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.
A fully nonlinear, closed-form mapping from time series of pressure measurements at an arbitrary depth in the fluid column to time series of surface displacement is derived from the full Stokes boundary value problem for water waves. The formula is implicit and nonlocal and must be solved numerically, albeit to machine precision.
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.
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.
The classical Cahn-Hilliard equation  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).
In this talk, intended for general mathematical audience, I will explain how complex analytic methods have been used to study the distribution of mathematical objects of number theoretic interest in the last couple of centuries, starting with Riemann's approach to proving the Prime Number Theorem. At the end I will sketch some recent progress.
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