Colloquium Archives for Academic Year 2014

The fundamental group of an algebraic variety, and hyperbolic complex manifolds

When: Wed, September 10, 2014 - 3:15pm
Where: Math 3206
Speaker: Burt Totaro (UCLA) -
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.)

Annual Department Welcome

When: Wed, September 17, 2014 - 3:15pm
Where: MTH 3206
Speaker: Dr. Scott Wolpert (Chair, Dept. of Mathematics, UMCP) -

Existence of a solution to an equation arising from the theory of Mean Field Games

When: Wed, September 24, 2014 - 3:15pm
Where: Math 3206
Speaker: Wilfrid Gangbo (Georgia Institute of Technology) -
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).

The Malliavin Calculus and its Applications

When: Wed, October 1, 2014 - 3:15pm
Where: Math 3206
Speaker: David Nualart (University of Kansas) -
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.

On null singularities for the Einstein vacuum equations and the strong cosmic censorship conjecture in general relativity

When: Wed, October 8, 2014 - 3:15pm
Where: Math 3206
Speaker: Mihalis Dafermos (Princeton University) -

On the work of Fields Medalists Artur Avila and Maryam Mirzakhani

When: Fri, October 10, 2014 - 3:15pm
Where: Math 3206
Speaker: Giovanni Forni (University of Maryland) -
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.

No Colloquium

When: Wed, October 22, 2014 - 3:15pm
Where: Math 3206
Speaker: () -

No Colloquium

When: Wed, October 29, 2014 - 3:15pm
Where: Math 3206
Speaker: () -

An introduction to the theory of noncommutative motives

When: Fri, October 31, 2014 - 3:15pm
Where: Math 3206
Speaker: Goncalo Tabuada (MIT) -
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.

Some geometric mechanisms for Arnold Diffusion.

When: Wed, November 5, 2014 - 3:15pm
Where: Math 3206
Speaker: Rafael de la Llave (Georgia Institute of Technology) -
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

The interplay between geometric modeling and simulation of partial differential equations - Aziz Lecture

When: Wed, November 12, 2014 - 3:15pm
Where: Math 3206
Speaker: Annalisa Buffa (IMATI-CNR Pavia) -
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.

Some connections between Minimal surfaces and Free Boundary problems

When: Fri, November 14, 2014 - 3:15pm
Where: Math 3206
Speaker: Luis Caffarelli (University of Texas at Austin) -
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.

Integrable probability

When: Wed, November 19, 2014 - 3:15pm
Where: Math 3206
Speaker: Alexei Borodin (MIT) -
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.

Recent developments on certain dispersive equations as infinite dimensional Hamiltonian systems

When: Wed, December 10, 2014 - 3:15pm
Where: Math 3206
Speaker: Gigliola Staffilani (MIT) -
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.

Cohomology jump loci

When: Wed, January 28, 2015 - 3:15pm
Where: Math 3206
Speaker: Nero Budur (KU Leuven) -
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.

Dynamics of polynomial automorphisms of C^2

When: Wed, February 11, 2015 - 3:15pm
Where: Math 3206
Speaker: Mikhail Lyubich (Stony Brook University) -

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.

Machine Learning for Quantum Mechanics

When: Wed, February 18, 2015 - 3:15pm
Where: Math 3206
Speaker: Mattias Rupp (University of Basel) -
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.

Conformal blocks and the nef cone of the moduli space of curves

When: Wed, February 25, 2015 - 3:15pm
Where: Math 3206
Speaker: Angela Gibney (University of Georgia)
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.

The hypoelliptic Laplacian

When: Wed, March 4, 2015 - 3:15pm
Where: Math 3206
Speaker: Jean-Michel Bismut (Universite' Paris Sud) -
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.

Cherednik algebras and torus knots

When: Wed, March 11, 2015 - 3:15pm
Where: Math 3206
Speaker: Pavel Etingoff (MIT) -
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.

Quadratic interaction potential and Lagrangian representation for conservation laws

When: Wed, March 25, 2015 - 3:15pm
Where: Math 3206
Speaker: Stefano Bianchini (SISSA - International School for Advanced Studies) -
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

The Future of Cryptography

When: Fri, March 27, 2015 - 3:15pm
Where: Math 3206
Speaker: Michael Wertheimer (University of Maryland) -
Abstract: Mathematical challenges in cybersecurity abound. However a number of phase changes are occurring in the subdomain of cryptography that need new mathematical solutions. The demands are both near and long term. Among them is a shift of emphasis away from encryption to authentication, provably secure implementations of algorithms, and post-quantum cryptography. We will examine trends in privacy, security, economics, and computing and describe how they are conspiring to create exciting new mathematical research problems.

The speaker is the former Director of Research at the National Security Agency and author of “The Mathematics Community and the NSA” published in the February 2015 edition of the Notices of the American Mathematical Society. He will be available to answer questions regarding this article.

