Upcoming MAMS Colloquium Series
*All colloquia will be held in Yost 306 on Fridays at 3:15PM, unless otherwise noted.
November 5, 2021
Title: The Bergman Reproducing Kernel and the Asymptotic Properties of Polynomials Orthogonal over Planar Domains
Speaker: Erwin Mina Diaz, University of Mississippi
Abstract: The Bergman reproducing kernel of a domain $D$ of the complex plane is a function of two variables $K_D(z,\zeta)$, analytic in $z$ and anti-analytic in $\zeta$, with the property that for every analytic and square integrable function $f$ over $D$, $f(z)=\int_Df(\zeta)K(z,\zeta)dA(\zeta)$, $z\in D$. The existence of this kernel is a general fact in functional analysis, and it can be explicitly computed in a number of cases. If $(f_n)_{n=0}^\infty$ is a complete orthonormal system, then $K(z,\zeta)=\sum_{n=0}^\infty f_n(z)\overline{f_n(\zeta)}$. When the orthonormal system is chosen to be a sequence $(p_n)_{n=0}^\infty$ of orthonormal polynomials, the relation $K(z,\zeta)=\sum_{n=0}^\infty p_n(z)\overline{p_n(\zeta)}$ yields a simple and useful way to approximate the reproducing kernel. Reciprocally, in this talk we explain a method that in certain situations allows us to extricate from the kernel $K_D(z,\zeta)$ the behavior of $p_n(z)$ as $n\to\infty$.
November 12, 2021
Title: Revealing the simplicity of high-dimensional objects via pathwise stochastic analysis.
Speaker: Ronen Eldan, Princeton University
Abstract: A common motif in high dimensional probability and geometry is that the behavior of objects of interest is often dictated by their marginals onto a fixed number of directions. This is manifested in the fact that several classical functional inequalities are dimension free (i.e., have no explicit dependence on the dimension), the extremizers of those inequalities being functions that only depend on a fixed number of variables. Another related example comes from statistical mechanics, where Gibbs measures can often be decomposed into a small number of “pure states” which exhibit a simple structure that only depend on a small number of directions in space.
In this talk, we present an analytic approach that helps reveal phenomenona of this nature. The approach is based on pathwise analysis: We construct stochastic processes, driven by Brownian motion, associated with the high-dimensional object which allow us to make the object more tractable, for example, through differentiation with respect to time.
I will try to explain how this approach works and will briefly discuss several results that stem from it, including functional inequalities in Gaussian space, high dimensional convexity as well results related to decomposition Gibbs measures into pure states.
November 19, 2021
Title: Accurate Mathematical Tool for the Phase Control of Biological Oscillators
Speaker: Alberto Perez, Universidad Complutense de Madrid
Abstract: In this talk we present a numerical methodology for an accurate computation of the phase dynamics of a n-dimensional oscillator. Indeed, we compute the phase dynamics of an oscillator not just on its asymptotic state but also on its transient states. The methodology relies on the parameterization method which allows us to obtain a parameterization of the attracting invariant manifold of a limit cycle in terms of the phase amplitude variables. The talk will highlight the many advantages of computing this parameterization. On the one hand, it permits to compute two important foliations of the attracting manifold of the limit cycle: the isochrons and the isostables which provide a geometrical portrait of the oscillator. On the other hand, it also provides the infinitesimal Phase (Amplitude) Response Functions (iPRFs, (iARFs)), which describe the phase and amplitude dynamics beyond the asymptotic state. The computation of the iPRFs and iARFs, permits to extend the classical adjoint equation for points beyond the limit cycle, hence allowing us to study useful strategies to reduce the dimension of dynamics when applying external perturbations without losing accuracy. We illustrate our methods by applying them to different single neuron and population models in neuroscience.
The talk will end with a brief description of the extension of these tools for stochastic oscillators. An ongoing work in collaboration with Prof. Peter Thomas (Case Western Reserve University, Ohio) and Prof. Benjamin Lindner (Humboldt-Universität zu Berlin).