Seminars and Colloquia by Series

Rigorous validation of Radially Symmetric Stationary Solutions of PDEs.

Series
CDSNS Colloquium
Time
Monday, May 2, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
C.M. Groothedde – V.U. Amsterdam
We shall take a look at computer-aided techniques that can be used to prove the existence of stationary solutions of radially symmetric PDEs. These techniques combine existing numerical methods with functional analytic estimates to provide a computer-assisted proof by means of the so-named 'radii-polynomial' approach.

Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space

Series
PDE Seminar
Time
Friday, April 29, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Adrian Tudorascu – West Virginia University
SGSW is a third level specialization of Navier-Stokes (via Boussinesq, then Semi-Geostrophic), and it accurately describes large-scale, rotation-dominated atmospheric flow under the extra-assumption that the horizontal velocity of the fluid is independent of the vertical coordinate. The Cullen-Purser stability condition establishes a connection between SGSW and Optimal Transport by imposing semi-convexity on the pressure; this has led to results of existence of solutions in dual space (i.e., where the problem is transformed under a non-smooth change of variables). In this talk I will present recent results on existence and weak stability of solutions in physical space (i.e., in the original variables) for general initial data, the very first of their kind. This is based on joint work with M. Feldman (UW-Madison).

Some new non-asymptotic results about the accuracy of the weighted bootstrap

Series
Stochastics Seminar
Time
Thursday, April 28, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mayya Zhilova – School of Mathematics, Georgia Tech
The bootstrap procedure is well known for its good finite-sample performance, though the majority of the present results about its accuracy are asymptotic. I will study the accuracy of the weighted (or multiplier) bootstrap procedure for estimation of quantiles of a likelihood ratio statistic. The set-up is the following: the sample size is bounded, random observations are independent, but not necessarily identically distributed, and a parametric model can be misspecified. This problem had been considered in the recent work of Spokoiny and Zhilova (2015) with non-optimal results. I will present a new approach improving the existing results.

Global well-posedness for the Cubic Dirac equation in the critical space

Series
PDE Seminar
Time
Wednesday, April 27, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Ioan Bejenaru – University of California, San Diego
We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space. The theory we develop is the Klein-Gordon counterpart of the Wave Maps / Schroedinger Maps theory. This is joint work with Sebastian Herr.

The Z_2^n Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

Series
Analysis Seminar
Time
Wednesday, April 27, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vincent Genest – MIT
In this talk, I will discuss the n-dimensional Dirac-Dunkl operator associated with the reflection group Z_2^{n}. I will exhibit the symmetries of this operator, and describe the invariance algebra they generate. The symmetry algebra will be identified as a rank-n generalization of the Bannai-Ito algebra. Moreover, I will explain how a basis for the kernel of this operator can be constructed using a generalization of the Cauchy-Kovalevskaia extension in Clifford analysis, and how these basis functions form a basis for irreducible representations of Bannai-Ito algebra. Finally, I will conjecture on the role played by the multivariate Bannai-Ito polynomials in this framework.

Dynamical problems in Hamiltonian PDEs

Series
Research Horizons Seminar
Time
Wednesday, April 27, 2016 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Zhiwu Lin – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided before the seminar.

Many conservative PDE models can be written in a Hamiltonian form. They include Euler equations in fluids, Vlasov models for plasmas and galaxies, ideal MHD for plasmas, Gross–Pitaevskii equation for superfluids and Bose-Einstein condensates, and various water wave models (KDV, BBM, KP, Boussinesq systems etc). I will describe some dynamical problems of these models, from a more unifying point of view by using their Hamiltonian forms. They include: stability/instability of coherent states (steady solution, traveling waves, standing waves etc.), invariant manifolds near unstable states, and inviscid and enhanced damping in fluids and plasmas. It is a topic course that will be taught in the fall.

Uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n

Series
Dissertation Defense
Time
Monday, April 25, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chenchen Mou – Georgia Institute of Technology –
The main goal of the thesis is to study integro-differential equations. Integro-differential equations arise naturally in the study of stochastic processes with jumps. These types of processes are of particular interest in finance, physics and ecology. In the thesis, we study uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n.

Hurewicz maps for infinite loopspaces

Series
Geometry Topology Seminar
Time
Monday, April 25, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nicholas J. Kuhn – University of Virginia
In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1,2,4, 8. This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace. I have been studying Hurewicz maps for infinite loopspaces by showing how a filtration of the homotopy groups coming from stable homotopy theory (the Adams filtration) interacts with a filtration of the homology groups coming from infinite loopspace theory. There are some clean and tidy consequences that lead to a new proof of Milnor's theorem, and other applications.

Hamiltonian Instability in a Four-Body Problem

Series
CDSNS Colloquium
Time
Monday, April 25, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marian Gidea – Yeshiva Univ.
We consider a restricted four-body problem, modeling the dynamics of a light body (e.g., a moon) near a Jupiter trojan asteroid. We study two mechanisms of instability. For the first mechanism, we assume that the orbit of Jupiter is circular, and we investigate the hyperbolic invariant manifolds associated to periodic orbits around the equilibrium points. The conclusion is that the light body can undergo chaotic motions inside the Hill sphere of the trojan, or well outside that region. For the second mechanism, we consider the perturbative effect due to the eccentricity of the orbit of Jupiter. The conclusion is that the size of the orbit of the light body around the trojan can keep increasing, or keep decreasing, or undergo oscillations. This phenomenon is related to the Arnold Diffusion problem in Hamiltonian dynamics

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