# Doctoral projects in Mathematics and Applied Mathematics

Here you can find a description of the doctoral projects that you can apply for. The projects are connected to the research groups of the research environment MAM (Mathematics and Applied Mathematics).

## This is what you need to do

1. Please look through the projects and choose which of the proposed doctoral projects that you are most interested in taking part in.

2. Write a motivational letter where you briefly describe why you are interested in that/those project(s) and how they relate to your background and future goals.

3. Include the motivational letter in addition to the rest of your application.

Jobbannons och ansökan till Doktorand i matematik/tillämpad matematik External link.

## Algebra and Analysis with Applications

Read about the Algebra and Analysis with Applications research group.

### Possible doctoral projects

#### Catalogue of Mathematical Objects

The goal of this project is ultimately to create a new branch of mathematics literature, that focuses on examples and definitions rather than theorems. The reason for doing this is that having the right examples to study is often crucial for making progress on a research problem, but traditional publication puts little weight on examples, so finding one with sought properties in the literature can be close to hopeless.

The idea is to set up a database of mathematical objects — their constructions and properties, as well as the definitions of those properties, and whatever additional materials might be appropriate to get a grip on them – which is searchable online, and where mathematicians may publish such nuggets which would not fit as separate works in traditional journals. As with Wikipedia, content would be user-generated and user-reviewed, but unlike Wikipedia it would be open for claims not previously published elsewhere. Because examples conversely do not have to be new, the threshold for contributing to a field where one is not an expert is lowered, encouraging cross-disciplinary advancements.

The successful applicant would participate in building the infrastructure for the Catalogue. Parts of this will involve creating the website, and experience of this is a merit of note. Likewise experience of software development where there is not tight integration; various technologies to be employed in the Catalogue include TeX, graph databases of the RDF kind, and digital signatures. Familiarity with these is a plus, but on several points not expected.

The successful applicant would also conduct research in mathematics unrelated to the Catalogue; the area here is not constrained, except that it must be in some area for which the MDU math department can provide qualified supervision. The purpose of this is that the applicant should develop an understanding of and familiarity with the practices of the mathematics research community at large, since that will make the applicant better equipped to make sound decisions on the design of the Catalogue.

Moreover, this is a multidisciplinary position. The successful applicant will get a co-supervisor from outside the math department, with expertise in the development of (as appropriate) products, services, organisations, etc. Such methods shall be used to scientifically evaluate and improve how the Catalogue works together with its users, for example to identify workflows that are unnecessarily cumbersome or details that lend themselves to destructive behaviour. Part of the research time will be spent with the research group of this co-supervisor, to develop a practical understanding of the methods and theories used there.

It is expected that the resulting thesis will contain material both on the work with the Catalogue and from the unrelated mathematics research.

### Contact

Lars Hellström lars.hellstrom@mdu.se

#### Algebraic Combinatorial Computer Science

Matroid theory, a branch of algebraic combinatorics, captures the essence of various properties of many mathematical objects and has natural connections to a variety of mathematical areas, including graph theory, linear algebra, topology, algebraic and combinatorial geometry, algorithms, and optimization. An extension of matroids, known as polymatroids, particularly plays a significant role in combinatorial optimization by representing a class of submodular functions that enable efficient solutions for certain types of optimization problems. Polymatroids also have natural connections to several mathematical objects such as Shannon entropy and hypergraphs, which matroids do not. Matroid and polymatroid theory have undergone significant development in recent years and have been successfully applied to solve several problems across various areas of mathematics and computer science. In computer science, for instance, these theories have been used for problems in combinatorial algorithm optimization, flow control, and capacity allocation in network systems, analysis of large datasets for machine learning, and studies of properties of distributed storage in data networks and algebraic coding theory.

