Tue, 05 Nov 2019

15:30 - 16:30
L4

### Hilbert schemes of points of ADE surface singularities

Balazs Szendroi
(Oxford)
Abstract

I will discuss some recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge, and their connection to combinatorics (of coloured partitions) and representation theory (of affine Lie algebras and related algebras such as their W-algebra).

Mon, 11 Nov 2019

16:00 - 17:00
C1

### On Serre's Uniformity Conjecture

Jay Swar
(Oxford)
Abstract

Given a prime p and an elliptic curve E (say over Q), one can associate a "mod p Galois representation" of the absolute Galois group of Q by considering the natural action on p-torsion points of E.

In 1972, Serre showed that if the endomorphism ring of E is "minimal", then there exists a prime P(E) such that for all p>P(E), the mod p Galois representation is surjective. This raised an immediate question (now known as Serre's uniformity conjecture) on whether P(E) can be bounded as E ranges over elliptic curves over Q with minimal endomorphism rings.

I'll sketch a proof of this result, the current status of the conjecture, and (time permitting) some extensions of this result (e.g. to abelian varieties with appropriately analogous endomorphism rings).

Mon, 04 Nov 2019

16:00 - 17:00
C1

### What is Arakelov Geometry?

Esteban Gomezllata Marmolejo
(Oxford)
Abstract

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Arakelov theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

Mon, 28 Oct 2019

16:00 - 17:00
C1

### Cartier Operators

Zhenhua Wu
(Oxford)
Abstract

Given a morphism of schemes of characteristic p, we can construct a morphism from the exterior algebra of Kahler differentials to the cohomology of De Rham complex, which is an isomorphism when the original morphism is smooth.

Thu, 31 Oct 2019

13:30 - 14:30
L3

### Simplicity of Tannakian Categories (COW Seminar)

Martin Gallauer
(Oxford)
Abstract

Let A be a Tannakian category. Any exact tensor functor defined on A is either zero, or faithful. In this talk, I want to draw attention to a derived analogue of this statement (in characteristic zero) due to Jack Hall and David Rydh, and discuss some remarkable consequences for certain classification problems in algebraic geometry.

Tue, 22 Oct 2019

14:00 - 15:00
L6

### Homomorphisms from the torus

Matthew Jenssen
(Oxford)
Further Information

We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus Z_m^n, where m is even, to any fixed graph. Our main result establishes the "phase coexistence" phenomenon in a strong form: it shows that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some "dominant phase". This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper q-colourings of Z_m^n (so in particular, the discrete hypercube). For the proof we develop a `Cluster Expansion Method', which we expect to have further applications, by combining machinery from statistical physics, entropy and graph containers. This is joint work with Peter Keevash.

Mon, 04 Nov 2019

16:00 - 17:00
L4

### An optimal transport formulation of the Einstein equations of general relativity

Andrea Mondino
(Oxford)
Abstract

In the seminar I will present a recent work joint with  S. Suhr (Bochum) giving an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and  optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

Thu, 31 Oct 2019
11:30
C4

### Constructing geometries

Kobi Kremnitzer
(Oxford)
Abstract

In this talk I will explain a category theoretic perspective on geometry.  Starting with a category of local objects (of and algebraic nature), and a (Grothendieck)
topology on it, one can define global objects such as schemes and stacks. Examples of this  approach are algebraic, analytic, differential geometries and also more exotic geometries  such as analytic and differential geometry over the integers and analytic geometry over  the field with one element. In this approach the notion of a point is not primary but is  derived from the local to global structure. The Zariski and Huber spectra are recovered  in this way, and we also get new spectra which might be of interest in model theory.

Tue, 29 Oct 2019
12:00
L4

### Motivic Galois Theory and Feynman integrals

Erik Panzer
(Oxford)
Abstract

Feynman integrals govern the perturbative expansion in quantum field theories. As periods, these integrals generate representations of a motivic Galois group. I will explain this idea and illustrate the 'coaction principle', a mechanism that constrains which periods can appear at any loop order.

Thu, 14 Nov 2019
16:00
L6

### Propinquity of divisors

Ben Green
(Oxford)
Abstract

Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n)^{log 2} divisors. Must some of these be close together? Hooley's Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)^c for some c = 0.353.... given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely.

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