Abstract: In this talk we will survey the classical topological degree theory, and its application to the existence of traveling wave solutions for some reaction diffusion equations and integral difference equations arising from mathematical biology. This talk is based on a series of seminars on the Leray-Schauder degree theory held in the year of 2008-2009. The talk will start from the definition of topological degree, then move on to Leray-Schauder theory. For its application in reaction diffusion equations we will follow the theory developed by A. Volpert, V. Volpert and V. Volpert for systems of parabolic equations. The second application is an ongoing project, we will briefly present the problem, outline the steps to be done to prove the existence of traveling waves to the integral difference equations.

Abstract: Nerve cells convey information through patterns of electrical impulses. Endocrine cells secrete hormones in response to electrical impulses. In both cell types, the impulses often come in periodic bursts, during which a high-frequency series of spikes (active phase) is followed by a quiescent period (silent phase). The dynamic mechanism of bursting has been the focus of attention for more than two decades. In this seminar, I discuss how geometric singular perturbation theory, or fast/slow analysis, is used to understand bursting oscillations. We begin with simple relaxation oscillations, and then progress to ever more exotic behaviors, from bursting to phantom bursting, and finally, to compound bursting.

Abstract: If G is a group generated by U, does U necessarily generate G in a bounded way? More precisely, does there exist a number n such that every element of G can be given as a product of length at most n over U. In 2005 George Bergman proved that this is the case for the symmetric group, and the property became known Bergman's propertry. In this talk we will show that many natural semigroups exhibit the same property as the symmetric group, and we will discuss the related notion of the cofinality of a semigroup.

Abstract: The Tracy-Widom functions describe the limiting distribution of a variety of statistical quantities, including the largest eigenvalue of a random matrix drawn from the Gaussian orthogonal, symplectic, or unitary ensembles (GOE, GSE, or GUE), the longest increasing subsequence of a random permutation, and the outermost particle in a sea of non-intersecting Brownian particles. We obtain new formulas for the Tracy-Widom functions in terms of integrals of Painleve functions. Using these new formulas we find the complete asymptotic expansion of the left-hand tail of the GOE and GSE Tracy-Widom functions for the first time, as well as a second proof of the recently obtained result for the GUE case. We conclude by discussing progress on a new family of "incomplete" Tracy-Widom distributions corresponding to the largest observed eigenvalue if each eigenvalue has a fixed probability of being observed. This is joint work with Jinho Baik and Jeffery DiFranco.

Abstract: An association scheme can be viewed as a partition of a complete graph into regular subgraphs whose adjacency matrices together with the identity matrix form the basis of a matrix algebra. The theory of association schemes has proven useful in several areas of discrete mathematics such as coding theory, finite geometry, and design theory, to name a few. A distance-regular graph is a graph whose distance graphs form an association scheme. Much attention has been paid to association schemes which are generated by distance-regular graphs; however, the formal dual to this type of scheme, known as a "Q-polynomial" scheme, remains less understood. Though few examples of Q-polynomial schemes which do not arise from distance regular graphs are currently known (and most known examples are linked to exceptional lattices, simple groups, designs and codes) in recent years the number of known examples has been steadily growing. In this talk some introductory material on association schemes will be presented, followed by a description of recent progress toward understanding the structure of Q-polynomial schemes.

Abstract: I will split the talk into two parts. The first part will give an update of an old problem of Erdos. The second part will deal with statistical convergence - a summability method.

Abstract: Motivated by problems of current relevance for call centers, we consider a queuing system with a single pool of N identical servers that process incoming customers who have generally distributed service requirements, and abandon the queue if their waiting time exceeds their so-called patience time. We derive a first-order approximation of this system and study its asymptotic behavior, as the number of servers goes to infinity. We also discuss the implications of our analysis for the design of a call center. The analysis involves a range of mathematical tools, from measure-valued processes and renewal theory to partial differential equations.

