Fall 2016

The Department of Mathematics & Statistics colloquium speakers give 50-minute presentations on various mathematical and statistical topics. Colloquia are schedule from 2:30pm - 3:30pm unless otherwise noted.

If you would like to meet with a speaker, please contact math@unr.edu to schedule a meeting. We look forward to your participation in our upcoming colloquia! 

Colloquium Schedule
Sep 29 Paul Hurtado UNR Biological Dynamics: Research at the Interface of Biology, Statistics & Mathematics
Abstract for Biological Dynamics: Research at the Interface of Biology, Statistics & Mathematics

The now decades-old explosion in computing power has fueled a "quantitative revolution" in the biological sciences. Mathematics and statistics has played a big role in that revolution, and today more than ever before these disciplines are at the forefront of many critically important issues, including antibiotic resistance, cancer treatment and prevention, other areas of medicine, public and environmental health, agriculture, biodiversity, and human behavior. This synergy has lead to clear benefits to biology writ large, but has also lead to new mathematical and statistical challenges. In this talk, I will present an overview of some research projects, focusing primarily on two topics: First, the application of PDE models to investigate how multiple biological trade-offs drive the spatiotemporal evolution of pathogen virulence in a wildlife disease system. Second, I will introduce examples of using ODEs to model specific biological systems, discuss why we want to use those models as statistical models, and some of challenges of attempting to do so (e.g., the challenges of fitting ODE models to data, and evaluating those results). This talk is intended to be accessible to undergraduates as well as faculty.

AB 635
Oct 6 Vit Dolejší Charles University, Prague, Czech Republic Geometrical optimization of triangles with respect to interpolation error
Abstract for Geometrical optimization of triangles with respect to interpolation error

We present a technique which solves the following problem:
Let u be a sufficient regular function and p > 0 a given polynomial degree. We seek the shape of a triangle K with a given area a such that the error of the p degree polynomial interpolation on K is minimal. Furthermore, we present the use of this technique for the adaptive numerical solution of partial differential equations.

Oct 13 Vincent Martinez Tulane University Asymptotic coupling in hydrodynamic equations and its applications to data assimilation and dynamical systems
Abstract for Asymptotic coupling in hydrodynamic equations and its applications to data assimilation and dynamical systems

In their 1967 seminal paper, Foias and Prodi demonstrated that solutions to the two-dimensional (2D) in-compressible Navier-Stokes equations (NSE) has a finite number of "determining modes," i.e., that they satisfy the following property: if the difference between a sufficiently high Galerkin projection of two so-lutions converge to 0 asymptotically in time, then the complementary projection also converges to 0 as-ymptotically in time. In other words, knowledge of the low modes for all large times suffice to "enslave" the high-modes asymptotically in time. This remarkable property has since lead to several developments in the long-time behavior of solutions to the NSE, particularly to the mathematics of turbulence, and more recently, to data assimilation. In this talk, we will discuss recent studies in an approach to data assimila-tion proposed by Azouani, Olson, and Titi, which exploits precisely this asymptotic coupling property. We will also discuss an application inspired by this approach to data assimilation that involves reducing the dynamics of the original model to that of an ordinary differential equation, known as a "determining form." Indeed, the determining form may be viewed as a surrogate for the inertial manifold, whose exist-ence remains an outstanding open problem for the 2D NSE. We will discuss these issues in the context of the 2D NSE, as well as a geophysical equation known as the 2D surface quasi-geostrophic equation; the results presented in these studies is joint work with A. Biswas, M.S. Jolly, E.J. Olson, and E.S. Titi.

Oct 20 He Wang UNR Formality properties in topology and group theory
Abstract for Formality properties in topology and group theory

Formality is a topological property that arises from the rational homotopy theory developed by Quillen and Sullivan in 70’s. Roughly speaking, the rational homotopy type of a formal space is determined by its cohomology algebra. A closely related property of a finitely generated group is 1-formality, which allows one to reconstruct the rational pro-unipotent completion of the group solely from the cup products of degree 1 cohomology classes. In this talk, we will separate 1-formality into two complementary properties: graded formality and filtered formality, by studying various Lie algebras over a field of characteristic 0 attached to such group, including the associated graded Lie algebra, the holonomy Lie algebra, and the Malcev Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups, fundamental groups of Seifert fibered manifolds, and pure virtual braid groups. If time permits, we will introduce another approach to formality properties (proposed by Dror Bar-Natan) provided by Taylor expansions from the group to the completion of the associated graded algebra of the group ring. This talk is about joint work with Alex Suciu.

Oct 27 Cancelled (Faculty Meeting)
Nov 3 Brandon Levine University of Chicago Serre’s conjecture on modular forms
Abstract for Serre’s conjecture on modular forms

The Langlands program is a far-reaching set of conjectural connections between analytic objects (e.g., modular forms) and arithmetic objects (e.g., elliptic curves). In 1987, Serre made a bold conjecture about modular forms in the spirit of a characteristic p Langlands program. Serre's conjecture (now a Theorem due to Khare-Wintenberger and Kisin) has a number of interesting consequences including Fermat's Last Theorem. In this talk, I will introduce through examples modular forms and their arithmetic counterpart elliptic curves and discuss the relationship between Serre's conjecture and Wiles' famous proof of Fermat's Last Theorem. Since 1987, there have been a number of conjectures generalizing Serre's original conjecture to higher dimensions. After introducing these higher dimensional analogues, I will describe recent progress towards the weight part of these conjectures. This is joint work with Daniel Le and Bao V. Le Hung.

AB 635
Nov 10 Alain Valette Institut De Mathematiques, Switzerland The Kadison-Singer problem
Abstract for The Kadison-Singer problem

In 1959, R.V. Kadison and I.M. Singer asked whether each pure state of the algebra of bounded diagonal operators on ℓ2, admits a unique state extension to B(ℓ2). The positive answer was given in June 2013 by A. Marcus, D. Spielman and N. Srivastava, who took advantage of a series of translations of the original question, due to C. Akemann, J. Anderson, N. Weaver,... Ultimately, the problem boils down to an estimate of the largest zero of the expected characteristic polynomial of the sum of independent random variables taking values in rank 1 positive matrices in the algebra of n-by-n matrices.

AB 635
Nov 8 (Tues) Hong Wang TBA
Nov 15 (Tues) Emine Celik UNR Generalized Forchheimer Flows of Compressible Fluids in Heterogeneous Porous Media
Abstract for Generalized Forchheimer Flows of Compressible Fluids in Heterogeneous Porous Media

We study the generalized Forchheimer flows of slightly compressible fluids in heterogeneous porous media where the derived nonlinear partial differential equation for the pressure can be singu-lar and degenerate. Suitable weighted Lebesgue norms for the pressure, its gradient and time derivative are estimated . The continuous dependence on the initial and boundary data is established for the pressure with respect to those corresponding norms. We also obtain the estimates for the L -norms of the pressure in terms of the initial and the time-dependent boundary data. They are established by implementing De Giorgi's iteration in the context of the above weighted norms. The second part of this talk is focused on generalized Forchheimer flows for a larger class of compressible fluids. By using Muskat's and Ward's general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We estimate the L and W1,2-a (0<a<1) norms for the solution on the entire domain in terms of the initial and boundary data. It is carried out by using a suitable trace theorem and an appropriate modification of Moser's iteration. This is a joint work with Luan Hoang and Thinh Kieu.

AB 635