graphs.

View Abstract

Abstract: In the first part of the talk we will give a general outline of the two topics in Kahler geometry in the title, both growing out of work of Calabi. We will also discuss the parallels with affine differential geometry which arise when one studies toric manifolds. We will explain the standard conjectures in the field, relating the existence of these metrics to algebro-geometric notions of “stability”. In the last part of the talk we will say something about recent work with Chen and Sun which establishes this conjecture in the case of Kahler-Einstein metrics on Fano manifolds (Yau’s conjecture).

This is the first talk of the Calabifest, website: http://www.calabifest.org . Note special time and room. The colloquium tea will be AFTER the talk in the Math Rotunda.

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Abstract: Given any countable discrete group G and any countable left module M of the integral group ring of G, one may consider the natural action of G on the Pontryagin dual of M. Under suitable conditions, the entropy of this action and the L2-torsion of M are defined. I will discuss the relation between the entropy and the L2-torsion and indicate how this confirms the conjecture of Wolfgang Luck that any nontrivial amenable group admitting a finite classifying space has trivial L2-torsion. This is joint work with Andreas Thom.

View Abstract

Abstract: We may associate a meromorphic function of a complex
variable \zeta_F(s) with any number field F. If F is the field Q of
rational numbers, we obtain the classical Riemann zeta-function. For
well over a hundred years, mathematicians have been interested in
relating the behavior of \zeta_F(s) at integral values of s to
arithmetic invariants of the number field, starting with the famous
theorem of Dedekind which tells us that \zeta_F(s) has a simple pole
at s = 1, and computes the residue in terms of the class number, the
unit group, etc.. We can make good guesses as to what the answer
should be in general, and sometimes prove these guesses, but we are
very far from a complete understanding.

View Abstract

Abstract: The study of stable, or stationary, states of a physical system is a
well established field of applied mathematics. Less well known or
understood are “metastable” states. Such states are not fixed points
of the underlying equations of motion but are typically
a family of states which emerge relatively quickly, dominate the
evolution of the system for long times, and then ultimately give way
to the asymptotic state of the system (from which
they are typically distinct.) Their presence is a signal that multiple
time scales are important in the problem – for instance, one associated
with the emergence of the metastable state, one associated with the
evolution along the family of such states, and one associated with
the emergence of the asymptotic states.
In this talk I will discuss recent
research with Margaret Beck which proposes a dynamical systems
understanding of metastable behavior in Burgers equations and the
two-dimensional Navier-Stokes equation.

View Abstract

Abstract: Energy norm stability estimates for the finite element discretization of the Stokes problem follow easily from the variational formulation provided the discrete pressure and velocity satisfy a uniform inf-sup condition. But deriving uniform stability estimates in L^\infty is much more complex because variational formulations do not lend themselves to maximum norms. I shall present here the main ideas of a proof that relies on weighted L^2 estimates for regularized Green's functions associated with the Stokes problem and on a weighted inf-sup condition. The domain is a convex polygon or polyhedron. The triangulation is shape-regular and quasi-uniform. The finite element spaces satisfy a super-approximation property, which is shown to be valid for most commonly used stable finite-element spaces. Extending this result to error estimates and to the solution of the steady incompressible Navier-Stokes problem is straightforward.

View Abstract

Abstract: A surprising amount of physics is independent of the spacetime
metric. When expressed in the metric-independent, coordinate-free
language of differential forms, exterior derivatives and Lie derivatives,
the structure and generality of such physics appears simple and transparent.
In this talk I'll briefly review the relevant mathematics, and then illustrate
its natural role in a few juicy physics examples: Hamilton's equations,
Liouville's theorem and Poincare's integral invariants, Faraday's law of
electromagnetic induction, the "frozen in theorem" and conservation of
magnetic helicity in ideal magnetohydrodynamics, and Kelvin's circulation
theorem for fluid flow.

