Where: Math 3206

Speaker: Noel Walkington (Carnegie Mellon University) - www.math.cmu.edu/~noelw

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.

Where: Math 3206

Speaker: Bernard Shiffman (Johns Hopkins University) - http://www.math.jhu.edu/~shiffman/

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.

Where: Math 3206

Speaker: Willaim Goldman (University of Maryland) - www.math.umd.edu/~wmg

Abstract: This talk will describe several contributions of Bill Thurston (1946-2012),

and how they fundamentally changed how we think about mathematics.

Where: Math 3206

Speaker: David Vogan (MIT) - http://www-math.mit.edu/~dav/

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.

Where: Math 3206

Speaker: Robert Penner (Aarhus University and California Institute of Technology) - http://pure.au.dk/portal/en/persons/robert-penner%28a1eef621-f018-4121-a96c-3ea79123dc2a%29.html

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.

Where: Math 3206

Speaker: Boris Vainberg (University of North Carolina at Charlotte) - http://math.uncc.edu/~brvainbe

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.

Where: Math 3206

Speaker: Umberto Mosco (Worcester Polytechnic Institute) - http://www.wpi.edu/academics/facultydir/uxm.html

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.

Where: Math 3206

Speaker: Morris Hirsch (University of California and of Wisconsin) -

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

Where: Math 3206

Speaker: Michael Fisher (University of Maryland) - http://terpconnect.umd.edu/~xpectnil/

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.

Where: Math 3206

Speaker: Hillel Furstenberg (Hebrew University) - http://en.wikipedia.org/wiki/Hillel_Furstenberg

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.

Where: MATH 3206 (NOTE TIME)

Speaker: Prof. Qiang Du (Department of Mathematics and Department of Materials Sciences, Penn State University) -

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.

Where: Math 3206

Speaker: Lorenzo Zambotti (Laboratoire de Probabilites et Modeles Aleatoires, Universite Paris 6) - http://www.proba.jussieu.fr/~zambotti/

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.

Where: Math 3206

Speaker: Karin Melnick (University of Maryland) - http://www2.math.umd.edu/~kmelnick/

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.

Where: Math 3206

Speaker: Matthew Satriano (University of Michigan) - http://www-personal.umich.edu/~satriano/

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.

Where: Math 3206

Speaker: Thomas Strohmer (University of California, Davis) - http://www.math.ucdavis.edu/~strohmer/

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.

Where: Math 3206

Speaker: Matt Kerr (Washington University) - http://www.math.wustl.edu/~matkerr/

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]

Where: Math 3206

Speaker: Theodor Jacobson (University of Maryland) - http://terpconnect.umd.edu/~jacobson/

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.

Where: Math 3206

Speaker: Vivette Girault (Laboratoire Jacques-Louis Lions Universite Pierre e Marie Curie Paris, France) -

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.

Where: Math 3206

Speaker: Eugene Wayne (Boston University) - http://www.bu.edu/math/people/faculty/dynamical-systems/wayne/

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.

Where: Math 3206

Speaker: Stephen Lichtenbaum (Brown University) - http://www.math.brown.edu/faculty/lichtenbaum.html

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.

Where: Math 3206

Speaker: Hanfeng Li (SUNY at Buffalo) - http://www.nsm.buffalo.edu/~hfli/

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.

Where: EGR 1202

Speaker: Simon Donaldson (Imperial College) - http://www2.imperial.ac.uk/~skdona/

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.

Where: Math 3206

Speaker: Nalini Anantharaman (Orsay) - http://www.math.u-psud.fr/~anantharaman/

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.

Where: Math 3206

Speaker: Yann Brenier (Ecole Polytechnique) -

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.

Where: Math 3206

Speaker: Jeremy M Orloff (MIT) -

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.

Where: Math 3206

Speaker: Panagiotis E. Souganidis (University of Chicago) - http://math.uchicago.edu/~souganidis/

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.

Where: Math 3206

Speaker: Hans Feichtinger (University of Wien) - http://www.univie.ac.at/nuhag-php/home/fei.php

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.

Where: Math 3206

Speaker: Spotlight Talks (UMD) -

Abstract: Spotlight talks on Graduate Research will be held in Math 3206.

So there will be no colloquium this week.

Where: Math 3206

Speaker: Alessio Figalli (University of Texas at Austin) - http://www.ma.utexas.edu/users/figalli/

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.