Colloquium: Michaël Ulrich(Lycée Kléber)

Thursday, February 2nd

4pm 135 TMCB

Refreshments Served

Everyone Welcome!

Compact quantum groups are a well-known way of generalizing the concept of a group in the framework of non-commutative mathematics.  Less known are dual groups.  Introduced by Voiculescu in the 80’s and also known as H-algebras, they are very similar to compact quantum groups but the computliplication has values this time in a free product instead of a tensor product.  We will introduce the notion of dual group, compare it to compact quantum groups and give some new results that allow us to understand them

Colloquium: Barry Simon (Cal Tech)

Title: Tales of Our Forefathers

Abstract: This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I’ll convince you they were also human beings and that, as the Chinese say, “May you live in interesting times” really is a curse.

Time: Nov 14th 4:00-5:00PM 3106 JKB

Colloquium: Brent Albrecht (University of California)

 

Title: Optimal Mass Transport and Curvature Bounds

 

Abstract: Have you ever been in a hurry at the grocery store? Have you ever wondered if you are collecting the items on your list in the most efficient way possible? This is a discrete case of the fundamental problem of optimal mass transport theory. As a consequence of its many and varied applications in both pure and applied mathematics, the problem of optimal mass transport theory has received considerable attention during the past few years. (Indeed, Cédric Villani, one of the prominent researchers in this area, was awarded the Fields Medal in 2010.) On the pure side, the theory of optimal mass transport has found important connections to nonlinear partial differential equations (such as the critically-important Monge-Ampère equations that appear within its foundations), dynamical systems, inequalities (including isoperimetric inequalities), and geometry. On the applied side, partial answers to the transport question have had significant implications in such fields as economics, statistics, computer vision, physics, fluid dynamics, engineering, biology, and even meteorology.

 

In this talk, we will introduce the basic notions of optimal mass transport theory as well as discuss some interactions between optimal mass transport theory and the geometry of the underlying (and other related) spaces. More particularly, we will present some results concerning the inherited geometric properties of the 2-Wasserstein metric space of probability measures (over its parent space) and state a special case of the so-called curvature-dimension condition arising in the study of optimal transport over Riemannian manifolds. We will also explore the possibility of extending L. Caffarelli’s celebrated contraction theorem to the setting of the standard sphere as well as examine a curvature bound obtained very recently by A. Kolesnikov. We will end with a generalization of a part of E. Calabi’s work concerning Hessian metrics.

 

The majority of this presentation should be accessible to all graduate students, especially those with coursework in differential geometry. All are welcome to attend.

Colloquium: Jaylan Jones (Michigan State University)

 

Title: The Three M’s of Scientific Computing: A High-Performance Solution to the FCH Equation

 

Abstract: Modern computing architecture is advancing our ability to approximate solutions to partial differential equations. I will present an example of how a graphics processing unit (GPU) can make a large problem tractable. The Functionalized Cahn-Hilliard (FCH) equation in three dimensions,

 

ut = Δ(ε2Δ – W″(u) + ε2η)(ε2 ΔuW′(u)),

 

describes pore network formation in a functionalized polymer/solvent system like those used in hydrogen fuel cells. The physical process is defined by multiple time-scales: short time phase separation, long time network growth, and slow evolution to steady state. This requires very long, accurate simulations to correctly describe the physics. I will present a fast, numerically stable, and time-accurate method for solving the FCH equation. Experimentation with several different methods for computing time evolution led to a Fourier spectral method in space with an exponential time integrator giving the desired qualities. Numerical results on a GPU show a 10x speedup over a fully-parallelized eight-core CPU.

Colloquium: Benjamin Webb (Laboratory of Statistical Physics at Rockefeller University)

 

Title: Dynamic Stability of Time-Delayed Networks

 

Abstract: Networks, including computer networks, social networks, and biological networks, have within the past few years attracted an enormous amount of interest. Due to both their physical size and finite processing speeds the dynamics of these networks are inherently time-delayed. In this talk we consider how the global stability of a dynamical network depends on its time-delays. We find that a network’s stability is unaffected by the introduction or removal of undistributed delays and that such delays can be characterized in terms of the network’s graph structure. By extending this technique we also describe how to remove the implicit delays in a network (or general dynamical system) to gain improved estimates of the network’s (system’s) stability. This approach of “restricting” a network is illustrated by applications to certain classes of Cohen-Grossberg neural networks.

