Mathematics Colloquium
All talks

Thursday, March 21, 2019, 12:30PM, NAC 6/114
Yung Choi (U. Conn), Stationary and traveling waves of the FitzHughNagumo EquationsThe talk will be structured for general colloquium audience.
Stationary waves are steady state solutions in unbounded domain, while traveling waves will appear steady when viewed in a moving frame. We start with stationary waves of a scalar reactiondiffusion equation, as they can be easily explained using a phase plane analysis. Next we survey some results on the FitzHughNagumo system. For some special parameter regimes, the solutions give discontinuous jump profiles as a certain parameter epsilon goes to zero. This can be analyzed using Γconvergence and gives rise to a geometric variational problem; its radially symmetric solutions (known as bubbles) and their local stability are completely classified for all parameters.
We next turn our attention to traveling waves. Again we start with a scalar equation and work towards the FitzHughNagumo systems. The use of a Γconvergence analysis to traveling wave has recently been achieved. It seems to be the first instance of extending this kind of analysis to nonstationary problems.

Thursday, February 21, 2019, 12:30PM, NAC 6/114
Vincent Martinez (Hunter College (CUNY)), Studies in analyticity for hydrodynamic and chemotaxis modelsIn their 1987 seminal paper, Foias and Temam established analyticity in both space and time for solutions of the two(2D) and threedimensional (3D) NavierStokes equations (NSE) by developing an energy method now known as the technique of Gevreynorms, i.e., a norm which characterizes real analyticity of a function. This approach has since become standard for establishing spatial analyticity of solutions to various parabolictype equations. In this talk, we shed light on the relation between the structure of the equation and its wellposedness theory in various analytic Gevreynorm regularity classes. We do so in the context of the supercritical SQG equation, the KellerSegel equation, and a coupled chemotaxishydrodynamic model through the notion of “criticality.”

Thursday, February 07, 2019, 12:30PM, NAC 6/114
Ethan Akin (CCNY), On Nontransitive DiceThere exist nonstandard intransitive 6sided dice. A nonstandard die is labeled with positive numbers and repeats are allowed. A die A beats die B if Prob(A > B) > 1/2. There exist such dice with A beats B, B beats C and C beats A. We show that with n Ksided dice (large K) any tournament can be modeled. A tournament on n is a choice of direction i beats j or j beats i for any pair i,j of distinct numbers between 1 and n. The talk should be accessible to undergrads.

Thursday, December 06, 2018, 12:30PM, NAC 6/114
Giulio Tiozzo (University of Toronto), Trees, entropy, and the Mandelbrot setThe notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.
In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomials as the entropy of a certain forwardinvariant set of the Julia set (the Hubbard tree).
As we shall see, the core entropy is a purely topological / combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we will relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston.

Thursday, November 29, 2018, 12:30PM, NAC 6/114
Philippe Sosoe (Cornell U.), Applications of CLTs and homogenization for Dyson Brownian Motion to Random Matrix TheoryI will explain how two recent technical developments in Random Matrix Theory allow for a precise description of the fluctuations of single eigenvalues in the spectrum of large symmetric random matrices. No prior knowledge of random matrix theory will be assumed.
(Based on joint work with B Landon and HT Yau)
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