University of Oxford

It is at first sight surprising that a minimizer of an integral of the calculus of variations may make the integrand infinite somewhere.

This talk will discuss some examples of this phenomenon, how it can be related to material defects, and related open questions from nonlinear elasticity and the theory of liquid crystals.

Cube Complexes with Coupled Links (CLCC) are a special family of non-positively curved cube complexes that are constructed by specifying what the links of their vertices should be. In this talk I will introduce the construction of CLCCs and try to motivate it by explaining how one can use information about the local geometry of a cube complex to deduce global properties of its fundamental group (e.g. hyperbolicity and cohomological dimension). On the way, I will also explain what fly maps are and how to use them to deduce that a CAT(0) cube complex is hyperbolic.

We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.

We will discuss topological and algebraic aspects of splittings of free groups. In particular we will look at the core of two splittings in terms of CAT(0) cube complexes and systems of surfaces in a doubled handlebody.

I will present Vidit Nanda's paper "Local homology and stratification" (https://arxiv.org/abs/1707.00354), and briefly explain how in my master thesis I am applying ideas from the paper to study word embedding problems.

Abstract of the paper:  We outline an algorithm to recover the canonical (or, coarsest) stratification of a given regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a finite sequence of categories whose colimit recovers the canonical strata via (isomorphism classes of) its objects. The entire process is amenable to efficient distributed computation.
 

CAT(0) spaces are defined as having triangles that are no fatter than Euclidean triangles, so it is no surprise that under special conditions  you find pieces of the Euclidean plane appearing in CAT(0) spaces. What is surprising though is how weak these special conditions seem to be. I will present some well known results of this phenomenon, along with detailed sketch proofs.

This is the continuation of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

I will talk about the properties of algebraic integers that can arise as stretch factors of pseudo-Anosoc maps. I will mention a conjecture of Fried on which numbers supposedly arise and Thurston’s theorem that proves a similar result in the context of automorphisms of free groups. Then I will talk about recent developments on the Fried conjecture namely, every Salem number has a power arising as a stretch factor. 

Let n be a positive integer. In this talk we provide a recipe to 
construct full orbit sequences in the affine n-dimensional space over a 
finite field. For n=1 our construction covers the case of the well 
studied pseudorandom number generator ICG.

This is a joint work with Federico Amadio Guidi.

I will discuss a wonderful structure theorem for finitely generated group containing a codimension one polycyclic-by-finite subgroup, due to Martin Dunwoody and Eric Swenson. I will explain how the theorem is motivated by the torus theorem for 3-manifolds, and examine some of the consequences of this theorem.