Mon, 10 May 2021

16:00 - 17:00
Virtual

An asymptotic expansion for the counting function of semiprimes

Dragos Crisan
(Oxford)
Abstract

A semiprime is a natural number which can be written as the product of two primes. Using elementary methods, we'll explore an asymptotic expansion for the counting function of semiprimes $\pi_2(x)$, which generalises previous findings of Landau, Delange and Tenenbaum.  We'll also obtain an efficient way of computing the constants involved. In the end, we'll look towards possible generalisations for products of $k$ primes.

Mon, 07 Jun 2021
14:15
Virtual

Stability of fibrations through geodesic analysis

Michael Hallam
(Oxford)
Abstract

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

Mon, 26 Apr 2021

16:00 - 17:00
Virtual

Motivic representations and finite rational points

Jay Swar
(Oxford)
Abstract

I will briefly introduce the Chabauty-Kim argument for effective finiteness results on "topologically rich enough" curves. I will then introduce the Fontaine-Mazur conjecture and show how it provides an effective proof of Faltings' Theorem.

In the case of non-CM elliptic curves minus a point, following work of Federico Amadio Guidi, I'll show how the relevant input for effective finiteness is provided by the vanishing of adjoint Selmer groups proven by Newton and Thorne.

Mon, 26 Apr 2021
12:45
Virtual

Calculation of zeta functions for one parameter families of Calabi-Yau manifolds

Philip Candelas
(Oxford)
Abstract

The periods of a Calabi-Yau manifold are of interest both to number theorists and to physicists. To a number theorist the primary object of interest is the zeta function. I will explain what this is, and why this is of interest also to physicists. For applications it is important to be able to calculate the local zeta function for many primes p. I will set out a method, adapted from a procedure proposed by Alan Lauder that makes the computation of the zeta function practical, in this sense, and comment on the form of the results. This talk is based largely on the recent paper hepth 2104.07816 and presents joint work with Xenia de la Ossa and Duco van Straten.

Mon, 26 Apr 2021
14:15
Virtual

Equivariant Seidel maps and a flat connection on equivariant symplectic cohomology

Todd Liebenschutz-Jones
(Oxford)
Abstract

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

Mon, 10 May 2021
14:15
Virtual

Hilbert schemes for fourfolds and Quot-schemes for surfaces

Arkadij Bojko
(Oxford)
Abstract

Counting coherent sheaves on Calabi--Yau fourfolds is a subject in its infancy. An evidence of this is given by how little is known about perhaps the simplest case - counting ideal sheaves of length $n$. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao--Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^{[n]})$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld--Johnson--Lim--Oprea--Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.

Wed, 27 Jan 2021

16:00 - 17:00

Multiplicative gerbes and H^4(BG)

Christoph Weis
(Oxford)
Abstract

The cohomology of a manifold classifies geometric structures over it. One instance of this principle is the classification of line bundles via Chern classes. The classifying space BG associated to a (Lie) group G is a simplicial manifold which encodes the group structure. Its cohomology hence classifies geometric objects over G which play well with its multiplication. These are known as characteristic classes, and yield invariants of G-principal bundles.
I will introduce multiplicative gerbes and show how they realise classes in H^4(BG) when G is compact. Along the way, we will meet different versions of Lie group cohomology, smooth 2-groups and a few spectral sequences.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

Mon, 08 Mar 2021

16:00 - 17:00
Virtual

Chen's theorem

Julia Stadlmann
(Oxford)
Abstract

In 1966 Chen Jingrun showed that every large even integer can be written as the sum of two primes or the sum of a prime and a semiprime. To date, this weakened version of Goldbach's conjecture is one of the most remarkable results of sieve theory. I will talk about the big ideas which paved the way to this proof and the ingenious trick which led to Chen's success. No prior knowledge of sieve theory required – all necessary techniques will be introduced in the talk.

Mon, 08 Feb 2021

16:00 - 17:00
Virtual

Recent progress on Chowla's conjecture

Joni Teravainen
(Oxford)
Abstract

Chowla's conjecture from the 1960s is the assertion that the Möbius function does not correlate with its own shifts. I'll discuss some recent works where with collaborators we have made progress on this conjecture.

Mon, 01 Feb 2021
14:15
Virtual

Leaf decompositions in Euclidean spaces

Krzysztof Ciosmak
(Oxford)
Abstract

In the talk I shall discuss an approach to the localisation technique, for spaces satisfying the curvature-dimension condition, by means of L1-optimal transport. Moreover, I shall present recent work on a generalisation of the technique to multiple constraints setting. Applications of the theory lie in functional and geometric inequalities, e.g. in the Lévy-Gromov isoperimetric inequality.

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