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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: Mathematical Biology Seminar

In the ocean, coherent vortices account for a large portion of the ocean turbulent kinetic energy and their presence significantly affects the dynamics and the statistical properties of ocean flows, with important consequences on transport processes. Mesoscale vortices also affect the population dynamics of phyto- and zooplankton, and are associated with secondary currents responsible for localized vertical fluxes of nutrients. The fact that the nutrient fluxes have a fine spatial and temporal detail, generated by the eddy field, has important consequences on primary productivity and the horizontal velocity field induced by the eddies has been suggested to play an important role in determining plankton patchiness. Owing to their trapping properties, vortices can also act as shelters for temporarily less-favoured planktonic species. In this contribution, I will review some of the transport properties associated with coherent vortices and their impact on the dynamics of planktoni ecosystems, focusing on the simplified conceptual model provided by two-dimensional turbulence.

Series: PDE Seminar

Shear flow instability is a classical problem in hydrodynamics. In particular, it is important for understanding the transition from laminar to turbulent flow. First, I will describe some results on shear flow instability in the setting of inviscid flows in a rigid wall. Then the effects of a free surface (or water waves) and viscosity will be discussed.

Series: Combinatorics Seminar

Let f be a polynomial or multilinear form in a large number of variables. A basic question we can ask about f is how dispersed it becomes as the number of variables increases. To be more concrete: If we randomly (and independently) set each entry to be either 1 or -1, what is the largest concentration of the output of f on any single value, assuming all (or most) of the coefficients of f are nonzero? Can we somehow describe the structure of those forms which are close to having maximal concentration? If f is a linear polynomial, this is a question originally examined by Littlewood and Offord and answered by Erdos: The maximal concentration occurs when all the nonzero coefficients of f are equal. Here we will consider the case where f is a bilinear or quadratic form.

Series: Stochastics Seminar

We consider a random field of tensor product type X and investigate the quality of approximation (both in the average and in the probabilistic sense) to X by the processes of rank n minimizing the quadratic approximation error. Most interesting results are obtained for the case when the dimension of parameter set tends to infinity. Call "cardinality" the minimal n providing a given level of approximation accuracy. By applying Central Limit Theorem to (deterministic) array of covariance eigenvalues, we show that, for any fixed level of relative error, this cardinality increases exponentially (a phenomenon often called "intractability" or "dimension curse") and find the explosion coefficient. We also show that the behavior of the probabilistic and average cardinalities is essentially the same in the large domain of parameters.

Series: Graph Theory Seminar

The problem of generating random integral tables from the set of all nonnegative integral tables with fixed marginals is of importance in statistics. The Diaconis-Sturmfels algorithm for this problem performs a random walk on the set of such tables. The moves in the walk are referred to as Markov bases and correspond to generators of a certain toric ideal. When only one and two-way marginals are considered, one can naturally associate a graph to the model. In this talk, I will present a characterization of all graphs for which the corresponding toric ideal can be generated in degree four, answering a question of Develin and Sullivant. I will also discuss some related open questions and demonstrate applications of the Four Color theorem and results on clean triangulations of surfaces, providing partial answers to these questions. Based on joint work with Daniel Kral and Ondrej Pangrac.

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: PDE Seminar

Optimization problems with PDE constraints are commonly solved in different areas of science and engineering. In this talk we give an introduction to this field. In particular we discuss discretization techniques and effective linear and nonlinear solvers. Examples are given from inverse problems in electromagnetics.

Series: CDSNS Colloquium

The connection between transport barriers and potential vorticity (PV) barriers in PV-conserving flows is investigated with a focus on zonal jets in planetary atmospheres. A perturbed PV-staircase model is used to illustrate important concepts. This flow consists of a sequence of narrow eastward and broad westward zonal jets with a staircase PV structure; the PV-steps are at the latitudes of the cores of the eastward jets. Numerically simulated solutions to the quasigeostrophic PV conservation equation in a perturbed PV-staircase flow are presented. These simulations reveal that both eastward and westward zonal jets serve as robust meridional transport barriers. The surprise is that westward jets, across which the background PV gradient vanishes, serve as robust transport barriers. A theoretical explanation of the underlying barrier mechanism is provided, which relies on recent results relating to the stability of degenerate Hamiltonians under perturbation. It is argued that transport barriers near the cores of westward zonal jets, across which the background PV gradient is small, are found in Jupiter's midlatitude weather layer and in the Earth's summer hemisphere subtropical stratosphere.

Series: Analysis Seminar

In any standard course of Analytical Mechanics students are indoctrinated that a Lagrangian have a profound physical meaning (kinetic energy minus potential energy) and that Lagrangians do not exist in the case of nonconservative system. We present an old and regretfully forgotten method by Jacobi which allows to find many nonphysical Lagrangians of simple physical models (e.g., the harmonic oscillator) and also of nonconservative systems (e.g., the damped oscillator). The same method can be applied to any equation of second-order, and extended to fourth-order equations as well as systems of second and first order. Examples from Physics, Number Theory and Biology will be provided.