# Mathematics

## About this team

### Team information

The research group in Mathematics can be informally divided into two subgroups, for pure and applied mathematics, respectively. Our research in pure mathematics can mainly be attributed to algebra, combinatorics and analytical number theory. In applied mathematics, we work on mathematical physics, numerical linear algebra, optimization, inverse problems and numerical methods.

Abstract algebra is not about numbers in the first place, but about abstract elements on which different operations can be performed that are more or less similar to the usual arithmetic operations. Among the algebraic structures constructed in this way, we study in particular Lie superalgebras describing symmetries in fundamental physics, partly in collaboration with Chalmers University of Technology. Research is also conducted within representation theory and category theory.

Combinatorics is primarily about studying and counting the number of possibilities to, for example, perform a certain operation. The research here is done within enumerative and algebraic combinatorics, with a special focus on permutations and Coxeter groups. Applications appear in bioinformatics, more specifically comparative genomics.

In analytical number theory, methods from complex analysis are used to investigate the properties of integers. In particular, we study Dirichlet series and zeta functions in order to get a better understanding of the prime numbers.

In mathematical physics we perform research on nonlinear Schrödinger equations, both within applied international collaborations and on theory. Stochastic ordinary differential equations can model the dynamics of more complicated partial differential equations. Here we work on applications, for example within quantum dynamics and heat transfer.

Our research in the field of numerical linear algebra includes a development of theory, algorithms, and software tools for analyzing problems that involve matrices, often with various symmetries and block structures. In collaboration with Umeå University, Catholic University of Louvain, and the University of São Paulo we work in perturbation theory of linear and non-linear eigenvalue problems. Growing needs of solving and analyzing large scale problems demand a better understanding of low rank operators. Research in this direction is done in collaboration with University Carlos III of Madrid and supported by Wenner-Gren Foundations.

An inverse problem is to determine unknown quantities in a mathematical model via given data. A current application that we are researching is carbon dioxide storage under the seabed and more specifically the problem of identifying possible leaks. This is done in collaboration with the University of Bergen, Norway.

Optimization is a very broad research area where we are most active in non-linear programming. An example is portfolio optimization where the problem is to find the best possible investment portfolio. Another current application is the learning of neural networks that are central to machine learning and artificial intelligence.

For a number of years, we have developed and applied a new methodology based on damped dynamical systems, Dynamical Functional Particle Method, based on the idea of solving equations by effectively solving a damped dynamic system. Based on this methodology, we further develop algorithms for solving matrix equations and eigenvalue problems.

Within the research group, we have significant experience of interdisciplinary projects and collaboration with companies. Suggestions are welcome!

## Research projects

### Active projects

- Eigenstructures of LOw-RAnk matrix polynomials (ELORA)
- Low-rank methods for systems of Sylvester-type matrix equations
- New algorithms for training artificial neural networks (ANN)
- Nonlinear Schrödinger equations
- Optimization of investment portfolios
- Stochastic simulations of partial differential equations
- Tensor hierarchy algebras and extended geometry