Title: Around Waring problem for homogeneous polynomials

Date: 11/07/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Waring problem for homogeneous polynomials (forms) asks to represent a given form of degree k*d as a sum of d-th powers of forms of degree k. The main objective is to find a presentation with a small number of summands. The classical case going back to J.J. Sylvester deals with k=1 and binary forms. We will survey some of the results in this area and pose some elementary looking open problems. No preliminary knowledge of the topic is required

Speaker: Guozhen Lu, University of Connecticut; Nobody Else

Title: Fourier analysis on hyperbolic spaces and sharp higher order Hardy-Sobolev-Maz'ya inequalities

Date: 11/07/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this talk, we will describe some recent works on
the sharp higher order Hardy-Sobolev-Maz'ya and Hardy-Adams inequalities on hyperbolic balls and half spaces. The relationship between the classical Sobolev inequalities and the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives will be established. Our main approach is to use the Fourier analysis on hyperbolic spaces and Green's function estimates.

Title: Topological Hochschild Homology and Higher Characters

Date: 11/08/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

In this talk I'll explain how Hochschild homology and duality theory in bicategories can be used to obtain interesting Euler characteristic-type invariants in a number of mathematical contexts (all of the terms in the previous sentence will be explained). A topological refinement, using THH, of this reasoning very easily yields interesting fixed point invariants, such as the Lefschetz trace and Reidemeister trace. Using this, one can show that the cyclotomic trace from algebraic K-theory is computing fixed point invariants. Time permitting, I'll explain how zeta functions relate to the above. Prerequisites: an appetite for category theory, and a belief in, but not knowledge of, the stable homotopy category.

Title: Singularity Formation in General Relativity

Date: 11/08/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The celebrated Hawking–Penrose theorems are breakdown results for solutions to the Einstein equations of general relativity, which are a system of highly nonlinear wave-like PDEs. These theorems show that, under appropriate assumptions on the matter model, a large, open set of initial data lead to geodesically incomplete solutions. However, these theorems are “soft” in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is tied to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness stemming from lack of information for how to uniquely continue the solution (this is roughly known as the formation of a Cauchy horizon). In various works, some joint with I. Rodnianski, we have obtained the first results in more than one spatial dimension that eliminate the ambiguity for an open set of initial data: for the solutions that we studied, the incompleteness is tied to the blowup of various spacetime curvature scalars along a spacelike hypersurface. Physically, this phenomenon corresponds to the stability of the Big Bang and/or Big Crunch singularities. From an analytic perspective, the main theorems are stable blowup results for quasilinear systems of elliptic-hyperbolic PDEs. In this talk, I will provide an overview of these results and explain how they are tied to some of the main themes of investigation by the mathematical general relativity community. I will also discuss the role of geometric and gauge considerations in the proofs, as well as intriguing connections to other problems concerning stable singularity formation.

Title: Optimal transport or: How I learned to stop worrying and grew my own shipping empire

Date: 11/09/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Optimal transport, a.k.a. the Monge-Kantorovich problem has been an active area of mathematics recently. It is an optimization problem, but has connections to PDEs, geometry, economics, image processing, kinetics, and probability, among other areas. I will start with the discrete optimal transport problem and explain some results about existence, duality, and computation. If time permits, I will discuss the continuous framework and some other variations of the problem.

Speaker: Wenrui Hao, Pennsylvania State University

Title: Computational modeling for cardiovascular risk evaluation

Date: 11/09/2018

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Atherosclerosis, the leading cause of death in the United State, is a disease in which a plaque builds up inside the arteries. The LDL and HDL concentrations in the blood are commonly used to predict the risk factor for plaque growth. In this talk, I will describe a recent mathematical model that predicts the plaque formation by using the combined levels of (LDL, HDL) in the blood. The model is given by a system of partial differential equations within the plaque with a free boundary. This model is used to explore some drugs of regression of a plaque in mice, and suggest that such drugs as used for mice may also slow plaque growth in humans. Some mathematical questions, inspired by this model, will also be discussed. I will also mention briefly some related projects about abdominal aortic aneurysm (AAA) and red blood cell aggregation, which would have some potential blood biomarkers for diagnosis of AAA.