## Further Simulations of Sustainable AI​

Use the newly created meta-system based on the classical artificial-life-plattform Avida as a base to set up, simulate and analyse evolving population of increasingly more competent “creatures” and a guardian interacting organism that is preprogrammed to let the creatures develop, but not reach a specified high competence.

Further information

Contact: Torbjörn Lundh (contact)

## A game theoretic approach to match racing starts in sailing

This project is based on a fleet race simulation created by Sebastian Berg in the summer and fall of 2019, where he wrote an agent based system (in Python) for a fleet of identical boats, but where each of the boats had a preset starting tactics. The system was run and the tactics distribution were evolved towards an evolutionary stable state. The new project will take its starting point in the above described system but make a stand alone system aimed at the completely different situation of a match race start (with only two non-identical boats).

Further information

Contact: Torbjörn Lundh (contact)

## Augmented reality simulation of visual guidence for minimally invasive tumour operations in livers

A continuation of a collaboration project with Niclas Kvarnström and Mårten Falkenberg at Sahlgrenska and also with Klas Modin here at MV. This is a direct continuation of the current master thesis work by Lisa Månsson during the fall 2019 where she sets the foundation for the suggested AR-system by creating several optical models. This thesis work suggestion will take the project closer to the clinic by creating models for user friendly image augmentation.

Further information

Contact: Torbjörn Lundh (contact)

## Spin-integralekvationer

Ett matematiskt problem med viktiga tillämpningar är att numeriskt beräkna hur en våg som träffar ett objekt sprids av detta. Det finns ett flertal metoder för att diskretisera och lösa randvärdesproblem för partiella differentialekvationer av detta slag, såsom finita element metoder, finita differensmetoder och randintegralekvationer. Projektet som föreslås använder den sistnämnda metoden, som löser den partiella differentialekvationen genom att omformulera den som en integralekvation på randen. Genom att diskretisera och approximera integraler med summor, får man en matrisekvation som man löser för att beräkna den spridda vågen ifråga. Fördelen gentemot de första två metoderna, är att det är lättare att diskretisera randkurvan eller randytan ifråga, eftersom området där vågorna fortplantar sig är av en dimension högre och dessutom i vårt fall ett obegränsat, icke-kompakt område. I många situationer visar sig integralekvationer ha prestanda som är överlägsen vanlig FEM, speciellt då randen är icke-slät. Bakgrunden till projektet är spridningsproblemet för tidsharmoniska elektromagnetiska vågor. Förvånansvärt är detta fortfarande ett aktivt forskningsområde, och de integralekvationer som vanligtvis används i industrin lider av allvarliga problem. L. Greengard vid Courant institute NYU, berömd bland annat för FMM, en av de mest framgångsrika numeriska algoritmerna någonsin, beskriver situationen här: https://www.math.nyu.edu/faculty/greengar/EMdesign/projects.html Ett allvarligt problem är så kallade falska resonanser, som innebär att integralekvationerna vid vissa frekvenser blir olösbara. Detta är en defekt av den använda integralformuleringen; I själva verket finns där alltid en fin entydig lösning. Handledaren har upptäckt en hel ny sorts integralekvationer som speciellt inte lider av sådana falska resonanser. Dessa ekvationer är hittills helt oprövade numeriskt, även om den teoretiska analysen visar att de har mycket bra egenskaper. Det föreslagna projektet handlar om att göra inledande numeriska inplementeringar av dessa nya så kallade spin-integralekvationer. Hur omfattande projektet görs, numeriskt och teoretiskt, bestäms av intresse och tidsramar.

Further information

Contact: Andreas Rosén (contact)

## An index theorem of Kasparov

A landmark result in modern mathematics is the Atiyah-Singer index theorem from the 60s. This index theorem computes the Fredholm index of an elliptic operator on a closed manifold in terms of algebraic topology thus tying together far apart branches of mathematics (topology, geometry and analysis). Since the 60s, the index theorem of Atiyah-Singer has seen a wide range of generalizations. The most powerful one is an index theorem of Kasparov which allows you to deduce index formulas in a broad range of settings with a flick of your wrist. The main idea in the index theorem of Kasparov is to (at a homological level) factor an elliptic operator in terms of local information. This masters thesis aims at studying the index theorem of Kasparov, and hopefully provide its first rigorously written up proof, taking its starting point in two recent papers of Baum-van Erp (https://arxiv.org/abs/1604.03502 and https://arxiv.org/abs/1604.03535).

Further information

Contact: Magnus Goffeng (contact)

## Variations of magnitude

An invariant that has attracted quite some attention in the last decade is the magnitude of a compact metric space. It gives a way of encoding the size of a metric space. For compact metric spaces of geometric origin (domains in Euclidean space or manifolds), the magnitude recovers geometric invariants such as volume and certain curvatures. This masters project aims at studying how magnitude varies when changing the underlying space. A concrete problem is to show that when varying a domain in Euclidean space in a real analytic fashion, the magnitude depends real analytically on small variations.

