The periods of a Calabi-Yau manifold are of interest both to number theorists and to physicists. To a number theorist the primary object of interest is the zeta function. I will explain what this is, and why this is of interest also to physicists. For applications it is important to be able to calculate the local zeta function for many primes p. I will set out a method, adapted from a procedure proposed by Alan Lauder that makes the computation of the zeta function practical, in this sense, and comment on the form of the results. This talk is based largely on the recent paper hepth 2104.07816 and presents joint work with Xenia de la Ossa and Duco van Straten.

The elliptic Hall algebra has appeared in many different contexts in representation theory and geometry under different names.  We will explain how this algebra is categorified by the quantum Heisenberg category.  This diagrammatic category is modelled on affine Hecke algebras and can be viewed as a deformation of the framed HOMFLYPT skein category underpinning the HOMFLYPT link invariant.  Using the categorification of the elliptic Hall algebra, one can construct large families of representations for this algebra.

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.

We consider the problem of clustering in two important families of networks: signed and directed, both relatively less well explored compared to their unsigned and undirected counterparts. Both problems share an important common feature: they can be solved by exploiting the spectrum of certain graph Laplacian matrices or derivations thereof. In signed networks, the edge weights between the nodes may take either positive or negative values, encoding a measure of similarity or dissimilarity. We consider a generalized eigenvalue problem involving graph Laplacians, with performance guarantees under the setting of a signed stochastic block model. The second problem concerns directed graphs. Imagine a (social) network in which you spot two subsets of accounts, X and Y, for which the overwhelming majority of messages (or friend requests, endorsements, etc) flow from X to Y, and very few flow from Y to X; would you get suspicious? To this end, we also discuss a spectral clustering algorithm for directed graphs based on a complex-valued representation of the adjacency matrix, which is able to capture the underlying cluster structures, for which the information encoded in the direction of the edges is crucial. We evaluate the proposed algorithm in terms of a cut flow imbalance-based objective function, which, for a pair of given clusters, it captures the propensity of the edges to flow in a given direction. Experiments on a directed stochastic block model and real-world networks showcase the robustness and accuracy of the method, when compared to other state-of-the-art methods. Time permitting, we briefly discuss potential extensions to the sparse setting and regularization, applications to lead-lag detection in time series and ranking from pairwise comparisons.

The first studies of the friction at the base of glacier were done by the pioneers Weertman and Lliboutry in the 1950s, who proposed theories under assumptions that have not been questioned for decades. Among these assumptions, the most questionable are the 2d geometry of the bumps, the pure sliding at the ice-bed interface and the hypothesis of stationary water pressure. In this seminar, I will present recent works using local modelling of basal friction with the finite element method that explore the validity of the proposed friction laws when these assumptions do not hold any more. 

The outstanding issue of a non-singular extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein’s equations is solved completely. The Misner’s method of obtaining the extension for Taub-NUT spacetime is generalized in a non-singular manner. The Killing vectors that define non-singular spaces of non-null orbits are derived and applied. The global structure of spacetime is discussed. The non-singular conformal geometry of theinfinities is derived. The Killing horizons are present.

Anabelian geometry asks how much we can say about a variety from its fundamental group. In 1997 Shinichi Mochizuki, using p-adic hodge theory, proved a fundamental anabelian result for the case of p-adic fields. In my talk I will discuss representation theoretical data which can be reconstructed from an absolute Galois group of a field, and also types of representations that cannot be constructed solely from a Galois group. I will also sketch how these types of ideas can potentially give many new results about p-adic Galois representations.

Abstract: In this work, we analyze a class of stochastic inverse optimal control problems with entropy regularization. We first characterize the set of solutions for the inverse control problem. This solution set exemplifies the issue of degeneracy in generic inverse control problems that there exist multiple reward or cost functions that can explain the displayed optimal behavior. Then we establish one resolution for the degeneracy issue by providing one additional optimal policy under a different discount factor. This resolution does not depend on any prior knowledge of the solution set. Through a simple numerical experiment with deterministic transition kernel, we demonstrate the ability of accurately extracting the cost function through our proposed resolution.

Joint work with Sam Cohen (Oxford) and Lukasz Szpruch (Edinburgh).