Existential decidability of a ring is the question as to whether an algorithm exists which determines whether a given system of polynomial equations and inequations has a solution. It is a classical result (``Hilbert's 10th problem'') that the ring of integers is not existentially decidable. Over the years there has been many results related to Hilbert 10th problem over different fields. For instance, the existential decidability of a Henselian valued field of mixed characteristic and finite ramification can be reduced to the positive existential decidability of its residue field, plus some additional structure.

An example of a mixed characteristic Henselian field is the fraction field of Witt Vectors. It is a construction analogous to the construction of the p-adic numbers from $\mathbb{F}_p$, and it takes a perfect field $F$ of characteristic $p$ and constructs a field with value group $\mathbb{Z}$ and residue field $F$. We will look at the existential decidability of the Henselian valued fields arising from finite extensions of the Witt vectors over a positive characteristic Henselian valued field. I will report on our progress so far, the problems that we have encountered, and the goals we are working toward.

Two hundred years ago, Robert Brown observed the statistics of the motion of grains of pollen in water. It took almost one hundred years for Einstein and others to develop an effective theory describing this motion as that of a random walker. In this talk, I will challenge a key implication of this well established theory. When studying systems with very large numbers of particles diffusing together, I will argue that the Einstein random walk theory breaks down when it comes to predicting the statistical behavior of extreme particles—those that move the fastest and furthest in the system. In its place, I will describe a new theory of extreme diffusion which captures the effect of the hidden environment in which particles diffuse together and allows us to interrogate that environment by studying extreme particles. I will highlight one piece of mathematics that led us to develop this theory—a non-commutative binomial theorem—and hint at other connections to integrable probability, quantum integrable systems and stochastic PDEs.

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Daniel Cortild is going to talk about: 'Regularization Methods for Hierarchical Programming'

We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. 

Joint work with Meggie Marschner, and Mathias Staudigl (University of Mannheim)