Oxford University

In this talk, I will present some recent results exploring the connections between dynamical systems and network science. I will particularly focus on large-scale structures and their dynamical interpretation. Those may correspond to communities/clusters or classes of dynamically equivalent nodes. If time allows, I will also present results where the underlying network structure is unknown and where communities are directly inferred from time series observed on the nodes.

I will present a variation of positive model theory which addresses the issues of approximations of conventional geometric structures by sequences of Zariski structures as well as approximation by sequences of finite structures. In particular I am interested in applications to quantum mechanics.

I will report on a progress in defining and calculating oscillating in- tegrals of importance in quantum physics. This is based on calculating Gauss sums of order higher or equal to 2 over rings Z/mZ for very specific m. 

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n+1 edges of each colour there must be a matching that uses each colour exactly once (such a matching is called rainbow). This conjecture recently have been proved asymptotically by Pokrovskiy. In this talk, I will consider the same question without the bipartiteness assumption. It turns out that in any multigraph with bounded edge multiplicities, that is properly edge-coloured by n colours with at least n+o(n) edges of each colour, there must be a matching of size n-O(1) that uses each colour at most once. This is joint work with Peter Keevash.

 The aim of this talk is to describe the classification problem for Higgs bundles and to explain how a combination of classical and Non-Reductive Geometric Invariant Theory might be used to solve this classification problem.
 
I will start by defining Higgs bundles and their physical origins. Then, I will present the classification problem for Higgs bundles. This will involve introducing the "stack" of Higgs bundles, a purely formal object which allows us to consider all isomorphism classes of Higgs bundles at once. Finally, I will explain how the stack of Higgs bundles can be described geometrically. As we will see, the stack of Higgs bundles can be decomposed into disjoint strata, each consisting of Higgs bundles of a given "instability type". Both classical and Non-Reductive GIT can then be applied to obtain moduli spaces for each of the strata.

In a recent preprint, Basterra, Bobkova, Ponto, Tillmann and Yeakel
defined operads with homological stability (OHS) and showed that after
group-completion, algebras over an OHS group-complete to infinite loop
spaces. This can in particular be used to put a new infinite loop space
structure on stable moduli spaces of high-dimensional manifolds in the
sense of Galatius and Randal-Williams, which are known to be infinite
loop spaces by a different method.

To complicate matters further, I shall introduce a mild strengthening of
the OHS condition and construct yet another infinite loop space
structure on these stable moduli spaces. This structure turns out to be
equivalent to that constructed by Basterra et al. It is believed that
the infinite loop space structure due to Galatius--Randal-Williams is
also equivalent to these two structures.

Given a graph $G$, we can form a hypergraph $H$ whose edges correspond to the triangles in $G$. If $G$ is the standard Erdős-Rényi random graph with independent edges, then $H$ is random, but its edges are not independent, because of overlapping triangles. This is (presumably!) a major complication when proving results about triangles in random graphs.  However, it turns out that, for many purposes, we can treat the triangles as independent, in a one-sided sense (and losing something in the density): we can find an independent random hypergraph within the set of triangles. I will present two proofs, one of which generalizes to larger complete (and some non-complete) subgraphs.

Topologists have the Steenrod squares, a collection of additive homomorphisms on the Z/2 cohomology of a space M. They can be defined axiomatically and are often be regarded as algebraic operations on cohomology groups (for many purposes). However, Betz and Cohen showed that they could be viewed geometrically. 

Symplectic geometers have quantum cohomology, which on a symplectic manifold M is a deformation of singular cohomology using holomorphic spheres.

The geometric definition of the Steenrod square extends to quantum cohomology. This talk will describe the Steenrod square and quantum cohomology in terms of the intersection product, and then give a description of this quantum Steenrod square by putting these both together. We will describe some properties of the quantum squares, such as the quantum Cartan formula, and perform calculations in certain cases.

Manifolds, the main objects of study in Differential Geometry, do not have nice categorical properties. For example, the category of manifolds with smooth maps does not contain all fibre products.
The algebraic counterparts to this (varieties and schemes) do have nice categorical properties. 

A method to ‘fix’ these categorical issues is to consider C^infinity schemes, which generalise the category of manifolds using algebraic geometry techniques. I will explain these concepts, and how to translate to manifolds with corners, which is joint work with my supervisor Professor Dominic Joyce.

Quiver varieties are an attractive research topic of many branches of contemporary mathematics - (geometric) representation theory, (hyper)Kähler differential geometry, (symplectic) algebraic geometry and quantum algebra.

In the talk, I will define different types of quiver varieties, along with some interesting examples. Afterwards, I will focus on Nakajima quiver varieties (hyperkähler moduli spaces obtained from framed-double-quiver representations), stating main results on their topology and geometry. If the time permits, I will say a bit about the symplectic topology of them.

This talk will be a gentle introduction to braided fusion categories, with the eventual aim to explain a result from my thesis about symmetric fusion categories. 

Fusion categories are certain kinds of monoidal categories. They can be viewed as a categorification of the finite dimensional algebras, and appear in low-dimensional topological quantum field theories, as well as being studied in their own right. A braided fusion category is additionally commutative up to a natural isomorphism, symmetry is an additional condition on this natural isomorphism. Computations in these categories can be done pictorially, using so-called string diagrams (also known as ``those cool pictures''). 

In this talk I will introduce fusion categories using these string diagrams. I will then discuss the Drinfeld centre construction that takes a fusion category and returns a braided fusion category. We then show, if the input is a symmetric fusion category, that this Drinfeld centre carries an additional tensor product. All of this also serves as a good excuse to draw lots of pictures.