University of Oxford

In this talk I will discuss my extension of the Koszul duality theory of Beilinson, Ginzburg, and Soergel to the more general setting of quasi-abelian categories. In particular, I will define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders' embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of derived categories for certain categories of modules over Koszul monoids and their duals. The key examples of categories for which this theory works are the categories of complete bornological spaces and the categories of inductive limits of Banach spaces. These categories frequently appear in derived analytic geometry.

In this talk we cover recent work in collaboration with Diego Granziol and Steve Roberts where we study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian and derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. 

Knot projections for knots in the 3-sphere allow us to easily describe knots, compute invariants, enumerate all knots, manipulate them under Reidemister moves and feed them into a computer. One might hope for a similar representation of knots in general 3-manifolds. We will survey properties of knots in general 3-manifolds and discuss a proposed diagram-esque representation of them.

Amenable actions are answering the question: "When can we prevent things like the Banach-Tarski Paradox happening?". It turns out that the most intuitive measure-theoretic sufficient condition is also necessary. We will briefly discuss the paradox, prove the equivalent conditions for amenability, give some ways of producing interesting examples of amenable groups and talk about amenable groups which can't be produced in these 'elementary' ways.

Teaser question: show that you can't decompose Z into finitely many pieces, which after rearrangement by translations make two copies of Z. (I.e. that you can't get the Banach-Tarski paradox on Z.)

The aim is introducing the Bieri-Neumann-Strebel invariants and showing some computations. These are geometric invariants of abstract groups that capture information about the finite generation of kernels of abelian quotients.

In the 1980s, gauge theory was used to provide new invariants (up to
diffeomorphism) of orientable four dimensional manifolds, by counting
solutions of certain equations up to to a choice of gauge. More
recently, similar techniques have been used to study manifolds of
different dimensions, most notably on Spin(7) and G_2 manifolds. Using
dimensional reduction, one can find candidates for gauge theoretic
equations on manifolds of lower dimension. The talk will give an
overview of gauge theory in the 4 and 8 dimensional cases, and how
gauge theory on Spin(7) manifolds could be used to develop a gauge
theory on 5 dimensional manifolds.

Great progress has been made recently in exploiting categorical/topological/higher symmetries in quantum field theory. I will explain how the same structure is realised directly in the lattice models of statistical mechanics, generalizing Kramers-Wannier duality to a wide class of models. In particular, I will give an overview of my work with Aasen and Mong on using fusion categories to find and analyse lattice topological defects in two and 1+1 dimensions.  These defects possess a variety of remarkable properties. Not only is the partition function is independent of deformations of their path, but they can branch and fuse in a topologically invariant fashion.  The universal behaviour under Dehn twists gives exact results for scaling dimensions, while gluing a topological defect to a boundary allows universal ratios of the boundary g-factor to be computed exactly on the lattice.  I also will describe how terminating defect lines allows the construction of fractional-spin conserved currents, giving a linear method for Baxterization, I.e. constructing integrable models from a braided tensor category.

The global symmetries of a d-dimensional quantum field theory (QFT), and their ’t Hooft anomalies, are conveniently captured by a topological field theory (TFT) in (d+1) dimensions, which we may refer to as the Symmetry TFT of the given d-dimensional QFT. This point of view has a vast range of applicability: it encompasses both ordinary symmetries, as well as generalized symmetries. In this talk, I will discuss systematic methods to compute the Symmetry TFT for QFTs realized by M-theory on a singular, non-compact space X. The desired Symmetry TFT is extracted from the topological couplings of 11d supergravity, via reduction on the space L, the boundary of X. The formalism of differential cohomology allows us to include discrete symmetries originating from torsion in the cohomology of L. I will illustrate this framework in two classes of examples: M-theory on an ALE space (engineering 7d SYM theory); M-theory on Calabi-Yau cones (engineering 5d superconformal field theories).

In this talk, I will talk about the category of coherent sheaves on the affine Flag variety of a simply-connected reductive group over $\mathbb{C}$. In particular, I'll explain how the convolution product naturally leads to a construction of the Iwahori-Hecke algebra, and present some combinatorics related to computing duals in this category.

 I will present a scenario where the early universe is in a topological phase of gravity.  I will discuss a number of analogies which motivate considering gravity in such a phase. Cosmological puzzles such as the horizon problem provide a phenomenological connection to this phase and can be explained in terms of its topological nature. To obtain phenomenological estimates, a concrete realization of this scenario using Witten's four dimensional topological gravity will be used. In this model, the CMB power spectrum can be estimated by certain conformal anomaly coefficients. A qualitative prediction of this phase is the absence of tensor modes in cosmological fluctuations.