Abstract: Let G be an edge-colored graph where adjacent edges may be colored the same. A path P in G is a rainbow path if no two edges of P are colored the same. The graph G is rainbow-connected if every two vertices of G are connected by a rainbow path. Generalizations of these concepts are introduced.

Abstract: Graph coloring is the standard way to model many scheduling problems, and it has applications in areas such as register allocation, radio frequency assignment, secret sharing, and even Sudoku puzzles. We will review some major results in graph coloring and one of its popular variants, list-coloring. I will then discuss open problems in these areas and progress that I have made on them.

Abstract: Intersection multiplicity is an important invariant, arising naturally from the study of the intersection of two varieties. I will discuss a definition of it introduced by Serre and some long-standing conjectures related to it. Frobenius endomorphism is one of the useful tools to attack these problems and many other homological conjectures in the characteristic $p$ case. I will briefly discuss some recent results regarding understanding Frobenius endomorphism from homological point of view. No background in commutative algebra or homological algebra is assumed for the audience.

Abstract: The Homological Conjectures are an interconnected set of open problems in the homological theory of modules over commutative rings that have attracted a lot of attention over the past 40 years. In this talk I will discuss Hochster's Direct Summand Conjecture, the Vanishing Maps of Tor Conjecture and the Strong Direct Summand Conjecture. As a window to these problems, I will start with an application to invariant theory. In particular, I will discuss the problem of how to show that the ring of G-invariant polynomials for a group G is ``nice'' in some way. I will finish the talk by discussing some of my work on the Strong Direct Summand Conjecture.

Abstract: Early in the development of twentieth-century mathematics, van de Waerden, Dehn, and others asked questions about the existence of "explicit" solutions to many problems. With the introduction of precise definitions of algorithms in the 1930's, these questions were investigated for some time before the revival in recent decades of "computational mathematics."

In the present talk I will attempt to describe relationships between the classical logical discipline of "computable mathematics" and the modern field of "computational mathematics." Several non-equivalent definitions of computation will be used. Examples will include fields, rings of integers, homotopy groups, the classification of manifolds, and Serre's conjecture on free modules.

Abstract: In 1940, S. M. Ulam asked the following questions: Given a group G₁, a metric group G₂ with metric d(•,•) and a positive number ε does there exist a δ>0 such that if f:G₁→G₂ satisfies d(f(xy),f(x)f(y))≤δ for all x,y in G₁, then a homomorphism $φ:G₁→G₂$ exists with d(f(x),φ(x))≤ε for all x∈G₁? In this talk, I will present some old and recent results on this Ulam's problem.

Abstract: In this talk, we start with a simple SEIR model for malaria transmission dynamics, based on a system of ordinary differential equations, as our baseline model. We derive a formula for the reproductive number and investigate the existence of endemic equilibria. We then introduce a simple two-stage-structured mosquito population model where the mosquito population is divided into two classes. After a brief investigation on this simple stage-structured mosquito model, we incorporate it into the simple SEIR malaria model. We present basic analysis for the combined model and discuss how this combined model can help us study the impact of climate change on the transmission of malaria and other mosquito-borne diseases. We also show that, using the reproductive number as a bifurcation parameter, the simple malaria model and the mosquito-stage-structured model can have a backward bifurcation. We finally talk about the interaction between wild and transgenic mosquitoes and its impact on the malaria transmission.

Abstract: I will discuss a generalized branching random walk as a model for a population breeding and dying in a spatially structured environment. In this model particles are born with higher probability at sites which are not too crowded. I will compare this model with some classical mathematical models. Under general assumptions on the breeding rates we proved the existence of a phase where the population survives without exploding and constructed a nontrivial invariant measure for this case. The results are joint work with G.Posta and F.Zucca.

Abstract: The branching random walk on Z^d exhibits only two behaviors (depending on the breeding parameter): either the process dies out a.s. or with positive probability each site is visited infinitely many times (strong survival). On general graphs (such as trees) there might be values of the parameter such as the process eventually leaves any finite set but does not die out. This is what is called weak survival. Our aim is to describe the critical values of the breeding parameter and discuss the behavior of the process at these critical values, for branching random walks on weighted graphs. The results are joint work with D.Bertacchi.

