What is the second moment of a random smooth plane curve?  Is the multiset of eigenvalues of a random orthogonal matrix a Gaussian random variable? Is a random compact Riemann surface a Poisson process? In this talk, I will describe a categorified version of probability theory that makes these nonsense questions into precise mathematics and give some applications to the topology and arithmetic of moduli spaces in algebraic geometry. 

In this public talk, internationally acclaimed choreographer Reggie Wilson and math professor Jesse Wolfson will describe their decade+ collaboration exploring what math can do for dance and what dance can do for math.

Fourier analysis has been an extraordinarily powerful mathematical tool since its development 200 years ago, and currently has a wide range of applications in diverse scientific fields including digital image processing, forensics, option pricing, cryptography, optics, oceanography, and protein structure analysis. Like a prism that decomposes a beam of light into a rainbow of colors, Fourier analysis transforms signals into a mathematical spectrum of basic wave components of different amplitudes and frequencies, from which many hidden properties in the data can be deciphered. At the abstract mathematical level signals are represented by functions and their filtering and other operations on them by operators. From a functional analytical point of view, these objects are studied by decomposing them into elementary building blocks, some of which have wavelike behavior too. Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: "waves with very different frequencies are almost invisible to each other". Many of these useful techniques have been developed around the study of some particular operators called singular integral operators and,  recently, similar techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, null-forms, and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals and function spaces, and their applications to the development of the equivalent of the calculus Leibniz rule to the concept of fractional derivatives.
 

I intend to give a true introduction, accessible to beginning graduate students, to the topics in the title: what is tropical geometry? What is a moduli space? And how can one be used to study the other?  Then I'll discuss some aspects of joint work with Francis Brown, Søren Galatius, and Sam Payne, in which we identify a Hopf algebraic structure on the weight 0 subspace of the compactly supported cohomology of the moduli space of abelian varieties and deduce a number of consequences.