I will give an introduction to the Univalent Foundations, a new approach to foundations of mathematics, proposed by Voevodsky based on ideas from homotopy theory. The Univalent Foundations are meant to more closely (than set theory) reflect our intuitions about mathematical objects, while also making computer-aided formal verification of proofs essentially straightforward. As an example, I will present the development of category theory in the new foundations (jww Ahrens and Shulman) and will contrast it with that within set theory. I will also discuss possible applications to formal verification of cryptographic standards.

Let $M$ be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. Lempert-Szoke and Guillemin-Stenzel have given canonical methods to construct adapted complex structures in neighborhoods of the zero section, equipped with a solution of the Homegeous Complex Monge-Ampere equation (HCMA) related to the geometry of $M$. These complex manifolds are called Grauert tubes. This structure is called entire if the structure may be extended to the whole of $TM$. We describe a circle of problems related to determining whether an entire Grauert tube is an affine algebraic manifold with ring of polynomials intrinsically distinguished by the HCMA exhaustion. We also discuss the relationship of this construction to the Paley-Wiener type theorem of Boutet de Monvel, and the relationship to eigenfunctions of the Laplace operator on $M$. Finally, we discuss the smallest dimensional cases, namely the two sphere, and invariant metrics on the three sphere, thought of as $SU(2)$.

Speaker's web page: https://lsa.umich.edu/math/people/faculty/dburns.html

In Calculus courses we learn methods to integrate functions of a real variable. Sometimes they work, but other times they seem to fail. I will present a result (due to Liouville) that indeed the integral of many functions cannot be expressed "in elementary terms", i.e., in terms of exponentials, logarithms and trigonometric functions. Specifically, this includes the Gaussian distribution (Bell curve) and the logarithmic integral used for prime number counting.

There will be pizza after the talk.

A celebrated result of Marcus, Spielman, and Srivas asserts that there exist bipartite Ramanujan graphs of every degree and number of vertices. In joint work with Puder and Sawin, my coauthors and I gave a significant generalization of their result. To prove our result we introduced a family of polynomials one can attach to a finite undirected graph. We call them d-matchings polynomials because when d = 1 one obtains the so-called matchings polynomial of a graph. In this talk I will define these polynomials and give some of the remarkable properties they satisfy.