Abstract: From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that–they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: analysis, model theory, computer algebra, surreal numbers. This talk will be an introduction for the non-specialist mathematician.

Abstract: We survey our constructions of pseudo-random sequences (binary, Z₈, Z_2l, …) from Galois rings. Techniques include a local Weil bound for character sums, and several kinds of Fourier transform. Applications range from cryptography (boolean functions, key generation), to communications (multi-code CDMA), to signal processing (PAPR reduction). This is a joint work with Dmitrii Zinoviev.

Abstract: Rising carbon dioxide levels should increase crop yields. But what is their effect on the nutritional value of our food? It is known that elevating the level of carbon dioxide can significantly reduce the leaf nitrogen content and hence leaf mites' reproduction and renders pesticide unnecessary in greenhouse vegetable production. This raises the question of how resource quality impacts the population dynamics in general. Mathematical biologists have built on variants of the Lotka-Volterra equations and in almost all cases have adopted the physical science's single-currency (energy) approach to understand population dynamics. However, biomass production is essentially a mass transfer process that requires more than just energy. It is crucially dependent on the chemical compositions of both the consumer species and food resources. In this talk, we explore how depicting organisms as built of more than one thing, for example, C to represent energy, and an important nutrient, such as P (or N), to represent quality, results in qualitatively different and realistic predictions about the resulting dynamics.

Abstract: Louis Bachelier invented a mathematical model of Brownian motion in 1900 (five years before Einstein did the same albeit for very different reasons) in order to model the Paris stock market. Bachelier's work forgotten, in the 1960s Paul Samuelson waged a lonely but ultimately successful campaign to convince his peers to use probability to model the stock market, which had always been considered the consequence of actions, and not at all random. The evolution of the last 40 years of these models has been astounding, leading to huge advances in our understanding of risk in the sense of insurance of unusual and often innovative forms, known as financial derivatives. In this talk, we will explain the mathematical interpretation of the economics concept of arbitrage, and en passant we will explain the term and the role of martingales.

Abstract: Collective self-driven synchronized motion of self-propelled particles such as flocking of birds, schooling of fishes, swarming of bacteria, appears in many context in biological organisms, mobile networks and human networks etc. In this talk, I will present kinetic and fluid models derivable from Cucker-Smale's flocking model, and also discuss their mathematical structures and possible applications of flocking mechanism to phototaxis problem arising from biology.

Abstract: It is difficult to find a connected metric space which does not contain any non-trivial separable connected subspace. So far there have been only two such examples given: by R. Pol and P. Simon. We give the third which is a graph of a function from the reals into a non-separable Banach space. In fact this idea provides a certain more general technique for producing special spaces. We also show a positive result when a regular subset of a Banach space must be separably connected (i.e. each two points are contained in a separable connected subspace) in the weak* topology.

This talk is based on a joint paper with Ph.D student Michal Wojcik: "Nonseparably Connected and Punctiform Spaces and Connected Graphs of Functions".

Abstract: In this paper, we discuss several models for the design, analysis and real-time control of intra-plant logistical problems. With a warehouse as the setting, we present (1) a large scale, mixed-integer programming model that can allocate products to areas in a warehouse and thus determine the size of each area; (2) a queuing network model that can analyze designs quickly and accurately with respect to important operational performance measures; and (3) an intelligent agent-based control mechanism to help a material handling system adapt in real-time and effectively to disturbances - external and internal - to the system. Effectiveness of this approach is illustrated with numerical examples and a real-world application.

