Georgia Institute of Technology College of Sciences

Endowing a finite combinatorial graph with lengths on its edges defines singular 1-dimensional Riemannian manifolds known as metric graphs. The spectra of their Laplacians have been widely studied. We show that these spectra have a structured linear part described in terms of arithmetic progressions and a nonlinear "random" part which is highly linearly and even algebraically independent over the rationals. These spectra give rise to exotic crystalline measures ("Generalised Poisson Summation Formulae") and resolve various open problems concerning the latter. This is a joint work with Pavel Kurasov.

A harmonic function of two variables is the real or imaginary part of an analytic function. A harmonic function of $n$ variables is a function $u$ satisfying

$$

\frac{\partial^2 u}{\partial x_1^2}+\ldots+\frac{\partial^2u}{\partial x_n^2}=0.

$$

We will first recall some basic results on harmonic functions: the mean value property, the maximum principle, the Liouville theorem, the Harnack inequality, the Bocher theorem, the capacity and removable singularities. We will then present a number of more recent results on some conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville theorems, Harnack inequalities, and Bocher theorems.

Combinatorics was conceived, and then developed over centuries as a discipline about finite structures. In the modern world, however, its applications increasingly pertain to structures that, although finite, are extremely large: the Internet network, social networks, statistical physics, to name just a few. This makes it very natural to try to think of the "limit theory" of such objects by pretending that "very large" actually means "infinite". This mathematical abstraction turns out to be very useful and instructive.

After briefly reviewing the basics of the theory (graphons and flag algebras), I will report on some more recent developments. Time permitting, we will discuss the most general form of the theory suitable for arbitrary combinatorial structures (peons and theons), its applications to the theory of quasi-randomness and its applications to machine learning.

The first two topics are based on joint work with L. Coregliano, and the third one on a recent paper by Coregliano and Malliaris.