Oxford

In this talk, I will introduce a new framework for working with moduli stacks in enumerative geometry, aimed at generalizing existing theories of enumerative invariants counting objects in linear categories, such as Donaldson–Thomas theory, to general, non-linear moduli stacks. This involves a combinatorial object called the component lattice, which is a globalization of the cocharacter lattice and the Weyl group of an algebraic group.

Several important results and constructions known in linear enumerative geometry can be extended to general stacks using this framework. For example, Donaldson–Thomas invariants can be defined for a general class of stacks, not only linear ones such as moduli stacks of sheaves. As another application, under certain assumptions, the cohomology of a stack, which is often infinite-dimensional, decomposes into finite-dimensional pieces carrying enumerative information, called BPS cohomology, generalizing a result of Davison–Meinhardt in the linear case.

This talk is based on joint works with Ben Davison, Daniel Halpern-Leistner, Andrés Ibáñez Núñez, Tasuki Kinjo, and Tudor Pădurariu.

Einstein’s equations are difficult to solve and if you want to compute something in holography knowing an explicit metric seems to be essential. Or is it? For some theories, observables, such as on-shell actions and free energies, are determined solely in terms of topological data, and an explicit metric is not needed. One of the key tools that has recently been used for this programme is equivariant localization, which gives a method of computing integrals on spaces with a symmetry. In this talk I will give a pedestrian introduction to equivariant localization before showing how it can be used to compute the on-shell action of 6d Romans Gauged supergravity. 
 

I will describe some recent efforts to recreate the miraculous properties of perverse sheaves on complex analytic spaces in the setting of real stratified spaces.

We use Non-Reductive GIT to construct compactifications of Hilbert schemes of complete intersections. We then study ample line bundles on these compactifications in order to construct moduli spaces of complete intersections for certain degree types.

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

[arXiv:2501.03202]

Hoheisel's theorem states that there is some $\delta> 0$ and some $x_0>0$ such that for all $x > x_0$ the interval $[x,x+x^{1-\delta}]$ contains prime numbers. Classically this is proved using the Riemann zeta function and results about its zeros such as the zero-free region and zero density estimates. In this talk I will describe a new elementary proof of Hoheisel's theorem. This is joint work with Kaisa Matomäki (Turku) and Joni Teräväinen (Cambridge). Instead of the zeta function, our approach is based on sieve methods and ideas coming from additive combinatorics, in particular, the transference principle. The method also gives an L-function free proof of Linnik's theorem on the least prime in arithmetic progressions.