Navigated to

Ideas for degree projects

At the Department of Mathematics and Mathematical Statistics you can find a number of interesting ideas for degree projects.


Perhaps it's soon time to start writing your thesis. When choosing a thesis topic, you can consider:

Courses you have taken

Think back to the courses you have completed. Was there something in particular that you found interesting?

Here you can find courses offered by the Department of Mathematics and Mathematical Statistics.

People

Have you encountered a teacher/researcher you would like to talk to in order to discuss potential thesis topics?

Here you can find staff working at the Department of Mathematics and Mathematical Statistics.

Subject areas

Here you can find a number of interesting ideas for your degree project.

Computational Mathematics

Here you will find contact details of researchers active in the field. Feel free to contact any of them or visit their personal page for more information on potential thesis topics in Computational Mathematics.

Discrete Mathematics

Below you can read about possible degree projects in the field of Discrete Mathematics. If you want to know more, you are welcome to contact the respective person.

Combinatorics

Topic areas for bachelor or master theses:

  • Extremal graph theory
  • Ramsey theory
  • Combinatorial geometry
  • Linear algebraic methods in combinatorics

Example of possible thesis projects:

  • Finding regular subgraphs (Erdos-Sauer problem)
  • Ramsey theory of structured graph families (Erdos-Hajnal conjecture)
  • Finding large convex sets among points in general position (Erdos-Szekeres conjecture) and higher dimensional variants
  • Coloring geometric graphs
  • The Cap set problem and the slice-rank method
  • Non-vanishing linear maps (Alon-Jaeger-Tarsi conjecture) and hyperplane covers

This list showcases various topics in combinatorics with recent developments that I am interested in, and is not complete. I am open to negotiating several other projects as well. The only prerequisite is basic understanding of graph theory, combinatorics, and linear algebra.

Graph theory, discrete probability, extremal combinatorics

I can supervise theses at all levels on the following topics:

  • Random graphs, random graph models, percolation theory
  • Extremal graph theory, extremal problems for graphs, hypergraphs and set systems
  • Combinatorial games, especially games on graphs
  • Graph processes and graph automata such as bootstrap percolation or the spread of a virus through a network
  • Combinatorial number theory
  • All aspects of graph theory
  • amongst others

Extremal and probabilistic combinatorics, my domain of expertise, is a highly active area of research, with many exciting developments just in the last few years. I thus always have at hand a dozen or so possible projects linked to recent advances and results. If you are interested in writing a thesis under my supervision, just send me an email or knock on my office door to arrange a meeting. We can then discuss your mathematical background and interests before I suggest 3-4 possible thesis topics.

Graph theory, hypergraphs, and computational combinatorics

I can supervise projects that connect directly to my own research or belong to the general areas of mathematics in which I work.

Some examples:

  • Graph theoretical problems of different kinds
  • Hyperraphs and design theory
  • Problems in probabilitity theoy and random graphs
  • Mathematical physics, either with connections to statistical mechanics or quantum computing
  • Algorithms and complexity theory
  • The theory of combinatorial games. This includes both underlying theory and algorithms

Discrete geometry, coding theory and cryptography

Topic areas for bachelor thesis:

  • Geometry, in particular geometric combinatorics and projective geometry over finite fields.
  • Combinatorics, in particular graphs and other combinatorial objects such as hypergraphs, simplicial complexes, incidence geometries, matroids and designs.
  • Algebraic tools in geometry and combinatorics.
    Group theory, in particular symmetry groups acting on discrete and/or geometric objects.
  • Coding theory, cryptography, and other applications to computer science.

Discrete mathematics

Topic areas for bachelor theses:

  • Graph theory
  • Design theory
  • History and philosophy of mathematics
  • Axiomatic basis of mathematics

Example of possible thesis projects:

  • Writing an exposition of the Bruck-Ryser-Chowla theorem, with expanded definitions and explanations, and carefully worked examples
  • Writing a survey on progress in the open problem of the existence of biplanes
  • Generating complete lists of small combinatorial designs using computer programming

Representation theory and number theory

Topic area suggestions for a BSc or MSc thesis:

  • Computing with nilpotent orbits in classical Lie algebras
  • Integration with p-adic numbers. For a fun introduction to p-adic numbers, see: https://youtu.be/3gyHKCDq1YA
  • Survey of the classification of Lie algebras and Lie groups
  • How lattice models from statistical mechanics describe special functions in representation theory
  • Representations of the general linear group, Young tableaux and Kashiwara crystals
  • Fourier coefficients of automorphic forms
    Solvable lattice models and quantum groups

Many of the projects are suitable for a mixture of smaller research problems and a survey of existing literature depending on thesis-level and preference.

Functional analysis and operator theory

The study of various spaces of analytic functions is a topic active ongoing research. The most common spaces are different variants of Hardy, Bergman, BMOA and Bloch spaces. The basic theory of all of these provides good topics for thesis projects that can be adapted to the student's interests and ambitions. There are also many natural concrete operators acting on these spaces, such as Toeplitz, Hankel, Volterra and composition operators.