Abstract: Divisible codes were introduced by H. N. Ward in 1981. A q-ary divisible code is a linear code over the field of q elements whose codewords all have weights divisible by some integer Δ>1, where Δ is called a divisor of the code. Ward proved a divisible code bound on dimension of a divisible code when the weight spectrum is given. However, bound on dimension of divisible codes in terms of code length and divisibility level answers the most fundamental question in coding theory for divisible codes. In this talk, we will give an exact upper bound for the dimension of binary divisible codes in this sense and prove the uniqueness up to equivalence of the code attaining this bound, given the hypothesis that a certain nonzero weight exists. We will also see that the hypothesis is true for level 3 codes of maximum dimension with relatively short lengths.

Abstract: I will discuss recent results on the stability of stationary solutions for the Cahn-Hilliard equation in ℝ^d, d≥1. For the case d = 1, there are precisely three types of non-constant bounded stationary solutions, periodic solutions, pulse-type (reversal) solutions, and monotonic transition fronts. These solutions can be categorized as follows: the periodic and reversal solutions are both spectrally unstable, while the transition fronts are nonlinearly (phase-asymptotically) stable. The cases d≥2 are more complicated, and I will discuss what is known about stationary solutions in these cases. Particular emphasis will be placed on planar transition front (or "kink") solutions.

Abstract: Diaconis-Sturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families, based on the algebraic theory of toric ideals. This algorithm is applied to categorical data analysis through the notion of Markov bases. Initiated with its application to Markov chain Monte Carlo approach for testing statistical fitting of the given model, many researchers have extensively studied the structure of Markov bases for models in computational algebraic statistics. In the Markov chain Monte Carlo approach for testing statistical fitting of the given model, a Markov basis is a set of moves connecting all contingency tables satisfying the given margins.

It has been well-known that for two-way contingency tables with fixed row sums and column sums the set of square-free moves of degree two forms a Markov basis. However when we impose an additional constraint that the sum of a subtable is also fixed, then these moves do not necessarily form a Markov basis. Thus, in this paper, we show a necessary and sufficient condition on a subtable so that the set of square-free moves of degree two forms a Markov basis.

The paper on which this talk is based can be found at http://arxiv.org/abs/0708.2312. Slides will be made available at http://www.ms.uky.edu/~ruriko/pdf/Louisville.pdf

Abstract: In this lecture, I will review the role of mathematics in epidemiology and proceed to outline some of its contributions to the challenges posed by emergent diseases like SARS, tuberculosis and influenza.

Abstract: Every digraph of size m has a directed cut of size at least m/4 + Θ(sqrt(m)). This bound eventually improves when a restricted subfamily of digraphs is taken into consideration. For instance, if the maximum outdegree of the digraph is k, then it has a cut of size at least m/4 + m/(8k + 4). We investigate the size of the maximum directed cut for the larger family of digraphs in which each vertex has either indegree at most k or outdegree at most k.

Abstract: Self-dual codes over finite fields are a widely studied subject. Recently a great deal of attention has been given to self-dual codes over a variety of rings. It is known that all self-dual codes over Z_m can be found by applying the Chinese Remainder Theorem to self-dual codes over Z_p^e for p a prime. Hence, it is important to classify self-dual codes over the integers modulo prime powers, since this classification will give the classification over Z_m. This presentation will consider self-dual codes over the rings Z₈ and Z₉. Various weights and weight enumerators over these rings will be described. The torsion codes over these rings will be examined to characterize the structure of self-dual codes. Finally, the classification of self-dual codes of small lengths over Z₈ and Z₉ will be given.

Abstract: Some economic models, such as the cash-in-advance model of money of overlapping generations model, have the property that the dynamics are ill-defined going forward in time, but well defined going backward in time, often via a continuous function on a compact metric space called the "backwards map". We analyze such models using inverse limit spaces. Specifically, we recall a construction of a measure on the inverse limit space induced by an invariant measure on the factor space. We show (1) that if the measure on the factor space is "natural", then so is the induced measure on the inverse limit space; and (2) that integration of continuous functions from the inverse limit space to the reals makes sense with respect to the induced measure. We then compute the integral of a utility function associated with the economic model, thus obtaining an expected value for the utility function; and make some conclusions about what this says for the economic model.