Colloquia

Mathematical definitions, formally speaking

This talk will give an introduction to my current project, which aims to write all the theorems and definitions of mathematics in a computer-readable form.   By "computer-readable", we mean much more than TeX, Maple, or Sage.  We mean that the math is expressed in terms of the rules of logic and foundations of mathematics. This project is expected eventually to encompass all branches of mathematics.

Recurrence for IP Systems with Polynomial Wildcards

Shortly after Szemeredi proved that a set of natural numbers with positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof using ergodic theory. This major event gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed with ergodic theory. In this talk we give a brief survey of some of the successes in the field, leading into a description of an interesting, yet still unproved result, which would provide a generalization of many earlier results.

Machine learning for the discovery of governing equations and optimal coordinate systems

A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data and the best representation of an accompanying coordinate system. This problem is made more difficult by the fact that many systems of interest exhibit parametric dependencies and diverse behaviors across multiple time scales.

Fractional Cahn-Hilliard Equation(s): Analysis, Properties and Approximation

The classical Cahn-Hilliard equation [1] is a nonlinear, fourth order in space, parabolic partial differential equation which is often used as a diffuse interface model for the phase separation of a binary alloy. Despite the widespread adoption of the model, there are good reasons for preferring models in which fractional spatial derivatives appear [2,3]. We consider two such Fractional Cahn-Hilliard equations (FCHE).

Endpoint Sobolev inequalities for vector fields

Sobolev embeddings control the integrability of some power of a function by an
integral of the derivative of the function at a lower power. The limiting case
where the latter power is taken to be 1 due to Gagliardo and Nirenberg, is
inaccessible to classical methods of harmonic analysis and turns out to be a
functional version of the isoperimetric inequality. If one considers vector
fields instead of functions, one can hope that some redundancy in the
derivative would allow to obtain estimates with an integrand that does not