Quasimap invariants, Gromov-Witten invariants, and mirror maps

When: Fri, April 3, 2015 - 3:15pm
Where: Math 3206
Speaker: Ionut Ciocan-Fontanine (University of Minnesota) -
Abstract: Algebraic Gromov-Witten invariants are (virtual) counts of holomorphic maps from compact Riemann surfaces to complex quasiprojective manifolds. In the early 1990's, physicists used Mirror Symmetry to make stunning predictions about these counts in all genera for many Calabi-Yau threefolds. Namely, given such a threefold X and its "mirror" Calabi-Yau threefold Y, the generating function of genus g Gromov-Witten invariants of X is obtained from an (often explicit) "B-model partition function of Y" via a change of variables known as the mirror map. Some of these predictions have been since proved, many are still conjectural.
Quasimap theory is a more recent alternative way of map counting, and a natural question to ask is how it is related to Gromov-Witten theory. The answer turns out to be that for Calabi-Yau threefolds the generating functions of quasimap invariants and of Gromov-Witten invariants are related precisely by the mirror map. Put it differently, the quasimap theory of X is conjecturally _equal_ to the physicists' B-model partition function of the mirror threefold Y. This is joint work with Bumsig Kim.

Differential Galois Groups and Patching

When: Wed, April 8, 2015 - 3:15pm
Where: Math 3206
Speaker: Julia Hartmann (UPenn) -
Abstract: Differential Galois theory studies (linear, homogeneous) differential equations by means of their symmetry groups, the differential Galois groups, which are linear algebraic groups. In analogy to the inverse problem in ordinary Galois theory, one would like to know which linear algebraic groups occur as differential Galois groups over a given differential field, since this provides information about the field and its extensions. Over the complex numbers, this so-called "inverse problem" is related to the classical Riemann-Hilbert problem and is known to have a positive solution.

The talk gives an introduction to differential Galois theory and then turns toward the inverse problem. We give a complete answer in the case when the constant field is a Laurent series field. While Laurent series fields are never algebraically closed, they share with the complex numbers the property of being complete with respect to a metric topology. Our approach to the inverse problem is based on a recent version of patching methods for algebraic objects over such fields.

Waves in random media: the story of the phase (Aziz Lecture)

When: Wed, April 15, 2015 - 3:15pm
Where: Math 3206
Speaker: Prof. Lenya Ryzhik (Department of Mathematics Stanford University) -
Abstract: The macroscopic description of wave propagation in random media typically focuses on the scattering of the wave intensity, while the phase is discarded as a uselessly random object. At the same time, most of the beauty in wave scattering come from the phase correlations. I will describe some of the miracles, as well as some limit theorems for the wave phase.

Rayleigh quotients and equivariant family of measures

When: Mon, April 20, 2015 - 3:15pm
Where: Math 3206
Speaker: Francois Ledrappier (University of Notre Dame) -
Abstract: The universal cover of a compact negatively curved manifold has strong
convergence properties at infinity. In this talk we present such
properties related to the bottom of the spectrum of the Laplacian.
Most of the talk will be devoted to presenting and discussing the
notion of regular equivariant family of measures at the boundary that
play a major role in our results. This is joint work with Seonhee Lim.

Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes

When: Wed, April 22, 2015 - 3:15pm
Where: Math 3206
Speaker: Douglas Hofstadter (Indiana University at Bloomington) -
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.

Spectral Bisection of Graphs and Connectedness

When: Fri, April 24, 2015 - 3:00pm
Where: MATH 3206
Speaker: John Urschel (Baltimore Ravens) -
Abstract: We present a refinement of the work of Miroslav Fiedler regarding bisections of irreducible matrices. We consider graph bisections as defined by the cut set consisting of characteristic edges of the eigenvector corresponding to the smallest non-zero eigenvalue of the graph Laplacian (the so- called Fiedler vector).
We provide a simple and transparent analysis, including the cases when there exist components with value zero. Namely, we extend the class of graphs for which the Fiedler vector is guaranteed to produce connected subgraphs in the bisection. Furthermore, we show that for all connected graphs there exist Fiedler vectors such that connectedness is preserved by the bisection, and estimate the measure of the set of connectedness
preserving Fiedler vectors with respect to the set of all Fiedler vectors.

The Unreasonable Effectiveness of Thin Groups

When: Wed, April 29, 2015 - 3:15pm
Where: Math 3206
Speaker: Alex Kontorovich (Rutgers University) -
Abstract: Thin groups are certain arithmetic subgroups of Lie groups whose quotient manifolds have infinite volume. We will describe a number of problems in pseudorandom numbers, numerical integration, and homogeneous dynamics which, it turns out, are all the same problem when viewed through the lens of thin groups.

Mathematical challenges in kinetic models of dilute polymers: analysis, approximation and computation (Aziz Lecture)

When: Wed, May 6, 2015 - 3:15pm
Where: Math 3206
Speaker: Endre Suli (Oxford University) -