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.

The uncertainty in future market dynamics is an important consideration when developing strategies for hedging derivatives, particularly data driven strategies such as deep hedging. Deep market generators can produce higher fidelity training data than classical models, but, like those, typically require frequent recalibration to new market data. The resulting strategies are thus susceptible to underperformance if there is a mismatch (distributional shift) between training data and live data. We present a framework to train a modified deep hedger which displays a form of ambiguity aversion, henceforth termed an Ambiguity-Averse Deep Hedger (AADH). The modeller has full control over exactly which aspects of distributional shifts the AADH is to be robust to, through selection of features relevant to the trading strategy which are used to cluster the training data, allowing for the evaluation of a loss function motivated by the theory of smooth ambiguity aversion.

TBA

Lorenzo Lazzarino will talk about: 'Error estimations for randomized low-rank approximations'

Randomized algorithms in numerical linear algebra have proven to be effective in ameliorating issues of scalability when working with large matrices, efficiently producing accurate low-rank approximations. A key remaining challenge, however, is to efficiently assess the approximation accuracy of randomized methods without additional expensive matrix accesses.

In this talk, we discuss a posteriori error estimation strategies for randomized low-rank approximations, with a focus on estimators that can be constructed from the same data used to compute the approximation or without matrix global accesses. These can serve both as certification tools and as algorithmic building blocks, enabling adaptive approximations and informed trade-offs between accuracy and computational cost. As a motivation and a case study, we include a discussion on spectromicroscopy experiments.