Historical examples illustrate how dramatic such events can be: during the so-called “Flash Crash” of 2010, U.S. equity markets lost nearly a trillion dollars in value within minutes before rapidly rebounding. Episodes like this highlight how quickly instability can emerge – often without clear, advance warning.

Being able to detect early signs of instability is therefore crucial for investors and institutions. However, identifying these warning signals is far from straightforward. Crashes are rare, their precursors are often subtle, and different markets may exhibit entirely different types of behaviour. A strategy that works well for traditional equity indices may fail completely when applied to cryptocurrencies or emerging assets. This diversity makes the problem both challenging and highly relevant. This project aims to develop mathematical and computational tools for identifying anomalous patterns in financial time series that may indicate an upcoming market crash.

Student background: Participants should have a working knowledge of a programming language (e.g. Python, R, Julia, or MATLAB). Familiarity with time series analysis, statistics, or stochastic processes would be advantageous.

In this project, we will use phase field methods to model and simulate the swelling of a particle. Such methods utilize an auxiliary function, the phase field, to describe a moving interface as a level set of a smoothly varying function in contrast to applying boundary conditions on a sharp interface. The advantage is that there is no need to employ moving meshes and complicated boundary conditions, while the main challenge is to formulate the correct phase field equations to get the correct dynamics of the interface.

Depending on the background and skills of the group, students will build on existing models and code to model and simulate swelling of spherical particles. Several extensions and developments are possible to adapt the model to different materials, and focus may be put more towards theory of the model equations, or towards coding and simulations.

Students' background: A course in PDE:s is useful, including numerical methods for PDE:s. Interest in modelling of physical processes and some programming experience (e.g. Python, Matlab or C++) is also welcome.

This project studies optimal investment decisions under uncertainty using a real options framework.  Our model assumes that new and improved technologies emerge over time due to technological progress. Therefore, when considering an investment, firms should realize that a better opportunity could arise in the near future, which provides an incentive to delay the investment. However, waiting risks a competitive disadvantage and the possibility that the innovation will take much longer than anticipated.

The project combines analytical techniques (value-matching and smooth-pasting conditions) with numerical methods to characterize optimal investment thresholds and optimal scale choices. Depending on the skills of the group, several extensions may be addressed.

Student background: ...

The European organisation ENTSO-E has decided that all major grid operators should implement PRA in 2027. One of the major concerns is how probabilistic uncertainty can be effectively modelled in such systems. For example, uncertainty in load parameters can lead to a probability of overloading power lines or generators. To address this problem, the proposed techniques for this project involve data preprocessing, graph theory, probability theory and Monte-Carlo simulation. Knowledge on power grid modeling and learning is appreciated but not a requirement and the project is open for other techniques and methodologies. A fundamental question is therefore how uncertainty in one or in multiple components lead to the probability of the violation of operational constraints. In this project the goal is to find effective ways of modelling and characterizing this probabilities. 

Students' background: This project is fit for a broad range of students within STEM, for instance students in Mathematics, Physics, and Electrical, Physical, Computer, or Industrial Engineering programs.

To address these limitations, the water treatment industry is shifting toward optimized "combined disinfection" strategies. By integrating ultraviolet (UV) radiation with reduced chlorine dosages, it is possible to achieve superior pathogen inactivation while simultaneously lowering the chemical footprint of the treatment process.

The primary objective of this research is to develop and validate a comprehensive mathematical model that characterizes the temporal dynamics of bacterial populations and disinfectant concentrations. This model aims to provide a robust framework for understanding the complex interactions between microbial life, UV light, and chemical agents, ultimately facilitating more precise and safer water treatment protocols.

Student background: This project is fit for a broad range of students within STEM, for instance students in Mathematics and other basic sciences like Physics, Biology or Chemistry, with a clear focus on using mathematical tools to make progress in real-world problems. Students from all branches of engineering can also fit into this project. In particular, students are expected to:

• Formulate a Model: Create a system of equations that accounts for direct chemical inactivation, UV-induced mortality, and complex interactions like chlorine photolysis.

• Analyze Qualitative Dynamics: Use dynamical systems theory to predict long-term outcomes and ensure the model remains physically plausible.

• Perform Parameter Estimation: Calibrate the model against experimental data.

• Propose experimental procedures: Design experiments which will potentially allow us to learn relevant parameter values.

In this project, you will work with a semi-synthetic observational dataset to model individual treatment effects; identify distinct subgroups of patients based on the effects and develop treatment policies that could improve overall outcomes. A key focus will be on creating interpretable policies that are understandable to clinicians. The main methods involved include machine learning-based estimators for Conditional Average Treatment Effects (CATE) and rule-extraction techniques for defining treatment policies.

Student background: Students are expected to have basic knowledge of a programming language (Python or R) and statistics. Familiarity with Machine Learning is helpful but not required

Two modelling approaches are possible. One option is to formulate traffic flow as a system of conservation laws on a network, where each road segment is described by a partial differential equation and the main challenge is to define suitable coupling conditions at intersections. Another option is to use the Cell Transmission Model, a discrete-time and discrete-space approximation that is often more directly accessible computationally. Ambitious groups may also compare the two approaches.

The project combines applied mathematics, scientific computing, graph-based modelling, and data-driven reasoning. A successful outcome could include formulation of a traffic model on the given network, implementation of a prototype simulator, visualization of traffic patterns and bottlenecks, and comparison of baseline and disturbance scenarios. The work may also provide a basis for future integration into DTCC’s digital twin software environment.

Student background: This project is open to students from a broad range of STEM backgrounds. Useful preparation includes some experience with differential equations, numerical methods, scientific computing,graph  theory, optimization, or programming in Python, MATLAB, Julia, or similar environ-ments. Students interested in mathematical modelling, simulation, transportation systems, or urban applications are especially welcome. The project can accommodate both students who prefer theoretical analysis and students who prefer implementation and computational experiments.

Digital pathology-systems capture many small, overlapping image patches of a slide and combine them into a single high-resolution overview image. Due to slight distortions, misalignments, and illumination differences, these patches cannot be directly merged without visible seams or artifacts.

The goal of this project is to develop methods for stitching such images into a seamless, visually consistent overview that supports smooth zooming and panning. The core task will focus on aligning and blending a small grid of images (e.g., 2×2 or 3×3), using the overlap between neighboring patches to estimate geometric transformations and reduce visible boundaries.

A key modelling challenge is how to combine local pairw

ise alignments into a globally consistent reconstruction. This can be approached using tools from optimization, geometry, or graph-based methods. Once a working method is established, it can be extended to larger grids.

Since both the input data and the final image can be large, it is also important to consider computational efficiency. Students are encouraged to reflect on memory usage and scalability, for example through tiling strategies or efficient representations.

Relevant techniques include image processing, computer vision, linear algebra, and numerical optimization. The project is open to different methodological approaches.

Student background:
Suitable for students in Mathematics, Computer Science, Engineering Physics, Electrical Engineering, Biomedical Engineering, or related fields. Interest in image analysis, programming or computational methods is an advantage.