A Keisler measure is a finitely additive probability measure on definable sets in a structure. Keisler measures are an important model-theoretic tool in a variety of contexts, but are especially significant for the analysis of definable groups. We will describe a construction for producing Keisler measures on bounded perfect PAC fields, an important class of algebraic examples within model theory. We will explain how the existence of these measures entails that all groups definable in bounded perfect PAC fields, and even in unbounded Frobenius fields, are definably amenable. This work builds on our earlier constructions of measures for e-free PAC fields and a related construction due to Will Johnson. This is joint work with Zoé Chatzidakis.

Suppose P and Q are polynomials in several variables over the complex numbers. Say that P defines Q if the graph of Q is definable (with parameters) in the first-order language with P as its only non-logical symbol (interpreted on the complex numbers in the obvious way). Then say that P and Q are interdefinable if each defines the other. For example, one can show that x+y and x-y are interdefinable, but x+y and xy are not. In this talk, we will first experiment with some toy examples of the interdefinability notion above. Then, based on joint work with Chieu-Minh Tran, I will present a complete classification of multivariable complex polynomials up to interdefinability. (The surprise: there are not as many classes as you might expect). Time permitting, I will outline the proof strategy, which uses a mix of results from logic, combinatorics, and algebraic geometry.

In this talk I will discuss my research in the field of ''tame topology.'' I will start by situating my work within the larger study of tame topology by first discussing o-minimality. Then I'll introduce a generalization of o-minimality called d-minimality (and later its cousin weak d-minimality). I'll present some examples of d-minimal structures, and use those to motivate this particular choice of generalization. With all the necessary definitions in hand, I'll state my two main results which focus on proving the equivalence of d-minimality and weak d-minimality as well as giving a version of cell decomposition for d-minimal structures. I'll present a proof for the $\mathbb{R}^2$ case, and end by discussing future work such as generalizing into the $\mathbb{R}^n$ case and applications to a d-minimal Pila-Wilkie theorem.