- You are here:
- GT Home
- Home
- News & Events

Series: Analysis Seminar

Solutions of the simplest of the Painleve equations, PI, y'' = 6y^2+x, exhibit surprisingly rich asymptotic properties as x is large. Using the Riemann-Hilbert problem approach, we find an exponentially small addition to an algebraically large background admitting a power series asymptotic expansion and explain how this "beyond of all orders" term helps us to compute the coefficient asymptotics in the preceding series.

Series: PDE Seminar

The Cauchy problem for the Poisson-Nernst-Planck/Navier-Stokes model was investigated by the speaker in [Transport Theory Statist. Phys. 31 (2002), 333-366], where a local existence-uniqueness theory was demonstrated, based upon Kato's framework for examining evolution equations. In this talk, the existence of a global distribution solution is proved to hold for the model, in the case of the initial-boundary value problem. Connection of the above analysis to significant applications is discussed. The solution obtained is quite rudimentary, and further progress would be expected in resolving issues of regularity.

Series: Combinatorics Seminar

The entropy function has a number of nice properties that make it a useful
counting tool, especially when one wants to bound a set with respect to the set's
projections. In this talk, I will show a method developed by Mokshay Madiman, Prasad
Tetali, and myself that builds on the work of Gyarmati, Matolcsi and Ruzsa as well as
the work of Ballister and Bollobas. The goal will be to give a black-box method for
generating projection bounds and to show some applications by giving new bounds on
the sizes of Abelian and non-Abelian sumsets.

Series: CDSNS Colloquium

I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.

Series: Geometry Topology Seminar

Joint meeting at Emory

A k--dimensional Dehn function of a group gives bounds on the volumes of (k+1)-balls which fill k--spheres in a geometric model for the group. For example, the 1-dimensional Dehn function of the group Z^2 is quadratic. This corresponds to the fact that loops in the euclidean plane R^2 (which is a geometric model for Z^2) have quadratic area disk fillings. In this talk we will consider the countable sets IP^{(k)} of numbers a for which x^a is a k-dimensional Dehn function of some group. The situation k \geq 2 is very different from the case k=1.

Series: Geometry Topology Seminar

Joint meeting at Emory

Recall that an open book decomposition of a 3-manifold M is a link L in M whose complement fibers over the circle with fiber a Seifert surface for L. Giroux's correspondence relates open book decompositions of a manifold M to contact structures on M. This correspondence has been fundamental to our understanding of contact geometry. An intriguing question raised by this correspondence is how geometric properties of a contact structure are reflected in the monodromy map describing the open book decomposition. In this talk I will show that there are several interesting monoids in the mapping class group that are related to various properties of a contact structure (like being Stein fillable, weakly fillable, . . .). I will also show that there are open book decompositions of Stein fillable contact structures whose monodromy cannot be factored as a product of positive Dehn twists. This is joint work with Jeremy Van Horn-Morris and Ken Baker.

Series: Combinatorics Seminar

I will present an approximation algorithm for the following problem: Given a graph G and a parameter k, find k edges to add to G as to maximize its algebraic connectivity. This problem is known to be NP-hard and prior to this work no algorithm was known with provable approximation guarantee. The algorithm uses a novel way of sparsifying (patching) part of a graph using few edges.

Series: SIAM Student Seminar

Suppose b is a vector field in R^n such that b(0) = 0. Let A = Jb(0) the Jacobian matrix of b at 0. Suppose that A has no zero eigenvalues, at least one positive and at least one negative eigenvalue. I will study the behavior of the stochastic differential equation dX_\epsilon = b(X_\epsilon) + \epsilon dW as \epsilon goes to 0. I will illustrate the techniques done to deal with this kind of equation and make remarks on how the solution behaves as compared to the deterministic case.

Series: ACO Student Seminar

In the first part of the talk, I am going to give an introduction and overview of linear and semidefinite programming hierarchies. I will mostly review known integrality gaps for such programs and try to give an intuition of why we currently lack strong techniques for designing rounding algorithms. In the second part of the talk I will focus on the stronger SDP Lasserre hierarchy. In contrast with the previous LP and SDP hierarchies, very few examples of integrality gap instances are known to date. I will present a recent technique for designing such instances and discuss open problems in the area.

Series: School of Mathematics Colloquium

In this talk I will outline recent results of G-Q Chen, Dehua Wang, and me on the problem of isometric embedding a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Remarkably there is very pretty duality between this problem and the equations of steady 2-D gas dynamics. Compensated compactness (L.Tartar and F.Murat) yields proof of existence of solutions to an initial value problem when the prescribed metric is the one associated with the catenoid.