University of Oxford

It the talk, various conditions of local regularity of axisymmetric suitable weak solutions, including the so-called slightly supercritical ones, will be discussed.

Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals

Jane Coons

Gaussian graphical models are multivariate Gaussian statistical models in which a graph encodes conditional independence relations among the random variables. Adding colors to this graph allows us to describe situations where some entries in the concentration matrices in the model are assumed to be equal. In this talk, we focus on RCOP models, in which this coloring is obtained from the orbits of a subgroup of the automorphism group of the underlying graph. We show that when the underlying block graph is a one-clique-sum of complete graphs, the Zariski closure of the set of concentration matrices of an RCOP model on this graph is a toric variety. We also give a Markov basis for the vanishing ideal of this variety in these cases.

Topological persistence for multi-scale terrain profiling and feature detection in drylands hydrology

Gillian Grindstaff

With the growing availability of remote sensing products and computational resources, an increasing amount of landscape data is available, and with it, increasing demand for automated feature detection and useful morphological summaries. Topological data analysis, and in particular, persistent homology, has been applied successfully to detect landslides and characterize soil pores, but its application to hydrology is currently still limited. We demonstrate how persistent homology of a real-valued function on a two-dimensional domain can be used to summarize critical points and shape in a landscape simultaneously across all scales, and how that data can be used to automatically detect features of hydrological interest, such as: experimental conditions in a rainfall simulator, boundary conditions of landscape evolution models, and earthen berms and stock ponds, placed historically to alter natural runoff patterns in the American southwest.

The Gromov boundary of a hyperbolic group is a useful topological invariant, the properties of which can encode all sorts of algebraic information. It has found application to some algorithmic questions, such as finding finite splittings (Dahmani-Groves) and, more recently, computing JSJ-decompositions (Barrett). In this talk we will introduce the boundary of a hyperbolic group. We'll outline how one can approximate the boundary with "large spheres" in the Cayley graph, in order to search for topological features. Finally, we will also discuss how this idea is applied in the aforementioned results. 

We will record some surprising and lesser-known properties of free groups, and use these to give a model theoretic analysis of free group automorphisms and orbits under Aut(F). This will result in a neat geometric description of (a logic-flavoured analogue of) algebraic closures in a free group. An almost immediate corollary will be that elementary subgroups of a free group are free factors.

I will assume no familiarity with first-order logic and model theory - the beginning of the talk will be devoted to familiarize everyone with the few required notions.

When a reductive group G acts on a complex projective variety
X, there exist different methods for finding an open G-invariant subset
of X with a geometric quotient (the 'stable locus'), which is a
quasi-projective variety and has a projective completion X//G. Mumford's
geometric invariant theory (GIT) developed in the 1960s provides one way
to do this, given a lift of the action to an ample line bundle on X,
though with no guarantee that the stable locus is not empty. An
alternative approach due to Kapranov and others in the 1990s is to use
Chow varieties to define a 'Chow quotient' X//G. The aim of this talk is
to review the relationship between these constructions for reductive
groups, and to discuss the situation when G is not reductive.

Let $\Gamma$ be a finite subgroup of $SL(2, \mathbb{C})$. We can attach several different moduli spaces to the action of $\Gamma$ on $\mathbb{C}^2$, and we show how Nakajima's quiver varieties provide constructions of them. The definition of such a quiver variety depends on a stability parameter, and we are especially interested in what happens when this parameter moves into a specific ray in its associated wall-and-chamber structure. Some of the resulting quiver varieties can be understood as moduli spaces of certain framed sheaves on an appropriate stacky compactification of the Kleinian singularity $\mathbb{C}^2/\Gamma$. As a special case, this includes the punctual Hilbert schemes of $\mathbb{C}^2/\Gamma$.

Much of this is joint work with A. Craw, Á. Gyenge, and B. Szendrői.

In 1989, Bryant and Salamon constructed the first Riemannian manifolds with holonomy group $\Spin(7)$. Since a crucial aspect in the study of manifolds with exceptional holonomy regards fibrations through calibrated submanifolds, it is natural to consider such objects on the Bryant-Salamon manifolds.

In this talk, I will describe the construction and the geometry of (possibly singular) Cayley fibrations on each Bryant-Salamon manifold. These will arise from a natural family of structure-preserving $\SU(2)$ actions. The fibres will provide new examples of Cayley submanifolds.

Locally analytic representations of $p$-adic Lie groups are of interest in several branches of number theory, for example in the theory of automorphic forms and in the $p$-adic local Langlands program. To better understand these representations, Ardakov-Wadsley introduced a sheaf of infinite order differential operators $\overparen{\mathcal{D}}$ on smooth rigid analytic spaces, which resulted in several Beilinson-Bernstein style localisation theorems. In this talk, we discuss the current research on analogues of a Riemann-Hilbert correspondence for $\overparen{\mathcal{D}}$-modules, and what this has to do with complete convex bornological vector spaces.

A well-known conjecture of Ivanov states that mapping class groups of surfaces with genus at least 3 virtually do not surject onto the integers. Putman and Wieland reformulated this conjecture in terms of higher Prym representations of finite-index subgroups of mapping class groups. We show that the Putman-Wieland conjecture holds for geometrically uniform subgroups. Along the way we construct a cover S of the genus 2 surface such that the lifts of simple closed curves do not generate the rational homology of S. This is joint work with Markovic.

Group theoretic Dehn filling, motivated by Dehn filling in the theory of 3- manifolds, is a process of constructing quotients of a given group. This technique is usually applied to groups with certain negative curvature feature, for example word-hyperbolic groups, to construct exotic and useful examples of groups. In this talk, I will start by recalling the notion of word-hyperbolic groups, and then show that how group theoretic Dehn filling can be used to answer the Burnside Problem and questions about mapping class groups of surfaces.