Postgraduate Seminars Summer 2018

See the map for further details. Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.

26th April 2018 at 12pm in Room 505

Speaker: Jessica Renton

Supervisor(s): Prof Karen Page

Title: Invasion dynamics on epithelia

Abstract:
Epithelia are the tissues which form surfaces and linings, such as skin. It is of particular medical interest to study the dynamics of mutant invasion in these structures as the majority of cancers originate in epithelial cells. While evolutionary graph theory provides a robust framework for modelling invasion in structured populations, it relies on several unrealistic assumptions: most importantly that the structure is static. It is therefore questionable whether it can provide insight into a system such as an epithelium which is inherently dynamic. In this talk I will describe the EGT model and compare it to a more realistic mechanical model of an epithelium (the Voronoi Tessellation model). In particular I will discuss results regarding the success of co-operators in a prisoners dilemma game in both models.

03 May 2018
No seminar

10th May 2018 at 12pm in Room 505

Speaker: Alexander Stokes

Supervisor(s): Prof Rod Halburd
Title: The Painlevé equations and what happens when applied maths meets algebraic geometry

Abstract:
I will talk about six extraordinary nonlinear second-order ODE’s known as the Painlevé equations. The study of these equations and their discrete counterparts forms one of the cornerstones of the field of integrable systems. This began in the 1960’s with the mathematical discovery of solitons, and can be loosely described as the study of nonlinear systems that are remarkably ordered or in some sense exactly solvable. Though the study of the Painlevé equations began around the turn of the 20th century as part of the hunt for new special functions, the modern approach to them is through algebra and geometry. The talk should be accessible to anyone regardless of background, and the story will take detours through so many areas of mathematics that everyone in the audience should see a link to something they are interested in, whether that be something like special functions, travelling waves, quantum gravity, reflection groups, Dynkin diagrams, or blowups of algebraic surfaces.

17th May 2018 at 12pm in Room 505

Speaker: Albert Wood

Supervisor(s): Dr Felix Schulze

Title: A crash course on Mean Curvature Flow

Abstract:
Geometric flows are a really hot topic right now, since the Ricci flow was employed by Perelman to prove the long-standing Poincare conjecture. In this talk I will give a birds-eye view of a different flow that much of the UCL maths department is obsessed with - Mean Curvature Flow. I’ll try to motivate it’s study for non-experts, as well as show you some cool examples in more dimensions than you can count.

24th May 2018 at 12pm in Room 505

Speaker: Johnny Nicholson

Supervisor(s): Prof Frank Johnson

Title: An Introduction to Non-Simply Connected Topology

Abstract:
At the beginning of the 20th century, Henri Poincaré asked if homology, a new tool he had developed which attaches a sequence of abelian groups to a space, contained enough information to tell whether a given 3-manifold was a sphere. He soon found a counterexample and reformulated his question into what later became known as the Poincaré conjecture and, in doing so, discovered a way to assign a certain group, known as the fundamental group, to any topological space.

This separates the study of manifolds into the simply connected manifolds, i.e. those whose fundamental group is trivial, and the non-simply connected manifolds. In the simply connected world, many of the algebraic objects that are studied turn out to be commutative and thus amenable to standard techniques. This has led to many advancements and occupies much of the current research in algebraic topology. In the non-simply connected world however, these techniques break down and we are instead left to wrestle with hard problems in areas such as integral representation theory, algebraic k-theory and module theory.

In this talk, we will begin by giving an overview of the basic notions in algebraic topology for those that have no familiarity with the subject. We will then explore certain problems in topology for which the fundamental group plays a key role. In particular, we will point out certain phenomena which only have a chance of occurring when the fundamental group is large and complicated. If I have time at the end, I will talk briefly about some recent progress I have made on an algebraic problem which has its roots in (and hopefully some concrete applications to) some of the problems in topology that I will have mentioned.

I will aim to make this accessible to a wide audience, though some familiarity with what a group and a module are would be very helpful.

31st May 2018 at 12pm in Room D103

Speaker: Belgin Seymenoglu

Supervisor(s): Dr Steve Baigent

Title: Zeeman’s catastrophe machine

Abstract:
As mentioned in my Chalkdust article for Issue 07, Zeeman was keen on studying systems for which a continuous change of forces led to sudden, discontinuous effects - such effects are called “catastrophes”. To demonstrate Catastrophe Theory, he proposed a machine exhibiting the so-called catastrophic behaviour, and demonstrated it at the Royal Institution Christmas lectures in 1978.

I will bring along my replica of Zeeman’s catastrophe machine (made of cardboard, pins and rubber bands), so I can show you the dramatic “flipping” behaviour that sometimes happens as you gradually move the free rubber band around. But why does the machine flip? Can we find the bifurcation set governing the flipping? What’s the bifurcation diagram of the system? And (if time permits) what other scenarios did Zeeman try to apply Catastrophe Theory to?

After the talk is over, we can all have a play with the catastrophe machine!

7th June 2018 at 12pm in Room 505

Speaker: Sam Porritt

Supervisor(s): Prof Andrew Granville

Title: Three approaches to the Erdos-Kac Theorem

Abstract:
In 1939, Erdos and Kac proved that the number of prime divisors of a randomly chosen large integer is distributed approximately like a Gaussian random variable. I will highlight the underlying probability theory behind three different proofs of their theorem. No background in probability or number theory is required as, although I do want to talk about the actual proofs, the emphasis will be on the broad ideas rather than the technical details. I’ll also briefly mention something I’m working on and a multivariable analogue.