# Possible Master's Projects If you are a Master's student at KTH or SU and wish to do your Master's thesis with me, you are welcome to contact me. Below I outline a few possible projects. The purpose is to give you an idea of the kinds of problems we may tackle. They are not set in stone. ## Project Title: Field-Theoretic RG for Neural Networks ### Project Description: The human brain is an example of a biological neural network. Unlike the neural networks used in machine learning, the connections between neurons in biological networks are often local. It is thus possible to study a coarse-grained version of such networks using a neural field model, which is a nonlinear stochastic partial differential equation (PDE). Furthermore, a key difference between artificial and biological networks is that the latter is not homogeneous -- every neuron, even those of the same cell type, differs from others. A possible way to account for this heterogeneity is to include quenched noise in neural field models. In our [recent work](https://arxiv.org/abs/2503.21605), we introduced field-theoretic renormalization group (RG) techniques to study the large-scale properties of one such model. This has opened up the possibility to explore other related models and investigate certain key questions within these models, e.g., entropy production. The student needs to be familiar with either quantum field theory or statistical field theory. --- ## Project Title: Spectral Code on GPUs ### Project Description: I will first introduce two seemingly unrelated problems. Linear waves appear in many hydrodynamic problems in fluids and plasmas. For example, ocean waves, Alfvén waves, drift waves, etc. Under typical physical conditions, the nonlinear effects in these problems are small but not zero. Even a small nonlinearity can give rise to wave turbulence in such problems. Typically, such problems are studied in one or two dimensions. Neural networks are nonlinear systems driven by stochastic noise. Biological neural networks are often locally connected and can be described by PDEs called neural field models, e.g., Wilson-Cowan equations. Typical neural field problems are studied in two dimensions. There are similar key questions in both of these problems: 1. Scaling of the energy spectrum? 2. Non-Gaussian statistics or intermittency. In practice, numerically, these are very difficult problems. It is possible to write a pseudo-spectral code in two dimensions to study both of these problems, which can be run on a single GPU to achieve very high resolutions. This project is appropriate for students with experience in numerical computations. The code will be released as an open-source project. --- ## Project Title: Field Theories for Wave Turbulence ### Project Description: Linear waves appear in many hydrodynamic problems in fluids and plasmas. For example, ocean waves, Alfvén waves, drift waves, etc. Under typical physical conditions, the nonlinear effects in these problems are small but not zero. A small nonlinearity can give rise to wave turbulence in such problems. Typically, such problems appear in one or two dimensions. The usual analytical approach is to develop a closed equation for the second moment of the wave function -- this is typically a nonlinear integral equation called the kinetic equation. We have taken an alternative approach using field-theoretic renormalization group techniques for this problem. This has opened up the possibility to go beyond the kinetic equation. The student needs to be familiar with either quantum field theory or statistical field theory. --- ## Project Title: Statistical Mechanics of a Möbius Strip ### Project Description: This project addresses a problem that is more "cute" than useful: studying thermal fluctuations of a surface in the form of a Möbius strip within a Monte Carlo framework. We already have a code to simulate membranes with fixed boundaries. Recently, a summer student generalized the code to include stress-free boundaries. The next goal is to use this code to study elastic ribbons -- membranes that are much longer in one direction than the othe -- and then extend this from ribbons to Möbius strips. We aim to determine the mean equilibrium configuration of a Möbius strip and compare it with existing results. Additionally, we will calculate the fluctuation spectra and response function (through the Green-Kubo relation). This project is appropriate for students with experience in numerical computations. The code will be released as an open-source project.