Seminars and Colloquia by Series

On Splash and splat singularities for incompressible fluid interfaces

Series
PDE Seminar
Time
Tuesday, February 24, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Diego Cordoba Gazolaz – ICMAT –
For the water waves system we have shown the formation in finite time of splash and splat singularities. A splash singularity is when the interface remain smooth but self-intersects at a point and a splat singularity is when it self-intersects along an arc. In this talk I will discuss new results on stationary splash singularities for water waves and in the case of a parabolic system a splash can also develop but not a splat singularity.

Loop Spaces, Operads and the Space of Positive Scalar Curvature Metrics

Series
Geometry Topology Seminar
Time
Monday, February 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Walsh – Wichita State
In this talk we will begin by discussing the problem of understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. Recently much progress has occurred in this topic. We will then look at an application of the theory of operads to this problem in the case when the underlying manifold is an n-sphere. In the case when n>2, this space is a homotopy commutative, homotopy associative H-space. In particular, we show that it admits an action of the little n-disks operad. Via theorems of Stasheff, Boardman, Vogt and May, this allows us to demonstrate that the path component of this space containing the round metric, is weakly homotopy equivalent to an n-fold loop space.

Commutator methods for the spectral analysis of time changes of horocycle flows

Series
CDSNS Colloquium
Time
Monday, February 23, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael Tiedra de Aldecoa – Pontificia Univ. Catolica de Chile
We show that all time changes of the horocycle flow on compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes of horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators and the unique ergodicity of the horocycle flow. www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf <http://www.mat.uc.cl/%7Ertiedra/download/Horocycles_Bordeaux_2014.pdf <http://www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf>>

Introduction to regularity theory of second order Hamilton-Jacobi-Bellman equations

Series
PDE Working Seminar
Time
Friday, February 20, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Andrzej Sweich – GeorgiaTech
I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

Vector Fields on Spheres

Series
Geometry Topology Student Seminar
Time
Friday, February 20, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta Kolay – Georgia Tech

Please Note: This is a project for Prof. Wickelgren's course on Stable Homotopy Theory.

In this talk, I will show using Clifford algebras that there are ρ(n)-1 linearly independent vector fields on the unit sphere in the n dimensional Euclidean space, where ρ(n) is the Radon-Hurwitz number.

Conformal mapping and optimal meshes

Series
Analysis Seminar
Time
Thursday, February 19, 2015 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris Bishop – SUNY Stony Brook
The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry. If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all angles \leq 120 degrees and all new angles \geq 60 degrees (small angles in the original polygon must remain).

On models of short pulse type in continuous media

Series
CDSNS Colloquium
Time
Thursday, February 19, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yannan Shen – Univ. of Texas at Dallas
We develop a mathematical model for ultra-short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. The fundamental equation in the model is the short-pulse equation (SPE) which will be derived in frequency band gaps. We use a multi-scale ansatz to relate the SPE to the nonlinear Schroedinger equation, thereby characterizing the change of width of the pulse from the ultra short regime to the classical slow varying envelope approximation. We will discuss families of solutions of the SPE in characteristic coordinates, as well as discussing the global wellposedness of generalizations of the model that describe uni- and bi-directional nonlinear waves.

Harmonic analysis and the geometry of fractals

Series
School of Mathematics Colloquium
Time
Thursday, February 19, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Izabella Laba – University of British Columbia
Singular and oscillatory integral estimates, such as maximal theorems and restriction estimates for measures on hypersurfaces, have long been a central topic in harmonic analysis. We discuss the recent work by the speaker and her collaborators on the analogues of such results for singular measures supported on fractal sets. The common thread is the use of ideas from additive combinatorics. In particular, the additive-combinatorial notion of "pseudorandomness" for fractals turns out to be an appropriate substitute for the curvature of manifolds.

Two combinatorial applications of smooth numbers

Series
Combinatorics Seminar
Time
Wednesday, February 18, 2015 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
We look at two combinatorial problems which can be solvedusing careful estimates for the distribution of smooth numbers. Thefirst is the Ramsey-theoretic problem to determine the maximal size ofa subset of of integers containing no 3-term geometric progressions.This problem was first considered by Rankin, who constructed such asubset with density about 0.719. By considering progressions among thesmooth numbers, we demonstrate a method to effectively compute thegreatest possible upper density of a geometric-progression-free set.Second, we consider the problem of determining which prime numberoccurs most frequently as the largest prime divisor on the interval[2,x], as well as the set prime numbers which ever have this propertyfor some value of x, a problem closely related to the analysis offactoring algorithms.

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