In an era of intensive research and demand for new communication and information technology, there are many intriguing questions in computer science to explore where new mathematics is needed. Matroids and polymatroids, for example, can be used to investigate issues related to vector spaces and entropy to analyze various data networks. Additionally, other general algebraic structures besides vector spaces, such as groups, rings, and modules, are of interest for several computer science problems. For these algebraic structures, LRW-lattices can generally be used for analysis as opposed to matroids and polymatroids. LRW-lattices are an interesting but relatively unexplored area that generalizes matroids and polymatroids. In this project, a doctoral candidate is sought with a focus on developing the theory of matroids, polymatroids, and LRW-lattices to solve computer science problems through algebraic combinatorics. It would also be very interesting to investigate how connections between algebraic geometry and matroids can be used and generalized to polymatroids and LRW-lattices to solve various computer science problems.

## Contact

Thomas Westerbäck thomas.westerback@mdu.se

#### Superalgebras and Supergravity

Understanding the basic structure of space, time, and matter is of paramount importance, not only from a theoretical point of view, but as technology is advancing towards smaller length scales, also from the point of view of applications. For a long time, physicists have struggled with what is perhaps the greatest challenge of all modern science, namely the unification of two of the most successful theories of the 20th century: general relativity and quantum mechanics. General relativity, famously discovered by Einstein, is a theory of gravity that has been successful in describing the universe at large length scales, while quantum mechanics becomes relevant at small length scales or high energies, where gravity can be neglected. *Quantum gravity* is the name given to the (yet unknown) theory that unifies them, and that is needed for *e.g.* answering questions about the early universe and the nature of black holes. A promising candidate for quantum gravity is M-theory, which has eleven-dimensional supergravity as its low energy limit.

In this interdisciplinary project, we want to investigate what non-associative superalgebras can tell us about supergravity, and about quantum gravity beyond its low energy limit.

More details about the project Pdf, 322.1 kB.

### Contact

Per Bäck per.back@mdu.se

#### Commutative Algebra and Robotics Motion

This is a brief proposal for a PhD project in mathematics, with interdisciplinary connections to robotics research and computer science. A key part to solving robotics motion problems involves solving polynomial systems, which can be done in a number of ways using tools from, for example, commutative algebra. One approach to solving robotics motion problems involves polynomial elimination theory, using tools such as resultants. Another way to solve these problems is by using methods from numerical algebraic geometry and commutative algebra, and other computational algebra techniques. This might include, for example, the use and study of Gröbner bases. Robotics motion also has relations to rigidity theory, a geometric aspect of mathematics that relates to graph theory, matroid theory, representation theory, and algebraic geometry.

One possible starting point, from a mathematical perspective, could be to study the Rees algebra of an ideal. Two fundamental problems in this study are to determine the defining ideals of the algebra, and to detect whether the algebras are Koszul. This has been considered recently, for Rees algebras and multi-Rees algebras, and is open for many classes of ideals in both settings. One way to address these questions involves an explicit computation of quadratic Gröbner bases, which is difficult in general. Research on this topic would include an in-depth study of methods within commutative algebra, homological algebra, graph theory, and algebraic geometry.

Such a PhD project would be appropriate for a student interested in both algebra and in applications to robotics. The PhD student would take courses and study topics both within pure mathematics and within applied robotics research or other aspects of computer science. This project could be heavily tailored to suit individual interests. One possibility includes having primary supervision within mathematics, aiming to study recent advances in the algebraic methods relevant to the study of robotics motion, which would be informed by co-supervision within the robotics research group.

### Contact

Peder Thompson, peder.thompson@mdu.se

#### Optimization adapted for Clifford Neural Networks

Recently, Clifford algebras have been used in the context of convolutional neural networks. Such convolutional neural networks, called Clifford neural layers, have been proved to be more efficient for solving certain problems such as PDE-based problems, e.g., the Navier–Stokes equations. The project aims to study Clifford neural layers and to examine their strengths and weaknesses. More specific goals of the project are:

- to apply Clifford analysis in the backpropagation step in Clifford neural networks. It has been shown that real partial derivatives are not as efficient as “quaternionic” partial derivatives in quaternionic neural networks, which are a special case of Clifford neural networks. The same holds true for complex convolutional layers;
- to design optimizers that work better with Clifford neural layers since a major challenge in this method is the optimization of the layers due to computational cost.

