# Postgraduate Seminar

Organisers: Luke Peachey & Anna Skorobogatova

#### Term 2 2018-19 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 1: Wednesday 8th January

**Hollis Williams** - Penrose Inequalities in General Relativity

In this talk we give a brief introduction to Lorentzian manifolds before deriving and discussing some solutions to the Einstein field equations. Lorentzian geometry has a relatively small number of core theorems, many of them from the 1960s. We finish by discussing two of these, the positive mass theorem and the Penrose conjecture, the former of which has been proved in full generality in certain dimensions and the latter of which has only been proved in the so-called 'time-symmetric' case.

Week 2: Wednesday 15th January

**Arjun Sobnack - **It's All Downhill From Here

Gradient flows form an interesting class of (ordinary and partial) differential equations, for which the Łojasiewicz Inequalities (1965) are one of many useful tools. I will softly motivate and introduce gradient flows and present some applications of the inequalities. Time permitting, I will introduce the more general Łojasiewicz-Simon Inequalities (1983) and state some results about their application to non-linear evolution equations.

Week 3: Wednesday 22nd January

**No seminar.**

Week 4: Wednesday 29th January

**Alex Wendland**** -** How to Generate a Fair Coin in an Unfair World

In a world where not every kid gets candy, you will be unsurprised to hear that coins are biased. Given our crooked setting can we straighten things up?

Formally, given a p-coin (one that lands head with probability p) for what functions f : (0,1) -> (0,1) can we generate an f(p)-coin. Moreover, how would you generate them? Complicated functions like f(x) = 1/2, and f(x) = x^2 can be solved using an agile mind but something like f(x) = 2^(1/2) p^3 / ( 2^(1/2) − 5) p^3 + 11p^2 − 9p + 3 may cause a headache!

Joint work with Giulio Morina, Krzysztof Latuszynski, and Piotr Nayar.

**Marco Linton -**Hierarchies for one-relator groups

The notion of a HNN-extension of a group was first introduced 71 years ago as one of the essential building blocks of infinite groups. A group splits as a HNN-extension over some subgroup if and only if its abelianisation is strictly positive. If we fix a class of groups, one may ask a few questions about these splittings: What form can the subgroups that these groups split over take? If they remain in our fixed class, do they also split? If so, under iteration will we terminate at something nice?

In this talk we will first see how the classes of finitely generated or finitely presented groups may be too general to answer these questions in any sensible form. If we restrict our attention to the much simpler class of one-relator groups, we will show that everything is (sort of) as nice as possible. Time permitting, asymptotic results for these chains of splittings may also be discussed.

Week 6: Wednesday 12th February

**Matteo Barucco - **Equivariant Elliptic Cohomology, Kan we build it?

Important cohomology theories are associated to one dimensional Algebraic groups. Ordinary cohomology is associated to the additive group, while K-Theory is associated to the multiplicative one. Following this correspondence it's therefore natural to consider cohomology theories associated to elliptic curves and try to construct equivariant versions of them. After an historical overview of the problem of building these theories, we will try to understand which requirements our cohomology theory should satisfy, and try to give a breezy introduction to G spectra and algebraic models for them, the natural place where we construct our model.

Week 7: Wednesday 19th February

**Tasos Stylianou -** A local limit theorem for hyperbolic rational maps

The periodic orbit structure of hyperbolic rational maps is a long studied subject. Inspired by famous results in number theory, various authors tried to explain the chaotic nature of repelling periodic orbits of such maps. Motivated by a classical result in probability theory, I will try to explain how we can calculate asymptotically the number of periodic orbits satisfying some special properties. No previous knowledge will be assumed.

Week 8: Wednesday 26th February

**Steven Groen -** A unique pair of triangles

Today our goal is to find all pairs consisting of a right triangle and an isosceles triangle, both with integer sides, whose area and perimeter are equal. In 2018 it was shown that only one such pair exists, using some neat tools in arithmetic geometry.

Week 9: Wednesday 4th March

**Jaromir Sant - **Need for Speed - Rates of Convergence to Equilibrium for 1 dimensional Diffusions

Ergodicity is a central theme in the study of diffusion processes, and ensures that the long time behaviour of a process can be studied meaningfully. I'll introduce the notions of ergodicity and Harris recurrence for general state space stochastic processes, show how the two notions are linked in a general setting, and illustrate how this allows one to relate time averages to state space averages via Birkhoff's ergodic theorem. I will then focus on obtaining rates of convergence in this theorem, specifically for the case of 1D diffusions where the scenario is much easier to deal with. Through the use of speed and scale one can control the rate of convergence in the ergodic theorem by means of moments of hitting times of singletons. I will briefly comment on the extra problems that need to be tackled in higher dimensions time permitting.

