## Postgraduate Seminars

### Summer 2018

These seminars (unless otherwise stated) will take place on **Thursday lunchtimes (times and rooms TBC)**, most likely somewhere in the Mathematics Department, on an (almost) weekly basis ** - **see the map for further details. Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.

### 26 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

### 10 May 2018 at 12pm in Room 505

#### Speaker: Alexander Stokes

**Supervisor(s): Prof Rod Halburd**

###### Title: The Painleve 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 Painlev\'e 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.

** **

### 17 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.

** **

### 24 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.

** **

### 31 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!

### 07 June 2018 (Room and Time TBC)

#### 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.