### Contact

Masood Aryapoor, masood.aryapoor@mdu.se

#### Efficient Counting with Group Actions

In geometry and arithmetic it is often very difficult to say even very basic things about an object of interest. Which is its dimension? Does it consist of several pieces? Do any solutions exist at all? It is then all the more remarkable that in some cases of interest, one can answer these questions (and much more!) by performing computations over finite sets. However, these finite sets become very large very quickly. To exploit this small miracle, one still needs to find clever ways to efficiently make computations in large sets. Moreover, one often needs to take symmetry into account (which can be both a blessing and a curse).

In this project, the aim is to attack two such problems:

- Counting points in general position over finite fields: Given a variety over a finite field, the set of all general -tuples of points on contains a lot of geometric and arithmetic information. Using a computer, we would like to efficiently count the number of such points (taking both the action of the symmetric group and the action of Frobenius into account).
- Given a finite set with an action of a group and a partial relation we would like to efficiently compute the Möbius function of (seen as a partially ordered set). Near optimal non-parallel algorithms are known if one does not take the -action into account but parallel algorithms and algorithms taking into account do not seem to have been considered.

Both these problems will require innovative combinations of ideas from mathematics and computer science (especially parallel programming and high-performance computing).

### Contact

Olof Bergvall, olof.bergvall@mdu.se

## Discrete Mathematics and Modelling of Behaviour and Culture

Read about the research group Discrete Mathematics and Modelling of Behaviour and Culture

### Possible doctoral projects

#### Cultural evolution and AI

The field of cultural evolution entails the formal study of how individual behaviours, worldviews and societies emerge, spread and change. When the capacity for culture evolved in humans, a new, more rapid, type of evolutionary processes – cultural evolution – started to shape our behaviours far beyond biological evolution.

There is an ongoing research project at MDU in collaboration with Stockholm University to develop theory, through mathematical models, for cultural evolution and how it is influenced by digital technology, especially AI. The research project covers questions from evolutionary transitions, where AI might impose the next transition, with new conditions for life on earth, to related more immediate societal changes such as polarisation, democracy, and knowledge dispersal. These analyses will be applied to answer the guiding research question: How do digital information technologies transform cultural evolution, and what are the likely effects on individuals and society?

### Contact

Fredrik Jansson fredrik.jansson@mdu.se

#### Polarisation in digital societies

The project entails the development of mathematical models to study polarisation and its influence by digital technology and AI. The models are inspired by research on cultural systems and belief systems conducted at AMF, and by psychology to study sociological issues. The main idea is that beliefs are interdependent and can be organised in graphs, for example by how compatible they are. When we acquire a belief, it influences what other beliefs we can acquire, since they need to fit and be compatible. If, for example, people acquire slightly different beliefs initially, then after filtering new beliefs through the previous ones, they can end up with completely different clusters, and thus polarize into different belief systems. Some open research questions are:

- What polarisation patterns can be generated by different belief structures?
- Which filtering processes realise which patterns?
- How is this influenced by information technology?
- How will AI filter and disperse information?

### Contact

Fredrik Jansson fredrik.jansson@mdu.se

#### Discrimination in employment processes (supported by new technology)

This project has been developed with the sociologist Moa Bursell at MDU. From a sociological perspective, inclusion in the labour market is a central theme, but some parts of the involved processes have received little attention, and the theory is underdeveloped.

The project aims to investigate under what conditions discrimination increases/decreases as a result of the supply and demand for labour, i.e., how it is affected by employer versus employee selection. For this, both new theory, which is developed with the help of mathematical models to predict relationships between firm level and macrolevel, and the development of statistical methods, e.g. to study rankings, are needed.

