# Past Junior Geometry and Topology Seminar

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.

In this talk I will discuss the problem of finding Einstein metrics in the homogeneous and cohomogeneity one setting.

In particular, I will describe a recent result concerning existence of solutions to the Dirichlet problem for cohomogeneity one Einstein metrics.

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.

Topological field theories (TFT's) are physical theories depending only on the topological properties of spacetime as opposed to also depending on the metric of spacetime. This talk will introduce topological field theories, and the work of Freed and Hopkins on how a class of TFT's called "invertible" TFT's describe certain states of matter, and are classified by maps of spectra. Constructions of field theories corresponding to specific maps of spectra will be described.

Despite its fame there appears to be little literature outlining Lurie's proof sketched in his expository article "On the classification of topological field theories." I shall embark on the quixotic quest to explain how the cobordism hypothesis is formalised and give an overview of Lurie's proof in one hour. I will not be able to go into any of the motivation, but I promise to try to make the talk as accessible as possible.