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Abstract: Hodge structures are the linear algebra objects that record the
periods of integrals of differential forms on a variety over the complex
numbers. This talk concerns the symmetries and asymptotics of families of
abstract Hodge structures, and their connection to the representation
theory of Lie groups and the complex geometry of flag domains. The object
which provides this link is called a Mumford-Tate domain, although (as we
shall see) it really goes back to Picard. Tradition in Hodge theory holds
that asymptotics should be studied via the so-called limiting mixed Hodge
structure, but considerable insight (and delightful pictures) can be
obtained by looking at the domain's topological boundary in its Zariski
closure. [based on joint work with Gregory Pearlstein]

View Abstract

Abstract: Phase retrieval is the century-old problem of reconstructing a function, such as a signal or image, from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems -- which arise in numerous areas including X-ray crystallography, astronomy, diffraction imaging, and quantum physics -- are notoriously difficult to solve numerically. They also pervade many areas of mathematics, such as numerical analysis, harmonic analysis, algebraic geometry, combinatorics, and differential geometry.

I will briefly review the phase problem and discuss key mathematical developments, including seminal work by Norbert Wiener. I will then introduce a novel framework for phase retrieval, which comprises tools from optimization, random matrix theory, and compressive sensing. In particular, we will see that for certain types of random measurements a signal or image can be recovered exactly with high probability by solving a convenient semidefinite program without any assumption about the signal whatsoever and under a mild condition on the number of measurements. Our method, known as PhaseLift, is also provably stable vis-a-vis noise. I will describe how this approach carries over to the classical phase retrieval setting using structured random illuminations. I conclude with some open problems.

View Abstract

Abstract: We will discuss a technique which allows one to approximate singular varieties by smooth spaces called stacks. As an application, we will address the following question, as well as some generalizations: given a linear action of a group G on complex n-space C^n, when is the quotient C^n/G a singular variety? We will also mention some applications to Hodge theory and to derived equivalences.

View Abstract

Abstract: The classical exponential map in Riemannian geometry has
the following very important implications: if an isometry f
fixes a point and has trivial derivative there, then f is trivial;
moreover, the differential gives a simple normal form for all
isometries fixing a given point. Conformal transformations
of a Riemannian manifold are required only to preserve
angles, not distances. These have no exponential map.
Nontrivial conformal transformations can have differential
equal the identity at a fixed point, but this occurrence has
very strong implications for the underlying manifold.

I will present this rigidity phenomonenon in conformal geometry and a wide range of
generalizations. The key to these results is the notion of Cartan geometry, which infinitesimally
models a manifold on a homogeneous space. This point of view leads to a normal forms
theorem for conformal Lorentzian flows. It also leads to a suite of results on a seemingly
widespread rigidity phenomenon for flows on parabolic geometries, a rich family of geometric
structures whose homogeneous models include flag varieties and boundaries of symmetric spaces.

View Abstract

Abstract: The aim of this talk is to review a few models of the evolution of
a droplet of water on a wall. There are several microscopical descriptions
which are suggested by statistical physics, and go under the name of
wetting models. We discuss the localization/delocalization phase transition,
due to the energy/entropy competition, which is well understood in the
static case. We introduce some relevant techniques allowing to treat
the scaling limit of the dynamical version of some of these models, namely
stochastic partial differential equations with obstacles, Dirichlet forms,
optimal transport, integration by parts formulae on path spaces. Finally we
explain why the recent advances on the Khardar-Parisi-Zhang equation
might be useful to solve a long-standing problem on the critical wetting model.

View Abstract

Abstract: We discuss mathematical and computational issues related to some nonlocal balance laws in this talk. We use peridynamic materials models and nonlocal diffusion as examples to provide physical motivations and to illustrate mathematical challenges. We present a nonlocal vector calculus as an attempt to study related nonlocal variational problems in more systematic and axiomatic ways. We also explore connections and differences between nonlocal and local models and address questions concerning efficient and reliable numerical approximations.

View Abstract

Abstract: Szemeredi's theorem in combinatorial number theory asserts that
any subset of the integers having positive density contains arithmetic
progressions of any length. It turns out that this is equivalent to a
"multiple" recurrence statement for measure preserving transformations.
Together with Eli Glasner we show that this has an analogue for group
actions that are only measure preserving "on the average". By analogy
the case of the integers, this multiple recurrence result leads to
a theorem guaranteeing existence of geometric progressions in non-
amenable groups. The result for a finitely generated free group can
be made quite explicit.

View Abstract

Abstract: The talk will discuss informally the charm and attractions of counter-examples in mathematical science, some of the cautions to be borne in mind, and their sometime valuable contributions in providing true insight. Examples from of the speakers long-ago work will be cited.

