Georgia Institute of Technology College of Sciences

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: The Parameterisation Method is a powerful body of theory to compute the invariant manifolds of a dynamical system by looking for a parameterization of them in such a way that the dynamics on this manifold expressed in the coordinates of such parameterization writes as simply as possible. This methodology was foreseen by Guillamon and Huguet [SIADS, 2009 & J. Math. Neurosci, 2013] as a possible way of extending the domain of accuracy of the phase-reduction of periodic orbits. This fruitful approach, known as phase-amplitude reduction, has been fully developed during the last decade and provides an essentially complete understanding of deterministic oscillatory dynamics.
In this talk, we pursue the "simpler as possible" philosophy underlying the Parameterisation Method to develop an analogous phase-amplitude approach to stochastic oscillators. Main idea of our approach is to find a change of variables such that the system, when transformed to these variables, expresses in the mean as the deterministic phase-amplitude description. Then, we take advantage of the simplicity of this approach, to develop interesting objects with the aim of further clarifying the stochastic oscillation.

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: In this talk, I will discuss problems and results in the rigorous statistical mechanics of particle systems in a one-dimensional lattice.
I will briefly describe the classical examples, such as the Ising model and its various generalizations concerning the
existence of the free energy, thermodynamic limit and the phase transition phenomenon.
Towards the end of the talk, I will talk about a recent work in collaboration with Jorge Littin, on a generalization of the
Khanin and Sinai model with random interactions for which one can prove that there exists a critical behavior in the free
energy for some parameters of the model and on the other side one can also have uniqueness of the equilibrium state.

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, $\{f_k\}_{k \in I} \subset H$, be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, $\{T^n \varphi\}_{n=0}^{\infty} \subset H$, to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame. This is joint work with Carlos Cabrelli.

Join Zoom meeting:  https://gatech.zoom.us/j/96113517745

The study of Egyptian fractions, representing rational numbers as the sum of distinct unit fractions, is one of the oldest areas of number theory. In this talk we will discuss some fascinating problems in the area, including both open problems and some recent progress, such as the solution to the Erdos-Graham conjecture: 1 can be written as the sum of unit fractions with denominators drawn from an arbitrary set of integers of positive density.