Abstract: 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

Abstract: 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.