Math Department Events Listing

Come schmooze with your fellow mathematicians-at-arms. Enjoy fine cookies, teas, and conversation, or try your hand at a variety of games.

This talk is a logical continuation of my earlier lectures on asymptotic reductions of stochastically perturbed gradient flows. Whereas the SDE case may be analyzed rigorously there still exist many difficulties in the treatment of the SPDE case. I will illustrate some of the methods and ideas on the example of reduction of stochastically perturbed Allen-Cahn-type equations to diffusion-annihilation processes (system of interacting Brownian particles annihilating on collision).

The McKay conjecture is one of the outstanding open problems in group thory. It says that for a given prime p, the number of characters of G of degree not divisible by p is equal to the number of characters of N of degree not divisible by p, where N is the normalizer of a Sylow p-subgroup of G. In this talk we will discuss the McKay conjecture and some of its many reformulations. Along the way, we'll see some block theory and some interesting results in the representation theory of finite groups.

There are many interesting open questions about the behavior of Ricci flow solutions that exist for all time but fail to converge in the usual sense. We will survey some of these problems and their applications. Recent work (e.g. of Baird--Danielo, Glickenstein, Lauret, and Lott) reveals that there is a class of reasonable model spaces for such solutions. We will describe some progress in proving the asymptotic stability of these models.

There will be no Mathematics Instruction Colloquium this week. We will resume on Tuesday, November 6.

All CoS Staff and Academic Professionals are invited to a “Dean Time” meeting. Dean Ruiz will give a brief State of the College overview and hold a question & answer session.

We will discuss particle configurations of a zig-zag path in a domino tiling of an Aztec diamond. The Lindstrom-Gessel-Viennot theorem, which counts the weight of a collection of non-intersecting paths using a determinant, will be presented and then proved. I will then use this result to show an equivalence with the probability of a given particle configuration of a random Aztec tiling and the measure on Krawtchouk ensemble of Orthogonal Polynomials, involving the square of a Van der Monde determinant; Hooray for Linear Algebra!

We consider a gradient interface model with interaction potential which is a non-convex pertubation of a convex potential. We show the strict convexity of the surface tension using a multiple scale analysis. This is an extension of Funaki and Spohn's result, where the strict convexity of potential was crucial in their proof. Using decoupling of the nearest neighbouring vertices, we also give an estimate of the covariances for the gradients Gibbs states. Joint work with J.D. Deuschel.

Mathworks will be presenting on-site training for Matlab. Detailed agenda is TBA.

Come schmooze with your fellow mathematicians-at-arms. Enjoy fine cookies, teas, and conversation, or try your hand at a variety of games.

Sobolev gradients are an efficient method of calculating solutions to a wide variety of systems of partial differential equations. Successful applications have been made to problems in transonic flow, Ginsburg-Landau equations for superconductivity, elasticity, minimal surfaces and oil-water separation problems.

I will discuss why the poor numerical performance of ordinary gradients and the good performance of Sobolev gradients give an instance of the first law of numerical analysis: “Analytical Difficulties and Numerical Difficulties Always Come in Pairs.”

I will show how the method can also be successfully applied to the hyperbolic fully non-linear Monge-Ampère equation: Det(D²z) = κ (1 + z_x² + z_y²)² on the unit square, where κ = −1, represents the Gaussian curvature of the surface described by z.

It is an open question whether every quadratic form in at least nine variables over the rational function field in one variable over a p-adic field has a nontrivial zero. We shall trace the history of this question and sketch a proof of an affirmative answer to this question for nondyadic p-adic fields. More generally, every quadratic form in nine variables over function fields of nondyadic p-adic curves has a nontrivial zero. (Joint work with V. Suresh)

It is an open question whether every quadratic form in at least nine variables over the rational function field in one variable over a p-adic field has a nontrivial zero. We shall trace the history of this question and sketch a proof of an affirmative answer to this question for nondyadic p-adic fields. More generally, every quadratic form in nine variables over function fields of nondyadic p-adic curves has a nontrivial zero. (Joint work with V. Suresh)

Fun, fun, fun!

In the circulatory system, red blood cells are not distributed uniformly. In fact, two vessels of exactly the same size may contain significantly different percentages of red blood cells. One reason for this nonuniformity is that at diverging vessel bifurcations, where blood flows from one large vessel into two smaller branch vessels, the percentage of total blood entering one of the smaller vessels is not usually proportional to the percentage of red blood cells entering that vessel.

I have used a two-dimensional model to produce trajectories of red blood cells traveling through small diverging vessel bifurcations. This model has allowed us to investigate where red blood cells travel in vessel bifurcations as a function of percentage of total blood entering either branch, the angles at which the vessels branch, and the relative size of the branches. Comparing our model predictions with experiments has offered us important insight into red blood cell motion and distribution.

We shall look at examples of classes of 2-dimensional fields which include 2-dimensional strict henselian fields and function fields of surfaces over algebraically closed fields. Arithmetic properties like `index=exponent' for central simple algebras hold for these fields. We shall explain how this leads to an understanding of principal homogeneous spaces under linear algebraic groups defined over these fields.

More than a decade ago, the phenomena of long distance propagation and filamentation were discovered in high-power, ultrashort light pulses propagating in gaseous and condensed bulk media. This instigated keen interest and earnest research, motivated by both the basic physics and many practical applications. A great deal of our knowledge concerning these phenomena originates in modeling and numerical simulations. Realistic numerical modeling continues to provide new insights into the underlying physics that would be extremely difficult to understand from experiments alone. Here, I present an overview of numerical methods capable to study the highly nonlinear processes that accompany optical filamentation, such as explosive broadening of spectrum, plasma generation and harmonic generation. I will also discuss an effective nonlinear weave mixing paradigm that provides us with an intuitive yet quantitative way to understand complex experimental data.