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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.

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo

An efficient (arguably the most efficient) way to study and describe dynamical systems with complex behavior is to code their orbits. To do so one considers a partition (physicists call it coarse graining) of the space of states (phase space) of the systems and identify location of a point with an element of such partition where this point belongs. . By doing so one gets an unified description of dynamical systems in Differential equations, Number theory, probability theory, Combinatorics, etc. I'll present an introduction to this Symbolic dynamics for interacting dynamical systems (dynamical networks) and formulate some open problems.This talk will also serve as a preparatory one for the next day Math Colloquium.