Tue, 03 Nov 2020
14:00
Virtual

Combinatorics from the zeros of polynomials

Julian Sahasrabudhe
(Cambridge)
Further Information

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

Let $X$ be a random variable, taking values in $\{1,…,n\}$, with standard deviation $\sigma$ and let $f_X$ be its probability generating function. Pemantle conjectured that if $\sigma$ is large and $f_X$ has no roots close to 1 in the complex plane then $X$ must approximate a normal distribution. In this talk, I will discuss a complete resolution of Pemantle's conjecture. As an application, we resolve a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for, so called, strong Rayleigh distributions. I will also discuss how these sorts of results shed light on random variables that arise naturally in combinatorial settings. This talk is based on joint work with Marcus Michelen.

Mon, 01 Jun 2020
14:15
Virtual

Homological mirror symmetry for log Calabi-Yau surfaces

Ailsa Keating
(Cambridge)
Abstract

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.

Tue, 25 Feb 2020
14:00
L6

Coordinate Deletion

Eero Räty
(Cambridge)
Abstract

For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.

Thu, 13 Feb 2020

15:00 - 16:00
C5

Jacobian threefolds, Prym surfaces and 2-Selmer groups

Jef Laga
(Cambridge)
Abstract

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

Thu, 21 Nov 2019

11:30 - 12:30
C4

On NIP formulas in groups

Gabriel Conant
(Cambridge)
Abstract

I will present joint work with A. Pillay on the theory of NIP formulas in arbitrary groups, which exhibit a local formulation of the notion of finitely satisfiable generics (as defined by Hrushovski, Peterzil, and Pillay). This setting generalizes ``local stable group theory" (i.e., the study of stable formulas in groups) and also the case of arbitrary NIP formulas in pseudofinite groups. Time permitting, I will mention an application of these results in additive combinatorics.

Mon, 25 Nov 2019
15:45
L6

Irrationality and monodromy for cubic threefolds

Ivan Smith
(Cambridge)
Abstract

The homological monodromy of the universal family of cubic threefolds defines a representation of a certain Artin-type group into the symplectic group Sp(10;\Z). We use Thurston’s classification of surface automorphisms to prove this does not factor through the genus five mapping class group.  This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds, as established by Clemens and Griffiths.
 

Thu, 17 Oct 2019

12:00 - 13:00
L4

Quasi-normal modes on asymptotically flat black holes

Dejan Gajic
(Cambridge)
Abstract

A fundamental problem in the context of Einstein's equations of general relativity is to understand precisely the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late time intervals and carry information about the nature of the black hole, much like the normal frequencies of a vibrating string. These frequencies are called quasi-normal frequencies or resonances and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will discuss a new method of defining and studying resonances for linear wave equations on asymptotically flat black holes, developed from joint work with Claude Warnick.

Mon, 29 Apr 2019
15:45
L6

Knots, SL_2(R) representations, and a total Lin invariant

Jacob Rasmussen
(Cambridge)
Abstract

X.S. Lin defined an invariant of knots in S^3 by counting represenations 
of the knot group into SU(2) with fixed meridinal holonomy. Lin's 
invariant was subsequently shown to coincide with the Levine-Tristam 
signature. I'll define an analogous total Lin invariant which counts 
repesentations into both SU(2) and SL_2(R). Unlike the SU(2) version, this 
invariant does not (as far as I know) coincide with other known 
invariants. I'll describe some applications to left-orderability of Dehn 
fillings and branched covers, as well as a curious connection with the 
Alexander polynomial. This is joint work with Nathan Dunfield.

Mon, 10 Jun 2019

14:15 - 15:15
L4

Moduli of polarised varieties via canonical Kähler metrics

Ruadhai Dervan
(Cambridge)
Abstract

Moduli spaces of polarised varieties (varieties together with an ample line bundle) are not Hausdorff in general. A basic goal of algebraic geometry is to construct a Hausdorff moduli space of some nice class of polarised varieties. I will discuss how one can achieve this goal using the theory of canonical Kähler metrics. In addition I will discuss some fundamental properties of this moduli space, for example the existence of a Weil-Petersson type Kähler metric. This is joint work with Philipp Naumann.

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