2015-2016

We consider the problem of determining the maximum number of common solutions of a bunch of polynomials over a finite field. The simplest case is of course of a single (nonzero) polynomial in one variable, where the degree usually gives the maximum number of solutions. In the general case of several polynomials in several variables, the problem is meaningful and interesting when the base field is finite and the solutions are sought in the corresponding affine or projective space over the given finite field. When these polynomials are assumed linearly independent and of a degree bounded by a fixed positive integer, the problem is equivalent to a problem in coding theory, namely, that of determining the generalized Hamming weights of Reed-­‐Muller codes. The known solution in this case, due to Heijnen and Pellikaan (1998) uses results in combinatorics such as the Kruskal-­‐Katona theorem.

The case of systems of linearly independent multivariate homogeneous polynomials, all of the same degree, where the zeros are considered in a projective space over the given finite field is perhaps even more interesting. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the maximum value when the degree of the homogeneous polynomials is not too large in comparison to the size of the finite field. Special cases of the conjecture are known to be true, thanks to the results of Serre (1991) and Boguslavsky (1997), but the general case has been open for quite some time.

We will give a motivated account of the above problem and its alternative formulations while briefly explaining the relevant background. We will then describe some recent developments that has led to significant new results concerning the general case.

Joint work with Mrinmoy Datta.

High-­‐throughput screening (HTS) is a large-­‐scale process that screens hundreds of thousands to millions of compounds in order to identify potentially leading candidates rapidly and accurately. There are many statistically challenging issues in HTS. In this talk, I will focus the spatial effect in primary HTS. I will discuss the consequences of spatial effects in selecting leading compounds and why the current experimental design fails to eliminate these spatial effects. A new class of designs will be proposed for elimination of spatial effects. The new designs have the advantages such as all compounds are comparable within each microplate in spite of the existence of spatial effects; the maximum number of compounds in each microplate is attained, etc. Optimal designs are recommended for HTS experiments with multiple controls.

At the beginning of my talk I will present an idea of numerical approach to the ordinary differential equations (ODE), based on a time discretization technique. Next, I will discuss a simple class of partial differential equations (PDE) which lead us to an elliptic problem in linear algebra. I will consider its Galerkin approximation and show a basic result providing a convergence of Galerkin method and its error estimates. I will also present an idea of finite element method (FEM) for elliptic problems. Finally I will pass to evolution problems and show, how to apply both time and spatial discretization technique in this case. In order to illustrate how it works practically, I will present my recent result concerning numerical analysis of an evolution contact problem in mechanics.

What do the Hindu/JainYantrams* and Greek mythologies have to do with industrial experimentations? Apparently, ideas from these can be used to generate some rich classes of complete-mixture experiments, where sum of the ingredients/components /explanatory variables must add to 1. The requirement of sum being 1 implies that all design points must be on the simplex and a complete mixture requires that all ingredients are used with nonzero proportions and hence all design points must lie in the interior of the simplex. We provide details of how to construct these designs, illustrate their wide applicability and their flexibility to adapt to the constrained feasible regions. Not only do these designs work well within interior of the simplex, they can also be easily refined to satisfy the constraints, if there were any, on the mixture components. Leverage values for such designs are more evenly distributed among interior points compared to simplex-lattice or simplex-centroid designs which tend to place higher leverages on the vertices or edge design points where the experiments may not be feasible. Finally, based on the idea of procrustation, we provide an approach alternative to extreme vertices designs. *Yantrams are essentially numeric configurations used in Vedic/Jain religious practices and are closely related to magic squares.

Leveraged exchange traded funds (LETF) are recent and very successful financial innovations in the financial industry. A leveraged exchange-traded fund is a publicly traded mutual fund whose goal is to generate daily returns that are a multiple of the daily returns on some benchmark. In this presentation we will first review the mechanics of the leveraged exchanged traded funds and then derive an empirical formula for the actual leverage ratio (as opposed to target leverage ratio) that actually results at the end of the day, under the assumption that we have a priori fixed percent of hedging demand at level c. Under this scenario, we will also study the dynamic of the compounded returns as well as volatility of LETF’s related to the path of underlying index.