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Abstract: http://www2.math.umd.edu/~pbrosnan/Colloquium/hirsch_abstract.pdf

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Abstract: Fractals and PDEs are subjects with rich and diversified connections. In our talk we show how to construct fractal (differential) operators in the limit of singular elliptic operators from the surrounding space. We explain why fractal operators have opposite intrinsic dimensional behavior in comparison with sub-elliptic operators of Hörmander's type. We then outline a unifying effective theory in the sense of homogenization, based on abstract harmonic analysis and Dirichlet forms. Finally, we consider second order heat transmission problems across a fractal layer and show related numerical simulations.

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Abstract: We will discuss wave propagation in a network of branched thin wave guides which shrink to a one- dimensional graph when the thickness parameter vanishes. We will show that asymptotically one can describe the propagating waves, the spectrum and the resolvent of the problem in the terms of solutions of ordinary differential equations on the limiting graph with the gluing conditions on the vertices of the graph determined by the scattering matrices related to individual junctions of the network.
The situation will be considered when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. If the Neumann condition is imposed at the boundary of the network and the spectral parameter is close to zero (which is the threshold in this case), then the gluing condition becomes the Kirchhoff one. The latter is the standard condition that appears after homogenization.
We will describe one of the possible practical implementations of the discussed results where a periodic network is used. All the results were obtained jointly with S. Molchanov.

View Abstract

Abstract: Recent work has analyzed proteins using combinatorial and geometrical techniques adapted from the the study of moduli spaces.

Specifically, a 3d rotation can be associated to each protein hydrogen bond, and these data already embedded in the Protein Data Bank can be analyzed. Nature is economical in exploiting only a small part of the conformational possibilities providing new constraints for simulation, refinement or design and hydrogen bonds accordingly classified. The geometrically exotic hydrogen bonds have uncanny abilities to predict protein functional and architecturally significant sites from 3d structure.

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Abstract: Suppose G is compact Lie group. The representations of G -possible ways of realizing G as group of matrices- provide a powerful way to study problems involving symmetry under G. For example, if G acts by isometries on a Riemannian manifold, each eigenspace of the Laplace operator is a representation of G. Knowing the possible dimensions of representations can therefore tell you about possible multiplicities of Laplacian eigenvalues.

When G is noncompact, there may be no realizations of G using finite matrices, and those involving arbitrary infinite matrices are too general to be useful. Stone, von Neumann, Wigner, and Gelfand realized in the 1930s that unitary operators on Hilbert spaces provided a happy medium: that any group could be realized by unitary operators, but that the possible realizations could still be controlled in interesting examples.

Gelfand's "unitary dual problem" asks for a list of all the realizations of a given group G as unitary operators. Work of Harish-Chandra, Langlands, and Knapp-Zuckerman before 1980 produced a slightly longer list: all realizations of G as linear operators preserving a possibly indefinite Hermitian form. I will describe a notion of "signatures" for such infinite-dimensional forms, and recent work of Jeff Adams' research group ''Atlas of Lie groups and representations" on an algorithm for calculating signatures. This algorithm identifies unitary representations among Hermitian ones, and so resolves the unitary dual problem.

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Abstract: This talk will describe several contributions of Bill Thurston (1946-2012),
and how they fundamentally changed how we think about mathematics.

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Abstract: We consider random ensembles of polynomials, and more generally sections of
holomorphic line bundles, and we study the properties of their zeros,
critical points and level sets. How much clustering occurs? How likely are
the "typical" distributions? What are the statistics of the sup norms and of
the topology of the level sets? The answers to these and other questions
depend on the properties of the corresponding Bergman kernels. We shall
describe the asymptotics of Bergman kernels for linear systems of increasing
degree and their impact on the distributions of zeros, critical points, and
norms.

View Abstract

Abstract: The Ericksen--Leslie model of nematic liquid crystals and the Oldroyd--B fluid are continuum models of fluids containing elastic molecules. In each instance the momentum equation is coupled to an equation governing the evolution of the elastic components, and the numerical simulation of these fluids is notoriously difficult.

The equations for both systems can be derived from Hamiltonian's principle which reveals a subtle balance between inertia, transport, and dissipation effects. While both fluids have been the studied extensively, the theory for the Ericksen--Leslie model is more complete. This talk will focus on the common structure of these two fluids, the insight this provides into why naive numerical schemes may fail (the high Weisenberg problem), and the ingredients required to construct stable numerical schemes.