Week 10: Wednesday 11th March

TBC

#### Term 3 2018-19 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 6:

Week 8:

Week 9:

Week 10:

**Previous talks:**

#### Term 1 2018-19

Week 1: Wednesday 2nd October

**Anna Skorobogatova **- Compensated Compactness: a brief introduction

Compactness by compensation originates from ground-breaking results by Murat and Tartar in the 1970s, showing that weak convergence of a sequence with an additional differential constraint can allow one to show convergence/lower semi-continuity of some non-linear quantities depending on this sequence. In general, compensated compactness refers to improving an a priori convergence assumption to a stronger form of convergence by imposing a PDE constraint. I will give an overview of the main known results in this field, and I will discuss the obstacles we face when trying to establish this theory in an L1 setting. Fourier analysis plays a key role, as we will see.

Week 2: Wednesday 9th October

**Yani Pehova** - Quasi-randomness in permutations

In combinatorial problems, it is often helpful to consider a continuous approximation of a large discrete object in order to capture some of its properties in a robust way. In this talk, I will give an introduction to the theory of permutation limits, and consider the problem of forcing the limit of a sequence of combinatorial structures to be the same as the limit of a sequence of randomly generated structures of the same type.

Week 3: Wednesday 16th October

**Luke Peachey** - On the Gauss-Bonnet-Chern Theorem

The Gauss-Bonnet Theorem is a simple relationship between the topological and geometric information of a Riemannian surface. We will explore this relationship further in higher dimensions and for general vector bundles. Only the basic definitions from geometry and topology will be assumed.

Week 4: Wednesday 23rd October

**Solly Coles** - Symbolic Dynamics for Hyperbolic Flows

Smale's Axiom A flows have been studied in the field of dynamical systems and ergodic theory since their introduction in 1967. They are in some sense a generalisation of geodesic flow on a Riemannian manifold with negative sectional curvature. The work of Bowen in the early 1970s shows they can be effectively modelled using symbolic dynamics, which is much more well-understood. We introduce Axiom A and the approach of Bowen, before describing a specific application to growth of closed geodesics.

Week 5: Wednesday 30th October

**Phil Hanson **- Some Models in Population Genetics

From Kingman and his coalescent in 1982 to Fisher modelling Mendelian inheritance in 1918 (which is the first known use

of the term standard deviation); the study of population genetics has a rich history. It is a field where not only can we

attempt to descrive biology with mathematical models but the qualitative aspects of those models in turn feed back into

the biology.

In this talk we introduce some of the basic models, their scaling limits, duality relations between forward and backward

in time models and measure-valued diffusions.

Week 6: Wednesday 6th November

N/A

Week 7: Wednesday 13th November

**Christoforos Panagiotis** - Convergence of square tilings to the Riemann map

A well-known theorem of Rodin and Sullivan, previously conjectured by Thurston, states that the circle packing of the intersection of a lattice with a simply connected planar domain $\Omega$ into the unit disc $\mathbb{D}$ converges to a Riemann map from $\Omega$ to $\mathbb{D}$ when the mesh size converges to 0. The aim of this talk is to sketch the proof of the analogous statement when circle packings are replaced by another discrete version of the Riemann mapping theorem, the square tilings of Brooks, Smith, Stone and Tutte. Joint work with Agelos Georgakopoulos.

Week 8: Wednesday 20th November

**Alessio Borzi** - Cyclotomic Numerical Semigroups

A numerical semigroup is a submonoid of the natural numbers $\mathbb{N}$, respect to addition, with finite complement in $\mathbb{N}$. The aim of this talk is to present a rather surprising connection with cyclotomic polynomials and to discuss a related conjecture.

Week 9: Wednesday 27th November

**Bogdan Alecu** - Geometric griddability of permutation classes

Week 10: Wednesday 4th December

**Elena Zamaraeva** - On the number of 2-threshold functions

We consider 2-threshold functions over a 2-dimensional integer grid of a fixed size MxN, that is the functions which can be represented as the conjunction of two threshold functions. The asymptotic on the number of threshold functions is known to be $\frac{6M^2N^2}{pi^2} + O(M^2N^2)$. We provide an asymptotic formula for the number of 2-threshold functions. To achieve this goal we establish a one-to-one correspondence between almost all 2-threshold functions and pairs of integer segments with specific properties. We expect this bijection to be useful in algorithmic studies of 2-threshold functions. Joint work with Prof. Jovisa Zunic.