Category Archives: Yr 4 Description 2021/2022

Please click Older Posts at the bottom of the page and subsequent pages for further project descriptions.

Blebbing on Oscillating Surfaces – Supervisor: Dr Thomas Woolley

2021/2022, Double, Yr 4 Description 2021/2022

Title of project:
Blebbing on Oscillating Surfaces

Code:
TW2122B

Supervisor:
Dr. Thomas E. Woolley

Project description:
We propose to construct a mechanical model of muscle stem cell motion combining solid mechanics, stochastic theory and numerical simulations. During healing muscle stem cells adopt an amoeboid form of translocation, known as blebbing, in which cells are round and display highly dynamic plasma membrane extensions and retractions. The model will be a novel testing bed for biological hypotheses of cellular motion and illuminate connections between the multiple scales, as well as provide new mathematical insights into the connection between motion and surface geometry.
The aim is to model this mechanism of cellular motion and characterize the relationship between cell movement and the surrounding environmental geometry e.g. size, shape and topology. This relationship is usually discounted in experiments as extracted cells are placed on flat plates, or gels, which do not match the complicated heterogeneous environments that cells would have to naturally contend with. Mathematical modelling provides us with a way to test these disregarded factors and challenge current biological knowledge as to their importance.

Project offered as double module, single module, or both:
Double

Number of students who can be supervised on this project:
1

Categories and Classifying Spaces – Supervisor: Dr Ulrich Pennig

2021/2022, Yr 4 Description 2021/2022

Code: UP2122A

Project Title: Categories and Classifying Spaces

Description:
A category consists of a class of objects together with a set of maps (called morphisms or arrows) between any pair of objects and an operation of composition of arrows. The composition has to be associative, and each object has to have an identity arrow with domain and codomain equal to the object itself that behaves like an identity element for the composition. You have already encountered several categories in your mathematical life without even knowing it: The category of vector spaces and linear maps is the central object of study in Linear Algebra, the category of topological spaces and continuous maps is another one. 

Algebraic topology provides a construction that associates to any (small) category a topological space, called the classifying space of the category. These spaces often have very interesting properties and reflect certain topological features of your starting category. As an example: Any discrete group G gives rise to a category that has a classifying space denoted by BG. It turns out that the fundamental group of BG is isomorphic to G and topological invariants of BG often measure interesting features of the group G.

We will start the project by exploring a bit of category theory first. After that we will look into simplicial sets, simplicial spaces and nerves of categories. These constructions form the intermediate steps between categories and classifying spaces. Depending on how ambitious the candidate for this project feels, we may then have a look at a central theorem in K-theory called Bott periodicity, which has a proof in terms of classifying spaces given by Harris. This part would involve reading and understanding a research paper.

Type: Year 4 MMath Project

Supervisor: Ulrich Pennig

Prerequisite modules:
– Algebraic Topology (MA3008)
– Groups (MA0213) or Algebra II: Rings (MA3014)

Max. number of students: 1

Euler Characteristic and the Classification of Surfaces – Supervisor: Dr Ulrich Pennig

2021/2022, Double, Yr 4 Description 2021/2022

Title of project:
Euler Characteristic and the Classification of Surfaces

Code:
UP2122B

Supervisor: 
Dr Ulrich Pennig

Project description:
In the first lecture of the Algebraic Topology course we encountered the Euler characteristic of polytopes as a topological invariant. Polytopes are special cases of combinatorially defined topological spaces called simplicial complexes and the definition of the Euler characteristic can be adapted to work for these as well. If we triangulate a surface, it gives rise to a simplicial complex and therefore always has an associated Euler characteristic, which is independent of the choice of triangulation as we will see. Another important property of surfaces is their orientability, which tells us whether we can pick a full set of coordinate patches, such that the handedness of the coordinate system does not change going from one patch to the other. The torus from the lecture is orientable, while the Klein bottle is not.

The goal of this project is to understand the Euler characteristic of simplicial complexes in full generality and of surfaces in particular. We will then look at the orientability of surfaces. It turns out that the pair of Euler characteristic and orientability of a surface without boundary provides a complete homeomorphism invariant. This means that two surfaces without boundary are homeomorphic if and only if their Euler characteristics agree and they are both either orientable or not orientable. After having understood the definitions we will focus on proving this theorem.

Prerequisite Modules:
Algebraic Topology (MA3008)
Groups (MA0213) or Rings and Fields (MA3003)

Number of students who could be supervised for this project:
1