Random graphs are at the intersection of probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices. Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices. The general model that describes this framework is called the exponential random graph model (ERGM). It is used in social network analysis and appears in statistical physics as in the ferromagnetic Ising model. It can also be thought of as a generalization of a p-spin infinite-range spin glass model. We characterize the parameters that determine when an ERGM has desirable properties (e.g., stable, Lorentzian) using a well-developed dictionary between probability distributions and their corresponding generating polynomials.

The equivariant cohomology of a space with an action of a group $G$ inherits a canonical module structure over the cohomology ring of $BG$. In this talk, we study pair of groups $K \subseteq G$ such that some algebraic properties of the $G$-equivariant cohomology are captured by the action of the subgroup $K$.

Motivated by compact connected Lie group, torus and cyclic group actions, we generalize these reductions into the notions of free and flat extension pairs in equivariant cohomology.

If time permits, we will discuss related results into the equivariant cohomology of canonical cyclic group actions on surfaces arising as real moment-angle complexes.

I will talk about the theoretical challenges to the numerical study of dynamical systems. I will broadly discuss what practitioners attempt to compute, and whether such computations are always possible. Such questions lead to interesting mathematics with surprising practical implications. As an instructive example of the limitations on our ability to compute things, I will describe a "nice" one-dimensional dynamical system for which a numerical approximation of the long-term statistical behavior of the orbits is not possible. In particular, the Monte Carlo simulation provably fails for it.