Let p be an odd prime. The distribution of the quadratic residues of p in the set {1, 2, ..., p−1} is a classical topic in number theory which began with Dirichlet's pioneering work in 1839-40, and has attracted intense interest ever since. In this talk, we will be concerned with measuring the size of the set of primes which have a sequence of quadratic residues in given arithmetic progressions. More specifically (but not exactly!), if 𝒜 denotes a given finite set of arithmetic progressions of positive integers such that the cardinality of 𝒜 is at least 2, if s ≥ 2 is a fixed integer, and if U is the union formed from certain arithmetic progressions of length s taken from each element of 𝒜, we wish to calculate the asymptotic density of the set of all primes p such that U is a set of quadratic residues of p. After making this more precise, we will describe an elegant geometric algorithm which performs that calculation.

The interplay between representation theory and combinatorics has been studied for many years. A more recent development in this connection is the notion of a crystal, due to Kashiwara, which may be represented as a colored, directed graph, with Young tableaux as its vertices. My goal is to give a brief introduction to crystals and explain why they're useful. As an application, we will explain how this Young tableaux realization of certain crystals applies to a important formula appearing in automorphic forms.

Metric regularity has been recognized as an important notion in nonlinear analysis, optimal control, and especially optimization due to its role in the justification of the stopping criteria and convergence properties in numerical algorithms. In this talk we focus on the metric regularity property of the subdifferential of a function and establish its relationships with second-order growth conditions, tilt stability of local minimizers, and positive-definiteness properties of the second-order subdifferential. Applications to complexity of algorithms such as forward-backward splitting methods and Newton’s methods for solving convex/nonlinear optimization problems are also obtained.

Cluster Algebra is a new branch in mathematics which grows rapidly and has far-reaching implications in many fields including representation theory, geometry, combinatorics, mirror symmetry of string theory, statistical physics, etc. Lots of research of cluster algebras focuses on construction of their natural bases. Interesting geometric objects, including the quiver Grassmannians and Nakajima's graded quiver varieties, are used to construct some natural bases. Various combinatorial models are discovered in the study of bases, including snake diagrams and perfect matching, Dyck paths and compatible pairs, and in particular, a recent surprising construction of theta bases by Gross, Hacking, Keel and Kontsevich using techniques developed in the study of mirror symmetry of string theory. In this talk, I will discuss some recent advances in the study of these geometries and combinatorics.

Many physical phenomena involve the interaction of a fluid and a poroelastic structure such as the blood vessel interaction. A simple mathematical model for this problem is a coupled problem where blood is described as a free fluid and the vessel wall as a poroelastic medium. The numerical method to solve this problem is based on a decoupling strategy. The coupled fluid-poroelastic system will be cast as a constrained optimization problem with a Neumann type control that enforces continuity of the normal components of the stress on the interface. The optimization objective is to minimize any violation of the other interface conditions. I will present numerical algorithms based on a residual updating technique and show some numerical results to validate the accuracy and efficiency of the proposed algorithms.

Some of the most challenging issues in the management of HIV infection within a host are establishment of latently infected cells, emergence of drug resistance, and opioid dependence. In this talk, I will present within-host mathematical models, consistent with experimental data, that can help address these issues. First, I will show that the latent infection can be limited by early antiretroviral therapy (ART) during acute HIV infection, and this effect may be influenced by the pharmacodynamics properties of antiretroviral drugs. Second, I will show that although administration of ART cannot suppress viral load in many patients due to the emergence of resistance, it can alter the viral fitness resulting in an increase of CD4+ T cell count, which should yield clinical benefits. This benefit depends on the cell proliferation rate, which, in some situations, produces sustained T-cell oscillations. Third, I will discuss how opioid dependence can alter viral dynamics and immune responses.

The talk gives a survey on convergence results for various Newton-type methods including inexact methods. Extensions to non-smooth equations and variational inequalities will be shown. An application to convex spline interpolation will be presented.

