Georgia Institute of Technology College of Sciences

Given data and a statistical model, the maximum likelihood estimate is the point of the statistical model that maximizes the probability of observing the data. In this talk, I will address three different approaches to maximum likelihood estimation using algebraic methods. These three approaches use boundary stratification of the statistical model, numerical algebraic geometry and the EM fixed point ideal. This talk is based on joint work with Allman, Cervantes, Evans, Hoşten, Kosta, Lemke, Rhodes, Robeva, Sturmfels, and Zwiernik.

The structure of non-archimedean curves X and their tame covers f:Y-->X is well understoodand can be adequately described in terms of a (simultaneous) semistable model. In particular, asindicated by the lifting theorem of Amini-Baker-Brugalle-Rabinoff, it encodes all combinatorialand residual algebra-geometric information about f. My talk will be mainly concerned with the morecomplicated case of wild covers, where new discrete invariants appear, with the different function being the most basic one. I will recall its basic properties following my joint work with Cohen and Trushin,and will then pass to the latest results proved jointly with U. Brezner: the different functioncan be refined to an invariant of a residual type, which is a (sort of) meromorphic differential form on the reduction, so that a lifting theorem in the style of ABBR holds for simplest wild covers.

Let X be a curve over a p-adic field K with semi-stable reduction and let $\omega$ be a meromorphic differential on X. There are two p-adic integrals one may associated to this data. One is the Vologodsky (abelian, Zarhin, Colmez) integral, which is a global function on the K-points of X defined up to a constant. The other is the collection of Coleman integrals on the subdomains reducing to the various components of the smooth locus. In this talk I will prove the following Theorem, joint with Sarah Zerbes: The Vologodsky integral is given on each subdomain by a Coleman integrals, and these integrals are related by the condition that their differences on the connecting annuli form a harmonic 1-cocyle on the edges of the dual graph of the special fiber.I will further explain the implications to the behavior of the Vologodsky integral on the connecting annuli, which has been observed independently and used, by Stoll, Katz-Rabinoff-Zureick-Brown, in works on global bounds on the number of rational points on curves, and an interesting product on 1-forms used in the proof of the Theorem as well as in work on p-adic height pairings. Time permitting I will explain the motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals.

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to non-abelian p-groups, we characterize the curves Y for which this lifting property holds and when it is unique, but over a more general ground field.