Title: A priori upper bounds for the inhomogeneous Landau equation

Date: 11/28/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

Speaker: Stan Snelson, University of Chicago

We consider the Landau equation, an integro-differential kinetic model from plasma physics that describes the evolution of a particle density in phase space. It arises as the limit of the Boltzmann equation when grazing collisions predominate. I will give an overview of prior work on the regularity theory of the Landau equation, and describe how to prove a priori upper bounds that decay polynomially in the velocity variable. The key ingredients are precise upper and lower bounds on the coefficients, and tracking how local estimates scale as the velocity grows. I will also explain why the polynomial decay cannot be improved to exponential decay. This talk is based on joint work with Stephen Cameron and Luis Silvestre.

Title: Nonembeddability and Tverberg-type Theorems in Combinatorics

Date: 11/29/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Robert Davis, MSU

It is well-known that there are exactly two 'minimal' non-planar graphs, that is, graphs which cannot be embedded in the plane. It should then come as no surprise that one would like to generalize this combinatorial result to the case of embeddability of simplicial complexes. Topological methods have proven to be very useful here, with one of the landmark theorems being the topological Tverberg theorem of 1966. In this sequence of talks, we survey nonembeddabililty results, a surprising recent development of the topological Tverberg conjecture, and notions related to these topics.

We will discuss the 'group extension' problem, which asks: given groups G and N, in what ways may we construct a group E which has N as a normal subgroup and quotient E/N isomorphic to G? A brief introduction to group cohomology will be given, along with its connections to the extension problem. Time permitting, we will also mention the analogous theory for Lie algebras (i.e. Lie algebra extensions and Lie algebra cohomology).

Title: New Results and New Conjectures on the Monomer-Dimer Problem

Date: 12/01/2016

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Paul Federbush, U. Michigan

The number of configurations of dimers on a large periodic lattice cube that cover a fraction p of the vertices is asymptotically of the form of the exponential of a function of p times the volume V, as V goes to infinity. This function is commonly called the entropy. For hyper-cubical lattices in d dimensions we establish an expression for this function of p, for small p, involving a (convergent) power series in p. With the conjecture that all the terms in this series are positive, confirmed by computation for the first 20 terms in all dimensions, the series is valid for 0<p<1. In fact all analogous terms so far computed for any bipartite lattices have turned out to be positive. Similar results hold for the Virial series, the expansion of the pressure as a power series in p, for a dimer lattice gas, on any lattice. Here there is no restriction to bipartite lattices. By either a spark of creative incite, or a fortuitous fumbling about, one was lead to consider for the monomer-dimer problem on connected regular bipartite graphs the quantity d(i)=ln(N(i)/r^i)-ln(N'(i)/(v-1)^i) where v is the number of vertices, r is the degree, and N(i) and N'(i) are the number of i-dimer configurations on the graph and on the complete graph with the same set of vertices, respectively. This quantity for a sequence of graphs converging to the lattice converges to a multiple of the sum in the entropy expression, if we set p ~ (2i/v). Then the fact that all derivatives of the sum in the entropy are positive (the conjecture holding) leads one to conjecture that all finite difference derivatives of d(i) are positive. The Virial series conjecture leads to a putative upper bound for the same finite difference derivatives. In fact it seems to be true that 'almost all' graphs satisfy these lower and upper finite difference bounds! Looking just at the 0th derivative case one gets new upper and lower conjectured bounds on N(i), higher derivatives yielding a rich set of new bounds to study.

Title: Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension.

Date: 12/01/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Wei Wu, Courant

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and Phi^4 models for d \geq 4. We describe a spin model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some 'continuum escaping probability'. Based on joint works with Greg Lawler and Xin Sun.

Insurance claims have deductibles, which must be considered when pricing for insurance premium. Deductibles may cause censoring and truncation to the observed insurance claims. For this type of data, the regression approach is often used with deductible amount included as an explanatory variable inside a frequency-severity model, so that the resulting coefficient can be used for an assessment of the relativities for deductibles. This approach has the advantage of incorporating the selection effect into deductible ratemaking. On the other hand, standard actuarial textbooks recommend the maximum likelihood approach for estimating parametric loss models, which can be used for calculating the coverage modification amounts due to the deductibles. In this paper, a comprehensive overview of deductible ratemaking is provided, and the pros and cons of various approaches under different parametric models are compared. The regression approach proves to have an advantage in predicting aggregate claims when deductible choices influence the frequency and severity distributions. The maximum likelihood approach becomes necessary for calculating theoretically correct relativities for deductible levels beyond those observed, for each policyholder. For demonstration, loss models are fit to the Wisconsin Local Government Property Insurance Fund data, and examples are provided for the ratemaking of per-loss deductibles offered by the fund. Selected parametric models from the generalized beta family distributions are compared. Models for specific peril types can be combined to improve the ratemaking, and estimation issues for such models under truncation and censoring are discussed.

Title: Modeling Molecular Interactions in Biomolecular Systems

Date: 12/02/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Speaker: Pengyu Ren, University of Texas at Austin

Noncovalent interactions, electrostatic in nature, are essential in biomolecular processes such as protein/RNA folding and binding. Recently we have been systematically investigating the fundamental electrostatic forces including short-range induction and penetration effects by using ab initio Symmetry Adapted Perturbation Theory (SAPT), in order to advance the accuracy and transferability in physics-driven classical mechanics model. In this talk, I will present the development of AMOEBA polarizable multipole based force field along with its applications in molecular dynamics simulations towards understanding protein-ligand binding thermodynamics.