Monday, March 7, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Brittany Froese – New Jersey Institute of Technology
The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a variety of problems. However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, Monge-Ampere equations, and non-continuous solutions of the prescribed Gaussian curvature equation.
Monday, March 7, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Qingtian Zhang – Penn State University
Abstract: In this talk, I will present the uniqueness of conservative
solutions to Camassa-Holm and two-component Camassa-Holm equations.
Generic regularity and singular behavior of those solutions are also
studied in detail. If time permitting, I will also mention the recent
result on wellposedness of cubic Camassa-Holm equations.
Friday, March 4, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Changhui Tan – Rice University
Self-organized behaviors are very common in nature and human societies:
flock of birds, school of fishes, colony of bacteria, and even group of people's
opinions. There are many successful mathematical models which capture the large
scale phenomenon under simple interaction rules in small scale. In this talk, I
will present several models on self-organized dynamics, in different scales: from
agent-based models, through kinetic descriptions, to various types of hydrodynamic
systems. I will discuss some recent results on these systems including existence of
solutions, large time behaviors, connections between different scales, and
numerical implementations.
I will give an elementary introduction to Majid's theory of braided groups and how this may lead to a more geometric, less quantum, interpretation of knot invariants such as the Jones polynomial. The basic idea is set up a geometry where the coordinate functions commute according to a chosen representation of the braid group. The corresponding knot invariants now come out naturally if one attempts to impose such geometry on the knot complement.
Friday, March 4, 2016 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 256
Speaker
Ben Cousins – Georgia Tech
I will give a tour of high-dimensional sampling algorithms, both from a theoretical and applied perspective, for generating random samples from a convex body. There are many well-studied random walks to choose from, with many of them having rigorous mixing bounds which say when the random walk has converged. We then show that the techniques from theory yield state-of-the-art algorithms in practice, where we analyze various organisms by randomly sampling their metabolic networks.This work is in collaboration with Ronan Fleming, Hulda Haraldsdottir ,and Santosh Vempala.
Friday, March 4, 2016 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Jiayin Jin – Georgia Tech
In this talk, I will state the main results of center manifold theory for finite dimensional systems and give some simple examples to illustrate their applications. This is based on the book “Applications of Center Manifold Theory” by J. Carr.
Thursday, March 3, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Trapa – University of Utah
Unitary representations of Lie groups appear in many guises in
mathematics: in harmonic analysis (as generalizations of classical
Fourier analysis); in number theory (as spaces of modular and
automorphic forms); in quantum mechanics (as "quantizations" of
classical mechanical systems); and in many other places. They have
been the subject of intense study for decades, but their
classification has only recently emerged. Perhaps
surprisingly, the classification has inspired connections with
interesting geometric objects (equivariant mixed Hodge modules on
flag varieties). These connections have made it possible to extend
the classification scheme to other related settings.
The purpose of this talk is to
explain a little bit about the history
and motivation behind the study of unitary representations and offer
a few hints about the algebraic and geometric ideas which enter into
their study. This is based on joint work with Adams, van Leeuwen,
and Vogan.
Thursday, March 3, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arnaud Marsiglietti – IMA, University of Minnesota
In the late 80's, several relationships have been established
between the Information Theory and Convex Geometry, notably
through the pioneering work of Costa, Cover, Dembo and Thomas.
In this talk, we will focus on one particular relationship. More
precisely, we will focus on the following conjecture of Bobkov,
Madiman, and Wang (2011), seen as the analogue of the
monotonicity of entropy in the Brunn-Minkowski theory:
The inequality
$$ |A_1 + \cdots + A_k|^{1/n} \geq \frac{1}{k-1} \sum_{i=1}^k
|\sum_{j \in \{1, \dots, k\} \setminus \{i\}} A_j |^{1/n}, $$
holds for every compact sets $A_1, \dots, A_k \subset
\mathbb{R}^n$. Here, $|\cdot|$ denotes Lebesgue measure in
$\mathbb{R}^n$ and $A + B = \{a+b : a \in A, b \in B \}$ denotes
the Minkowski sum of $A$ and $B$.
(Based on a joint work with M. Fradelizi, M. Madiman, and A.
Zvavitch.)
While the fields named in the title seem unrelated, there is a strong
link between them. This amazing connection came to life during a meeting
between Freeman Dyson and Hugh Montgomery at the Institute for Advanced
Study. Random matrices are now known to predict many number theoretical
statistics, such as moments, low-lying zeros and correlations between
zeros. The goal of this talk is to discuss this connection, focusing on
number theory. We will cover both basic facts about the zeta functions
and recent developments in this active area of research.