# Postgraduate Seminar

Organisers: Luke Peachey & Anna Skorobogatova

#### Term 1 2018-19 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

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.

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

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

**Hollis Williams**- TBC

Week 3: Wednesday 22nd January

TBC

Week 4: Wednesday 29th January

TBC

TBC

TBC

Week 7: Wednesday 19th February

TBC

Week 8: Wednesday 26th February

TBC

TBC

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

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