Library and information science as a discipline concerns itself with the ways users experience the world of information, in general, and the world of scholarly publications, in the context of academic disciplines. The landscape of journal publications in the sciences has been changing in recent years as more publishing models have emerged, multiplying the ways in which journals are owned, produced, and accessed. These new publishing models have affected what avenues we use for discovering newly published research and how we evaluate the trustworthiness of journals. This talk will focus on techniques and resources that can facilitate the evaluation of journals both for the purposes of being a discerning consumer of article publications and for making targeted journal selections as an author of article publications. Taking a detailed look at the world of journal publishing, I believe, will be immediately and practically relevant to graduate students and faculty in mathematics and statistics. In a broader sense, it will offer a detailed perspective on the structures of scholarship and research in which their own work lives.

Soldiers and marines in the theater of war depend on the vehicles designed and built by the U.S. Army and the American defense industry. As with many other industries, the military ground vehicle community has found modeling and simulation (M&S) to be a very valuable tool for the improvement of their vehicles. A lot of the new methods for improving military ground vehicles are based on M&S, instead of more traditional design-build-test-fix methods. Of course, modeling and simulation of military ground vehicles uses a lot of advanced mathematical and statistical techniques.

The scientists and engineers of the U.S. Army Tank-Automotive Research, Development and Engineering Center (TARDEC) in Warren, Michigan, are always improving ways to give our soldiers the tanks and trucks they deserve, the very best. You might be surprised by the variety of mathematics and statistics courses that play into this vital work. From linear algebra to differential equations, from probability distributions to copulas, from numerical methods to graph theory, there are dozens of classes in the Department of Mathematics and Statistics which are used by the M&S engineers. This demanding application of math and statistics can draw in almost any course in the catalog.

This talk will show some of the challenges in the modeling of military ground vehicles, and highlight a few mathematical and statistical methods which are critical to making the Army better. We will focus on the exacting problem of protecting against an underbody blast, where the M&S is giving us the most “bang for the buck”.

From cryptography to the proof of Fermat's Last Theorem, elliptic curves are ubiquitous in modern number theory. Much activity is focused on developing methods to discover their rational points (those points with rational coordinates). It turns out that finding a rational point on an elliptic curve is very much like finding the proverbial needle in the haystack. In fact, there is no algorithm known to determine the group of rational points on an elliptic curve.

Hyperelliptic curves are also of broad interest; when these curves are defined over the rational numbers, they are known to have finitely many rational points. Nevertheless, the question remains: how do we find these rational points?

I'll summarize some of the interesting number theory behind these curves and briefly describe a technique for finding rational points on curves using (p-adic) numerical linear algebra.

Hybrid­-electric and battery­-electric vehicles are a growing segment of the automotive market, offering improved fuel efficiency with fun-­to­-drive performance. Mathematical models address diverse needs in the automotive arena, such as searching for battery designs that provide greater energy and power or informing vehicle design. In addition, on­board energy management requires a real-­time battery model for state estimation and power prediction. This talk will outline the basic operating principles of lithium-­ion batteries and discuss some aspects of the applicable mathematical models. Special attention will be given to the equivalent circuit models that are used in vehicle powertrain control.

In my talk I will define an iterative construction that produces families of elliptically fibered Calabi­-Yau n­-folds with section from families of elliptic Calabi­-Yau varieties of one dimension lower by a combination of a quadratic twist and a rational base transformation encoded in a generalized functional invariant. Parallel to the geometric construction, we iteratively obtain for each family its Picard­-Fuchs operator and a solution that is holomorphic near the point of maximal unipotent monodormy through a generalization of the classical Euler transform for hypergeometric functions. In particular, our construction yields one­-parameter families of elliptically fibered Calabi­-Yau manifolds with section whose Picard­-Fuchs operators realize all symplectically rigid Calabi­-Yau differential operators that were classified by Bogner and Reiter. This is joint work with Charles Doran.