Mon, 17 Feb 2020
14:15
L4

### Twisted indices of 3d supersymmetric gauge theories and enumerative geometry of quasi-maps

Heeyeon Kim
(Oxford)
Abstract

I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. In the first part of the talk, I will show that the twisted index computes the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface, which can be understood algebraically as quasi-maps to the Higgs branch. I will explain 3d N=4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to these moduli spaces. Finally, I will present a wall-crossing formula for these invariants derived from the gauge theory point of view.

Tue, 03 Dec 2019
12:00
L4

### Lie polynomials and a Penrose transform for the double copy

Lionel Mason
(Oxford)
Abstract

This talk will explain how Lie polynomials underpin the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides).  ABHY have recently shown that Lie polynomials arise naturally also in the geometry of the space K_n of momentum invariants, Mandelstams, and can be expressed in the space of n-3-forms dual to certain associahedral (n-3)-planes. They also arise in the moduli space M_{0,n} of n points on a Riemann sphere up to Mobius transformations in the n-3-dimensional homology.  The talk goes on to give a natural correspondendence between K_n and the cotangent bundle of M_{0.n} through which the relationships of some of these structures can be expressed.  This in particular gives a natural framework for expressing the CHY and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories and goes some way to expressing their double copy relations.   This is part of joint work in progress with Hadleigh Frost.

Mon, 10 Jun 2019

16:00 - 17:00
C1

### The Golod-Shafarevich Theorem: Endgame

Jay Swar
(Oxford)
Abstract

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.

Mon, 03 Jun 2019

16:00 - 17:00
C1

### The Golod-Shafarevich Theorem

Jay Swar
(Oxford)
Abstract

The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.

Mon, 15 Oct 2018

16:00 - 17:00
C3

### Periods and the number Zagier forgot

(Oxford)
Abstract

A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.

We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.

Mon, 09 Dec 2019

16:00 - 17:00
C1

### TBA

Alyosha Latyntsev
(Oxford)
Mon, 02 Dec 2019

16:00 - 17:00
C1

### What the L! The surprising world of L-functions

George Robinson
(Oxford)
Abstract

L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.

Mon, 25 Nov 2019

16:00 - 17:00
C1

### When shifted primes do not occur in difference sets

Zoe Wang
(Oxford)
Abstract

Let $[N] = \{1,..., N\}$ and let $A$ be a subset of $[N]$. A result of Sárközy in 1978 showed that if the difference set $A-A = \{ a - a’: a, a’ \in A\}$ does not contain any number which is one less than a prime, then $A = o(N)$. The quantitative upper bound on $A$ obtained from Sárközy’s proof has be improved subsequently by Lucier, and by Ruzsa and Sanders. In this talk, I will discuss my work on this problem. I will give a brief introduction of the iteration scheme and the Hardy-Littlewood method used in the known proofs, and our major arc estimate which leads to an improved bound.

Mon, 18 Nov 2019

16:00 - 17:00
C1

### Erdős' primitive set conjecture

Jared Duker Lichtman
(Oxford)
Abstract

A subset of the integers larger than 1 is called $\textit{primitive}$ if no member divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ over $n$ in a primitive set $A$ is universally bounded for any choice of $A$. In 1988, he famously asked if this universal bound is attained by the set of prime numbers. In this talk we shall discuss some recent progress towards this conjecture and related results, drawing on ideas from analysis, probability, & combinatorics.

Mon, 20 Jan 2020

14:15 - 15:15
L4

### Symplectic geometry of Conical Symplectic Resolutions

Filip Zivanovic
(Oxford)
Abstract

Conical Symplectic Resolutions form a broad family of holomorphic symplectic manifolds that are of interest to mathematical physicists, algebraic geometers, and representation theorists; Nakajima Quiver Varieties and Hypertoric Varieties are known as their special cases. In this talk, I will be focused on the Symplectic Geometry of Conical Symplectic Resolutions, and its non-symplectic applications. More precisely, I will talk about my work on finding Exact Lagrangian Submanifolds inside CSRs, and work in progress (joint with Alexander Ritter) about the construction of Symplectic Cohomology on CSRs.

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