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Series: Research Horizons Seminar

Orthogonal polynomials play a role in myriads of problems ranging from approximation theory to random matrices and signal processing. Generalizations of orthogonal polynomials - such as biorthogonal polynomials, cardinal series, Muntz polynomials, are used for example, in number theory and numerical analysis. We discuss some of these, and some potential research projects involving them.

Series: Research Horizons Seminar

This talk is not an appetizer to pizza, but rather an appetizer to the main course: Hua Xu's and Trevis Litherland's thesis defenses which will respectively take place on Thursday the 30th of October and November the 6th, in Skiles 269, at 3pm. I will present the history and origins of the problems they have been tackling ("Ulam's problems"). Various interactions with other fields such as Analysis, Algebra (Young Tableaux) or Bioinformatics (Sequence Comparison) will be touched upon. Then, some elementary but rather useful probabilistic techniques will also be introduced and shown how to be applied.

Series: Research Horizons Seminar

We consider the pseudodifferential operators H_{m,\Omega} associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian when restricted to a compact domain \Omega in {\mathbb R}^d. When the mass m is 0 the operator H_{0,\Omega} coincides with the generator of the Cauchy stochastic process with a killing condition on \partial \Omega. (The operator H_{0,\Omega} is sometimes called the fractional Laplacian with power 1/2.) We prove several universal inequalities for the eigenvalues (joint work with Evans Harrell).

Series: Research Horizons Seminar

In the study of one dimensional dynamical systems it is often assumed that the functions involved have a negative Schwarzian derivative. However, as not all one dimensional systems of interest have this property it is natural to consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class, that is to functions with an eventual negative Schwarzian derivative. The property of having an eventual negative Schwarzian derivative is nonasymptotic therefore verification of whether a function has such an iterate can often be done by direct computation. The introduction of this class was motivated by some maps arising in neuroscience.

Series: Research Horizons Seminar

In this introduction to knot theory we will focus on a class of knots called rational knots. Here the word rational refers to a beautiful theorem by J. Conway that sets up a one to one correspondence between these knots and the rational numbers using continued fractions. We aim to give an elementary proof of Conway's theorem and discuss its application to the study of DNA recombination. No knowledge of topology is assumed.

Series: Research Horizons Seminar

A Turning point is where solutions to differential equations change behavior from exponential to oscillatory. In this region approximate solutions given by the powerful WKB method break down. In a series of paper in the 30's and 40's Langer developed a transformation (the Langer transformation) that allows the development of good approximate solutions (in terms of Airy functions) in the region of the Turning point I will discuss a discrete analog of this transformation and show how it leads to nice asymptotic formulas for various orthogonal polynomials.

Series: Research Horizons Seminar

Dynamics of spatially extended systems is often described by Lattice Dynamical Systems (LDS). LDS were introduced 25 years ago independently by four physicists from four countries. Sometimes LDS themselves are quite relevant models of real phenomena. Besides, very often discretizations of partial differential equations lead to LDS. LDS consist of local dynamical systems sitting in the nodes of a lattice which interact between themselves. Mathematical studies of LDS started in 1988 and introduced a thermodynamic formalism for these spatially extended dynamical systems. They allowed to give exact definitions of such previously vague phenomena as space-time chaos and coherent structures and prove their existence in LDS. The basic notions and results in this area will be discussed. It is a preparatory talk for the next day colloquium where Dynamical Networks, i.e. the systems with arbitrary graphs of interactions, will be discussed.

Series: Research Horizons Seminar

A plasma is a gas of ionized particles. For a dilute plasma of very high temperature, the collisions can be ignored. Such situations occur, for example, in nuclear fusion devices and space plasmas. The Vlasov-Poisson and Vlasov-Maxwell equations are kinetic models for such collisionless plasmas. The Vlasov-Poisson equation is also used for galaxy evolution. I will describe some mathematical results on these models, including well-posedness and stability issues.

Series: Research Horizons Seminar

I will explain and prove a beautiful and useful theorem of Alon and Tarsi that uses multivariate polynomials to guarantee, under suitable hypotheses, the existence of a coloring of a graph. The proof method, sometimes called a Combinatorial Nullstellensatz, has other applications in graph theory, combinatorics and number theory.

Series: Research Horizons Seminar

* Dr. Trotter: perspective of the hiring committee with an emphasis on research universities.
* Dr. Carroll: perspective of the applicant with an emphasis on liberal arts universities.
* Dr. Dieci: other advice, including non-academic routes.