Thu, 07 Jul 2022
12:00
C2

### Resonances and unitarity from celestial amplitude

Dr Jinxiang Wu
((Oxford University))

Note: we would recommend to join the meeting using the Zoom client for best user experience.

Abstract

We study the celestial description of the O(N) sigma model in the large N limit. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. This is a joint work with D. García-Sepúlveda, A. Guevara, J. Kulp.

Wed, 06 Jul 2022
12:00
C2

### Pushing Forward Rational Differential Forms

Robert Moermann
(University of Hertfordshire)

Note: we would recommend to join the meeting using the Zoom client for best user experience.

Abstract

The scattering equations connect two modern descriptions of scattering amplitudes: the CHY formalism and the framework of positive geometries. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry in the CHY moduli space. In this talk, I consider the general problem of pushing forward rational differential forms via the scattering equations. I will present some recent results (2206.14196) for achieving this without ever needing to explicitly solve any scattering equations. These results use techniques from computational algebraic geometry, and they extend the application of similar results for rational functions to rational differential forms.

Tue, 22 Feb 2022
14:00
C2

### Minimum degree stability and locally colourable graphs

Freddie Illingworth
(Oxford)
Abstract

We tie together two natural but, a priori, different themes. As a starting point consider Erdős and Simonovits's classical edge stability for an $(r + 1)$-chromatic graph $H$. This says that any $n$-vertex $H$-free graph with $(1 − 1/r + o(1)){n \choose 2}$ edges is close to (within $o(n^2)$ edges of) $r$-partite. This is false if $1 − 1/r$ is replaced by any smaller constant. However, instead of insisting on many edges, what if we ask that the $n$-vertex graph has large minimum degree? This is the basic question of minimum degree stability: what constant $c$ guarantees that any $n$-vertex $H$-free graph with minimum degree greater than $cn$ is close to $r$-partite? $c$ depends not just on chromatic number of $H$ but also on its finer structure.

Somewhat surprisingly, answering the minimum degree stability question requires understanding locally colourable graphs -- graphs in which every neighbourhood has small chromatic number -- with large minimum degree. This is a natural local-to-global colouring question: if every neighbourhood is big and has small chromatic number must the whole graph have small chromatic number? The triangle-free case has a rich history. The more general case has some similarities but also striking differences.

Wed, 02 Mar 2022

16:00 - 17:00
C2

### Amenable actions and groups

Paweł Piwek
(University of Oxford)
Abstract

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.)

Thu, 10 Mar 2022

15:00 - 16:00
C2

### Gauge theories in 4, 8 and 5 dimensions

Alfred Holmes
(University of Oxford)
Abstract

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.

Thu, 24 Feb 2022

15:00 - 16:00
C2

TBC
Thu, 17 Feb 2022

15:00 - 16:00
C2

### Torsion points on varieties and the Pila-Zannier method - TALK POSTPONED UNTIL WEEK 5

Francesco Ballini
(Oxford University)
Abstract
In 2008 Pila and Zannier used a Theorem coming from Logic, proven by Pila and Wilkie, to give a new proof of the Manin-Mumford Conjecture, creating a new, powerful way to prove Theorems in Diophantine Geometry. The Pila-Wilkie Theorem gives an upper bound on the number of rational points on analytic varieties which are not algebraic; this bound usually contradicts a Galois-theoretic bound obtained by arithmetic considerations. We show how this technique can be applied to the following problem of Lang: given an irreducible polynomial f(x,y) in C[x,y], if for infinitely many pairs of roots of unity (a,b) we have f(a,b)=0, then f(x,y) is either of the form x^my^n-c or x^m-cy^n for c a root of unity.
Mon, 07 Mar 2022

16:00 - 17:00
C2

Benjamin Bedert
Mon, 21 Feb 2022

16:00 - 17:00
C2