Seminars and Colloquia by Series

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.

Uniform bounds on rational points on curves of low Mordell-Weil rank

Series
Algebra Seminar
Time
Wednesday, February 18, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eric Katz – University of Waterloo
In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves. I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform. These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.

Conformal mapping and optimal meshes

Series
Analysis Seminar
Time
Wednesday, February 18, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
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).

Braid Theory: Burau and Gassner Representations

Series
Geometry Topology Student Seminar
Time
Wednesday, February 18, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jamie Conway – Georgia Tech
We will describe the Burau representation of the braid group and the related Gassner representation of the pure braid group. We will explore how the Burau representation is related to the Alexander polynomial.

Stability of Three-dimensional Prandtl Boundary Layers

Series
PDE Seminar
Time
Wednesday, February 18, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 170 (Special)
Speaker
Wang, Yaguang – Shanghai Jiaotong University
In this talk, we shall study the stability of the Prandtl boundary layer equations in three space variables. First, we obtain a well-posedness result of the three-dimensional Prandtl equations under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. Second, we give an instability criterion for the Prandtl equations in three space variables. Both of linear and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, which is an exact complement to the above well-posedness result for a special flow. This is a joint work with Chengjie Liu and Tong Yang.

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