Preliminary work has shown that some of this can be done analytically but otherwise through simulations, and the methods include, among other things, the development of variants of ML estimation using Fischer's information matrix. The work group for organization, work life, and leadership works with large-scale data collections related to working life, and may have useful data.

### Contact

Fredrik Jansson fredrik.jansson@mdu.se

#### Understanding how large language models learn language

In recent years, the development of AI systems for processing and producing language has progressed immensely. However, most of the methods and techniques underlying OpenAI ChatGPT and other similar systems require an immense amount of data for training and are in many aspects black boxes. In the quest of understanding human language, there is a need to explore cognitively plausible AI systems and to understand how they achieve their capabilities. In a funded VR project that I (Jérôme Michaud, mathematics MDU) have together with Dr. Anna Jon-And (linguistics SU), we are developing cognitive architectures with the aim to understand how human acquire language using a minimalistic approach. In our VR project, we are interested in the induction of grammar from exposition of linguistic data only. Our approach relies on principles of reinforcement learning and only includes domain general cognitive abilities, such as chunking, generalizing, or sequence perception. Our project currently focuses on the syntactic properties of language and future work will focus on including semantics as well. In contrast, large language models natively capture meaning using word embeddings but are much harder to understand. In particular, the acquisition phase of language by large language models has not been examined in any detail since most of these models are pretrained.

A possible project would be to investigate the training phase of some large language model and compare it with other artificial systems targeted at language learning and grammar induction such as the model developed in our VR project. Using newly developed explainability techniques, it should be possible to examine in more detail how large language model learn. A suitable student would be interested in language acquisition, have an interest in AI systems for natural language processing, and programming and mathematical skills to be able to implement and study the algorithms.

### Contact

Jérôme Michaud jerome.michaud@mdu.se

## Stochastic Processes, Statistics and Financial Engineering

Read about the research group Stochastic Processes, Statistics and Financial Engineering

### Possible doctoral projects

#### Option Pricing and Hedging using Artificial Intelligence (AI) Technology

In the field of financial engineering, option pricing and hedging is an important content of research. Machine learning, as a branch of Artificial Intelligene, focus on learning patterns and relationships from raw data. In the most recent studies on option pricing/hedging problems, machine learning techniques, for example, the neural network architectures, have shown great advantage and have become the "name of the game" in the field of quantitative finance. This project deals with option pricing and hedging problems by using different types of machine learning algorithms.

### Contact

Ying Ni ying.ni@mdu.se

#### Stochastic Control Problems - Development of ML-based Algorithms

The project will be about building Machine Learning based algorithms to solve stochastic control problems and backward stochastic differential equations. These algorithms will be developed using Artificial Intelligence techniques such as (different types of) neural networks and Monte Carlo simulations as well as regressions. The applications of these algorithms will be in finance (such as pricing and hedging options) and in energy finance (such as valuation of gas storage).

### Contact

Achref Bachouch achref.bachouch@mdu.se

## Engineering Mathematics

Read about the research group Engineering Mathematics

### Possible doctoral projects

#### Modelling of Developmental Regulatory Networks with Delay Differential Equations

The project is interdisciplinary in nature and combines different fields of Applied Mathematics and Computational Biology.

The main aim of the project is to study mathematical models of genetic regulations which play a fundamental role in the developmental processes, such as the body plan formation of an organism. As the case study, the model of spatio-temporal pattern formation by gene networks in the early development of the Drosophila embryos will be considered. In previous studies, such gene networks have been modeled with systems of nonlinear reaction-diffusion equations based on the assumption that the transcription of genes and the translation of mRNA into proteins occur instantaneously.

