One of the great insights of model theory is the observation that very mundane-looking "combinatorial configurations" carry a huge amount of geometric information about a structure. In this talk, I will explain what we mean by "combinatorial configuration," and then I will sketch out how configurations can be "smoothed out" to yield Ramsey classes, which can themselves be analyzed using model-theoretic tools. I will also discuss the kinds of model-theoretic dividing lines that can be defined just through the interaction of structures with Ramsey classes.

In this talk, we will be working in with the theory of differentially closed fields of characteristic zero; essentially this theory says that every differential equation which might have a solution in some field extension already has a solution in the differentially closed field. After introducing this theory in a bit of detail, we will sketch a proof of the strong minimality of the differential equation satisfied by the classical $j$-function starting from Pila's modular Ax-Lindemann-Weierstrass theorem. This resolves an open question about the existence of a geometrically trivial strongly minimal set which is not $\aleph _0$-categorical. If time allows, we will discuss some finiteness applications for intersections of certain sets in modular curves. This is joint work with Tom Scanlon.

In joint work with Lou van den Dries and Joris van der Hoeven, we are investigating algebraic and model-theoretic properties of certain valued differential fields in which the valuation and derivation interact in a very strong way. Prime examples of interest are various fields of "transseries," first introduced independently by J. Écalle in his work on Hilbert’s 16th Problem and by the model theorists Dahn and Göring in their work around Tarski’s problem on real exponentiation, as well as Hardy fields (such as fields of germs of functions definable in an o-minimal expansion of the real field). One important question in this subject is what the right notion of "differentially henselian" should be. In this talk I will propose an answer to this question; I will introduce the concept of "newtonian" valued differential field and explain some of its fundamental properties. Lurking in the background is a version of the Newton polygon process for differential polynomials, which I also hope to describe.

I will explain some connections between Berkovich spaces of degenerations of Calabi-Yau varieties and the Minimal Model Program in birational geometry. The central object in this theory is the so-called weight function on the Berkovich space. This function has some interesting properties that suggest that one can use it to contract the Berkovich space onto its canonical skeleton. I will also show how analogous properties of weight functions of hypersurface singularities yield a proof of a 1999 conjecture of Veys on poles of maximal order of motivic zeta functions. This is based on joint work with Mircea Mustata and Chenyang Xu.

In this talk, I give an overview of a few strong tameness conditions on NIP theories, including VC-minimality, dp-minimality, and a few intermediate properties.  Specifically, I discuss these properties in the context of ordered groups, abelian groups, fields, and valued fields.  Time permitting, I conclude by talking about the VC-density problem in VC-minimal theories, exhibiting several approaches to the problem, including the UDTFS method and combinatorial methods.  A portion of this work is joint with Chris Laskowski and another portion is joint with Joseph Flenner.