Oxford University

Given an endomorphism f:X --> X of a 'dualisable' object in a symmetric monoidal category, one can define its trace Tr(f). It turns out that the trace is 'universal' among the scalars we can produce from f. To prove this we will think of the 1d framed bordism category as the 'walking dualisable object' (using the cobordism hypothesis) and then apply the Yoneda lemma.
Employing similar techniques we can define 'hermitian' objects (generalising hermitian vector spaces) and prove that there is a 1-1 correspondence between Hermitian structures on a fixed object X and self-adjoint automorphisms of X. If time permits I will sketch how this relates to hermitian K-theory.

While all results of the talk hold for infinity-categories, they work equally well for ordinary categories. Therefore no knowledge of higher category theory is needed to follow the talk.

This talk will be a general introduction to perverse sheaves and their applications to the study of algebraic varieties, with a view towards enumerative geometry. It is aimed at non-experts.

We will start by considering constructible sheaves and local systems, and how they relate to the notion of stratification: this offers some insight in the relationship with intersection cohomology, which perverse sheaves generalise in a precise sense.

We will then introduce some technical notions, like t-structures, perversities, and intermediate extensions, in order to define perverse sheaves and explore their properties.

Time permitting, we will consider the relevant example of nearby and vanishing cycle functors associated with a critical locus, their relationship with the (hyper)-cohomology of the Milnor fibre and how this is exploited to define refined enumerative invariants in Donaldson-Thomas theory.

Equivariant cohomology is adapted from ordinary cohomology to better capture the action of a group on a topological space. In Floer theory, given an autonomous Hamiltonian, there is a natural action of the circle on 1-periodic flowlines given by time translation. Combining these two ideas leads to the definition of  $S^1$-equivariant symplectic cohomology. In this talk, I will introduce these ideas and explain how they are related. I will not assume prior knowledge of Floer theory.

Symplectic cohomology is a Floer cohomology invariant of compact symplectic manifolds 
with contact type boundary, or of open symplectic manifolds with a certain geometry 
at the infinity. It is a graded unital K-algebra related to quantum cohomology, 
and for cotangent bundle, it recovers the homology of a loop space. During the talk 
I will define symplectic cohomology and show some of the results on its (non) vanishing. 
Time permitting, I will also mention natural TQFT algebraic structure on it.

The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.

We give an alternative, statistical physics based proof of the Ω(d2^{-d}) lower bound for the maximum sphere packing density in dimension d by showing that a random configuration from the hard sphere model has this density in expectation. While the leading constant we achieve is not the best known, we do obtain additional geometric information: we prove a lower bound on the entropy density of sphere packings at this density, a measure of how plentiful such packings are. This is joint work with Felix Joos and Will Perkins.

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

The number of solutions of a given algebro-geometric configuration, when it is finite, does not change upon a small perturbation of the parameters; this persists 
even upon specializations that change the topology.    The precise formulation of this principle of Poncelet and Schubert   required, i.a., the notions of   algebraically closed fields, flatness, completenesss, multiplicity.     I will explain a model-theoretic version, presented in   quite different terms.  It applies notably to difference equations involving the Galois-Frobenius automorphism $x^p$, uniformly in a prime $p$.   In fixed positive characteristic, interesting technical problems arise that I will discuss if time permits.  

We are all familiar with the need for continuum mechanics-based models in physical applications. In this case, we are interested in large-scale water-wave problems, such as coastal flows and dam breaks.
When modelling these problems, we inevitably wish to solve them on a finite domain, and require boundary conditions to do so. Ideally, we would recreate the semi-infinite nature of a coastline by allowing any generated waves to flow out of the domain, as opposed to them reflecting off the far-field boundary and disrupting the remainder of our simulation. However, applying an appropriate boundary condition is not as straightforward as we might think.
In this talk, we aim to evaluate alternatives to so-called 'active boundary condition' absorption. We will derive a toy model of a shallow-water wavetank, and consider the implementation and efficacy of two 'passive' absorption techniques.