For better understanding of the functioning of gene networks, we propose a more realistic model that takes into account the time delays in the production of proteins and results in the system of delay differential equations. We will address the questions of priori and posteriori identifiability of model parameters. The problem of minimizing the amount of quantitative data needed for parameter estimation in our model will be formulated from the point of view of Optimal Experimental Design.

The new model is expected to give us a better understanding of the underlying mechanism of gene networks, i.e., it can remove remaining ambiguities in regulatory interactions. The reverse engineering approach which we will design in this project can be used to study developmental systems in other organisms.

### Contact

Maksat Ashyralyyev maksat.ashyralyyev@mdu.se

#### Modelling of Ocean Waves on Unstructured Grids

The primary objective of this Ph.D. research is to investigate and describe ocean waves using mathematical models executed on unstructured grids based on the concept of mesh density function. Specifically, this density function will be associated with bathymetric ocean data. This research is highly relevant to the fields of oceanography and ocean engineering, as it pertains to applications in climate research and renewable energy, such as offshore wind and wave energy conversion.

This Ph.D. research will contribute to the understanding of ocean wave dynamics and their impacts on coastal regions and offshore engineering activities. The key contributions include:

- Development and implementation of mathematical models for ocean wave simulations on unstructured grids.
- Investigation of the interactions between wind, waves, and currents in complex oceanic environments.
- Application of the research findings to climate research, offshore wind energy, and wave energy conversion.

The research will utilize the SWAN (Simulating WAves Nearshore) third-generation wave model to represent the effects of wind-generated surface gravity waves. Unstructured grids obtained through a generalized conforming bisection (GCB) method will be employed to achieve highly adaptable grid resolution in complex oceanic geometries.

The SWAN model will be used to simulate ocean wave propagation, accounting for shoaling, refraction, diffraction, and other wave transformations. The model will consider interactions between wind, waves, and currents, allowing for a more accurate representation of wave dynamics.

The GCB algorithm will be employed to generate unstructured grids based on mesh density functions derived from bathymetric data. This algorithm avoids the creation of hanging nodes, ensuring mesh conformity and adaptability. It will allow for refined grids in areas with abrupt bathymetric changes, enhancing simulation accuracy.

### Contact

Sergey Korotov, sergey.korotov@mdu.se

#### Complex Neural Networks for Quantum AI

Complex neural networks are a promising development in artificial intelligence (AI) that utilize complex numbers in their computations. This is in contrast with neural networks used in present-day AI applications, which rely solely on real-valued computations. The benefit of using complex numbers is that they naturally encode frequency-domain information such as the phase and amplitude of a signal. This makes them particularly efficient when applied to problems involving such information, such as temporal data analysis (e.g., speech recognition, object tracking) and medical imaging.

Beyond their classical applications, complex neural networks hold significant potential for integration with quantum computing. Quantum computers leverage the principles of quantum mechanics to perform calculations, utilizing “qubits” as their fundamental unit. Unlike classical bits (0 or 1), qubits exist in a superposition state, representing both states simultaneously. This unique property enables quantum computers to excel at solving specific problems that are intractable for classical computers. Notably, qubits themselves are inherently represented by complex numbers due to the underlying quantum physics. This creates a natural synergy between complex neural networks and quantum computing architectures.

This research project has two main objectives:

- Develop robust complex neural network architectures:

Specifically, the objective is to develop architectures that are resilient to noise and faults present in both real-world data and in the computational hardware, including future quantum hardware. - Explore applications of complex neural networks

The research will focus on two key domains:- Quantum AI

Quantum computing is still in its infancy, but the hardware development is currently progressing at an unprecedented rate. We will explore how to implement and adapt complex neural networks for quantum computation, specifically focusing on mitigating the high error rates inherent in current quantum hardware. - Advanced Signal Processing in Computer Vision

This research will delve into how complex neural networks can leverage temporal information within video sequences for tasks like object recognition and activity analysis.

- Quantum AI

### Contact

Ludvig af Klinteberg, ludvig.af.klinteberg@mdu.se