Colloquium: Amanda Folsom (Yale University)

 

Title: q-series and quantum modular forms

 

Abstract: While the theory of mock modular forms has seen great advances in the last decade, questions remain. We revisit Ramanujan’s last letter to Hardy, and prove one of his remaining conjectures as a special case of a more general result. Surprisingly, Dyson’s combinatorial rank function, the Andrews-Garvan crank function, mock theta functions, and quantum modular forms, all play key roles. Along these lines, we also show that the Rogers-Fine false theta functions, functions that have not been well understood within the theory of modular forms, specialize to quantum modular forms. This is joint work with K. Ono (Emory U.) and R.C. Rhoades (Stanford U.).

Colloquium: Nathan Grigg (University of Washington)

 

Title: Moduli spaces, deformation theory, and noncommutative varieties

 

Abstract: In algebraic geometry, a moduli space is a space that parametrizes a class of objects, with each object represented by a point in the moduli space.

 

Deformation theory is the study of the neighborhood of a point of the moduli space. For example, the deformation theory of an algebraic variety classifies which varieties are "near" a given algebraic variety in the moduli space.

 

In this talk, I will discuss some examples of things you can learn from basic deformation theory. I will also discuss some recent results in this area which use the tools of modern algebraic geometry to deform an algebraic variety in a noncommutative direction.

Colloquium: Alan Parry (Duke University)

 

Title: Mathematics Meets Astrophysics: An Overview of General Relativity and Dark Matter in Galaxies

 

Abstract: One of the most beautiful intersections of mathematics and physics is Einstein’s theory of gravity, general relativity. It has successfully resolved problems with Newton’s notion of gravity and made incredible predictions about the universe that were later verified by observations. Recently, mathematical motivations for general relativity have presented a possible and tantalizing description of dark matter, a mysterious and exotic form of matter that makes up approximately 23% of the energy density in the universe. In this talk, I will introduce for general audiences the theory of general relativity and the mathematics behind it, discuss the major successes of general relativity, and describe this model for dark matter and the recent advances that have been made in its study.

Colloquium: Christopher Nelson (University of California, San Diego)

 

Title: A Perfect Positivstellensatz in a Free Algebra

 

Abstract: A positivstellensatz is an algebraic certificate for a given polynomial p to have a specific positivity property. Such theorems date back in some form for over one hundred years for conventional (commutative) polynomials. For example, given polynomials p, q, and r, one might ask

 

(Q)                                            Does q(x) ≥ 0 and r(x) = 0 imply p(x) ≥ 0?

 

In this talk I will focus on free noncommutative polynomials p, q, and r and substitute matrices for the variables xj. In the case that the positivity domain D ∶={X | p(X) ⪰ 0} is convex, there exists a "perfect positivstellensatz", which gives an elegant answer to (Q) with strict degree bounds. This non-commutative positivstellensatz has a much cleaner statement than analogous commutative results. In addition, from this result follow a few corollaries, including a non-commutative nullstellensatz as well as results relating to convex optimization.

Colloquium: Mark Allen (Purdue University)

 

Title:  Free Boundary on a Cone

 

Abstract:   The first problems a student encounters in a partial differential equations class involve looking for an unknown function which is a solution to an equation and has prescribed boundary data over the boundary of a fixed domain.However, in many applications in the physical sciences, biology, and even geometry, it becomes necessary to study a PDE with both an a priori unknown function and unknown domain. These types of problems have come to be called "free boundary problems". In this talk I will give some examples of free boundary problems that are notable both for their applications and for their importance in the historical development of this area of mathematics. I will discuss the importance of studying free boundary problems on manifolds that are not necessarily "smooth". I will also discuss recent work on studying the Alt-Caffarelli free boundary problem on a two dimensional cone generated by a smooth curve γ on the sphere. We show that when length(γ) < 2π the free boundary avoids the vertex of the cone. When length(γ) ≥ 2π we provide examples of minimizers such that the vertex belongs to the free boundary. This result is analogous to a well known result for distance minimizing geodesics on a cone. Indeed, the Alt-Caffarelli free boundary is known to behave similarly to minimal surfaces, and our result establishes another similarity between this free boundary problem and minimal surfaces. This is joint work with H. Chang Lara.