These seminars (unless otherwise stated) will take place on Thursdays at 1pm in Drayton House Lecture Theatre B19 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.

## 25 April 2019

**Speaker: David Sheard**

Supervisor: Assoc Prof L Louder

##### Title: Can forgetful computers do group theory?

**Abstract:**

In 1931 Gödel proved that there were problems in mathematics which can never be solved...fiddlesticks! To add insult to injury in 1936 Church and Turing independently showed that even among those problems which are solvable, there are some which take an infinite amount of time to solve algorithmically...double-fiddlesticks!! Since this time, questions of computability (decision problems) have entered all corners of mathematics, some of the most well-known being posed by Dehn: the word, conjugacy, and isomorphism problems in group theory. I shall introduce a model for a particularly stupid type of computer (a Turing machine with no memory) called a finite state automaton, and discuss whether such computers can do group theory. If/when they can, I'll discuss how they can help solve some of Dehn's decision problems.

## 02 May 2019

**Speaker: Albert Wood**

Supervisor: Dr F Schulze

##### Title: Singularities of Mean Curvature Flow

**Abstract:**

This time last year, I gave a talk to the postgraduate seminar on the beauty of Mean Curvature Flow. This year marks the release of the much anticipated sequel: `Mean Curvature Flow - when it all goes wrong'. Join me as I give a tour of the darker side of this much-studied flow, right into the deepest nooks and crannies of singularities. Though the flow takes place in any number of dimensions, it will be projected onto a 2D screen for convenient viewing.

It isn't necessary to have watched the original to understand this sequel - I will be giving all the explanation necessary to understand the material at the start of the lecture.

## 09 May 2019

**Speaker: Ardavan Afshar**

Supervisor: Prof A Granville

##### Title: A brief introduction to integer partitions

**Abstract:**

I will give a quick introductory tour of the theory of partitions (ways of writing a number as a sum of positive integers), culminating in a discussion of a famous and beautiful formula of Ramanujan from almost exactly one century ago.

## 16 May 2019

**Speaker: Atheeta Ching**

Supervisor: Dr S Baigent

##### Title: Atheeta vs. Predator: Requiem

**Abstract:**

Relationships between different species can be complex. For competitive Kolmogorov models, there is a special hypersurface called the carrying simplex which essentially contains all the interesting dynamics. We extend this to non-competitive systems where species can co-operate or have a predator-prey type relationship. Our most recent work covers the 3-species case!

## 23 May 2019

**Speaker: Michele Giacomini**

Supervisor: Prof A Yafaev

##### Title: Folding numbers

**Abstract:**

The aim of the talk will be to answer the question "what points inside a blank square of paper can be precisely marked by folding alone?". To motivate this question and point out another connection between origami and maths I will briefly present a mathematical method that can aid in the design process of new origami models. The talk will include lots of pretty picture and some origami props.

## 30 May 2019

**Speaker: Federico Barbacovi**

Supervisor: Assoc Prof E Segal

##### Title: Geometry, Algebra and category theory

**Abstract:**

One of the basic ideas of Algebraic geometry is that sometimes, to obtain geometric information about a space X, one should attach to it some algebraic data. At some point, also category theory entered the game, and the algebraic data people tried to associate to the space X wasn’t anymore a group or a ring, but a category. However, it’s not always clear how one side of this correspondence recover information about the other. In the talk, I will focus on two examples that share the nice property of being (nearly) interpretable in terms of drawings.

## 06 June 2019

**Speaker: Georgina Al-Badri**

Supervisor: Dr N Ovenden

##### Title: A mathematical model of the formation of blood vessels in vitro

**Abstract:**

Engineered tissues, whether small grafts or whole organ replacements, require a vascular structure to provide essential nutrients for growth and function of the tissue. One effective method to ensure survival once implanted in vivo is pre-vascularisation, where endothelial cells are included in the tissue (the cells that line blood vessels), and cultured appropriately.

Endothelial cells can assemble into capillary-like structures in vitro, however the resulting network depends on many factors, including the scaffold material, and presence of chemical factors, oxygen concentration, and presence of mechanical forces. We are developing a mathematical model, in the form of coupled PDEs modelling the cell density, scaffold density, and chemical concentration, including these key influences, to aid the design and culture of pre-vascularised engineered tissues. By combining mathematical, computational, and experimental techniques, we can predict the optimum conditions for a functional network, cutting out many trial and error experiments bothin vitro and in animal models.