Possible topics:

  • Basic theory of Hardy and/or Bergman spaces
  • Bergman kernel and Bergman projection
  • Bloch space and conformal mappings
  • Theory of one or several concrete operators on these spaces
  • Carleson measures

It is also possible to make project of more functional analytic nature. These projects do not necessarily need to be linked to complex analysis or operator theory.

Possible topics:

  • Spectral theory and Fredholm operators
  • Topological vector spaces
  • Zorn's lemma in analysis
  • Other topics can also be discussed. Many of these topics are quite advanced for a BSc project; instead it possible to choose one these topics and study the basic theory needed for dealing with these.

Suggested literature (note that all of these books go way beyond the scope of a BSc/Msc project):

  • Zhu, Operator theory in function spaces
  • Zhu, Spaces of holomorphic functions in the unit ball
  • Böttcher and Silbermann, Analysis of Toeplitz operators
  • Pavlovic, Function classes in the unit disk
  • Rudin, Functional analysis
  • Horvath, Topological vector spaces and distributions

Mathematical Biology

Topic areas for bachelor theses:

  • Differential equations-based models of biological populations
  • Computational simulations of evolution
  • Spatial models of organismal growth
  • Network models of interactions

Example of possible thesis projects:

  • Modeling competition and cooperation in microbial populations
  • Simulating the growth and development of an organism following different programs in different environments
  • Applying physics-based or economics-based concepts to microbial populations

For more information, contact Eric Libby

Mathematical Finance and Economics

Topic areas for bachelor theses:

  • mathematical modeling and pricing of financial derivatives
  • modeling of natural resources and their extraction
  • financial risk management and insurance

Example of possible thesis projects:

  • applying Monte-Carlo simulation to price exotic derivatives contracts
  • use dynamic programming to determine optimal extraction rates of natural
  • resources such as minerals, forestry or fisheries
  • mathematical modeling of principal agent problems, incentives, both static and dynamics

For more information, contact Christian Ewald

Mathematical Foundations of Artificial Intelligence

Below you can read about possible degree projects in the field of Mathematical Foundations of Artificial Intelligence. At the bottom, you will find contact details of researchers active in the field. Feel free to contact any of them or visit their personal page for more information on potential thesis topics in Mathematical Foundations of Artificial Intelligence.

Compressive sensing

Prerequisites: Linear algebra, probability theory, optimization (optional), measure theory (optional).


Compressive sensing is a framework for solving underdetermined systems of (bi-)linear equations under structure constraints. The methods can be applied in practice to reconstruct images from e.g. a very small part of their Fourier spectrum. The algorithms developed in this framework can often only be proven to work for random instances of the problem, necessitating the use of random matrix theory. It is also possible to consider infinite-dimensional versions of the theory.

Distributed and federated optimization

Prerequisites: Optimization

In the age of the internet, we often find ourselves in situations where many users are interconnected, but cannot (or do not want to) send all their information to each other. Distributed and federated learning is a framework for developing methods for such communities of users to optimize a common function which they all only partially know.

Equivariance and neural networks

Prerequisites: Linear algebra, representation theory (optional) Lie groups/algebras (optional), machine learning (optional).

Equivariance is just a fancy way of saying that that a function respects a symmetry. In geometric deep learning, one investigates how neural networks can be designed to automatically obey symmetries.

Neural differential equations

Prerequisites: Differential equations, , Lie groups/algebras (optional).

Neural differential equations is a framework for modeling ‘infinitely deep’ neural networks using dynamical systems. Possible extensions include making the models equivariant to respect symmetries, or considering partial differential equations and connections to physics inspired models.

Operator splitting

Prerequisites: Optimization
Many optimization problems are of the form ‘minimize f(x) + g(x)’ , where f and g are two functions which have fundamentally different properties (one can be smooth and the other non-differentiable, for instance). Operator splitting schemes are methods for efficiently optimizing such functions.

Contact:

Mathematical Statistics

Here you will find contact details of researchers active in the field. Feel free to contact any of them or visit their personal page for more information on potential thesis topics in Mathematical Statistics.

Partial Differential Equations

Many fundamental laws of physics and chemistry can be formulated as ordinary or partial differential equations. In the social sciences, mechanics, optimization, control theory, economics and life sciences, differential equations and systems of differential equations are often used to model the behavior of complicated systems.

Example of possible thesis projects:

  • Management strategies for hydropower plants
  • Harvesting strategies and stability of fish population
  • Analysis of vibration in rotor systems
  • Hidden chaos in a Jeffcott rotor with clearance
  • The p-Laplace equation and the stochastic game tug-of-war
  • Growth of viscosity subsolutions of nonlinear PDEs in unbounded cylinders
  • Existence and uniqueness of solutions to systems of PDEs related to optimization

You may choose to work mainly with the application or with the mathematical theory, or both. If interested, contact:

 

Latest update: 2024-12-05

Man sitter och skriver med penna på papper

Examples of previous theses

Here you can find examples of previous theses in Mathematics and Mathematical Statistics at Bachelor's and Master's levels.