# Past Junior Geometry and Topology Seminar

A theorem of Gromov states that the number of generators of the fundamental group of a manifold with nonnegative

curvature is bounded by a constant which only depends on the dimension of the manifold. The main ingredient

in the proof is Toponogov’s theorem, which roughly speaking says that the triangles on spaces with positive

curvature, such as spheres, are thick compared to triangles in the Euclidean plane. In the talk I shall explain

this more carefully and deduce Gromov’s result.

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA.

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.