Further information

Contact: Magnus Goffeng (contact)

## L^2 index theorems

An elliptic operator A on a closed manifold satisfy that there are finitely many solutions to the equation Au=0. In particular, elliptic operators are Fredholm operators and their Fredholm index ind(A):=dim ker(A)-dim ker(A*) is well defined. The celebrated Atiyah-Singer index theorem computes ind(A) by geometric data. For manifolds that are not necessarily closed, it is sometimes possible to show that the space of solutions to Au=0 is small in a certain operator algebraic sense. If for instance the manifold carries a free, proper, cocompact action of a Lie group G then G-equivariant elliptic operators are Fredholm relative to the von Neumann algebra of G. For G discrete, the L^2-index theorem of Atiyah computes the G-Fredholm index of G-equivariant elliptic operators by means of elliptic operators on the closed quotient by the G-action. This masters project aims at studying index theory relative to von Neumann algebras and its geometric ramifications.

Further information

Contact: Magnus Goffeng (contact)

## Higher order scattering theory

Scattering theory is concerned with problems motivated by the study of how waves scatter against an obstacle. To explain the mathematics appearing, consider the case of a physical object occupying a compact K in Euclidean space, interacting with the waves by means of some boundary conditions B. Scattering theory studies the resolvent of the Laplacian on the exterior of K equipped with the boundary conditions B, in other words the operator valued function $R(lambda):=(-Delta_B-lambda)^{-1}$ where $Delta_B$ denotes the Laplacian with boundary conditions B. With some work, one can show that this function extends meromorphically to complex $lambda$ and the main object of study in scattering theory is the pole structure of this function. The poles will give rise to standing waves in the exterior. The purpose of this project is to study what happens for higher order operators. Concretely, we will consider three dimensions and the operator $(-Delta-lambda)^{2}$. The motivation for this problem comes from a recently emerged structure in the metric invariant known as magnitude where little is known about the poles.

Further information

Contact: Magnus Goffeng (contact)

## Right censured repeated measures over time – How to analyze change in pain thresholds.

In chronic pain research, it is common to measure pain thresholds. Most common is thermo and pressure pain thresholds. Due to safety reasons, a maximum and minimum temperature, respectively a maximum pressure, is required in clinical and research settings. If only analyzing one such right censured measurement, a possible approach would be to use Cox regression, and view it as a survival outcome with censuring. However, in the present project, we consider data with several measurements per individual, and we want to model the change over time. In the field of econometrics, where censored data is quite common, several methods are proposed, e.g. Tobit estimators and likelihood estimation proposed by Schnedler [1]. Note, that there is a great confusion between the terms censured and truncated in the literature. Aims: -Review the literature in mathematical statistics, but also other methodological literature, for possible existing solutions to the problem. -To compare the usefulness of existing solutions and/or to suggest improvements. -To suggest an analysis strategy for modelling right censured measurements over time in medical pain research. -Illustrate with simulated data, the effect of ignoring the censuring in scenarios similar to that in an existing material with pain thresholds, the project LoadPain. 1. Schnedler, W., Likelihood estimation for censored random vectors. Econometric Reviews, 2005. 24(2): p. 195-217.

Further information

Contact: Anna Grimby Ekman (contact)

## Efficient implementation of Helmholtz equation with applications in medical imaging

In this project we will apply the finite element method (FEM) for the solution of Helmholtz equation in two and three dimensions. Solution should be implemented and tested on different real-life models in C++/PETSc using existing software package WavES (waves24.com). The main goal of the project is efficient implementation of the solution of Helmholtz equation using finite element method, and testing of the obtained solver on the real data provided at https://uwcem.ece.wisc.edu/MRIdatabase/InstructionManual.pdf Visualization of the obtaind results will be done in Paraview/GID. It is expected that application of the obtained software will be for fast detection of small-size tumors using microwave imaging.

Further information

Contact: Larisa Beilina (contact)

## Parameter identification for a mathematical model describing tumour-macrophages interactions

The goal of this Master project is development of the optimization method for the solution of a parameter identification problem (PIP) for system of ordinary differential equations (ODE) which describes dynamics of the anti-tumour/pro-tumour immune responses generated by macrophages. Algorithm for the solution of the problem should be formulated and numerically tested in Matlab. In this thesis can be used ideas developed in the previous work where time-adaptive FEM was applied for another similar problem, see link below.

Further information

Contact: Larisa Beilina (contact)

## Determination of parameters in kinetic modelling in positron emission tomography (PET)

The goal of this Master project is development of the optimization method for the solution of a parameter identification problem arising in PET for system of ordinary differential equations which presents the kinetic model for measurement of glucose transport and phosphorylation rate. Reconstruction algorithms should be formulated and numerically tested in Matlab.

Further information

Contact: Larisa Beilina (contact)

## Application of deep learning and/or numerical analysis to Bayesian inference for state-space models

This project is an industry collaboration with Smartr. The projects marries deep learning with Bayesian inference via MCMC, with emphasis on studying state-space models expressed as ODEs and SDEs. The student(s) should start not earlier than 2020. Notice: applications should be sent directly to Adam Andersson, see the address in the further information. However you may contact both Adam Andersson and Umberto Picchini for informal queries.

Further information

Contact: Umberto Picchini (contact)

## Residual neural networks and continuous dynamical systems

Smartr https://smartr.se is an expert bureau within data and AI and we propose the following master thesis project.

Residual neural networks have become very popular in deep learning. They can be viewed as Euler discretizations of controlled ordinary differential equations and training can be viewed as an optimal control problem. In this thesis we want to evaluate different ways to train residual neural networks with or without algorithms from optimal control. The methods shall be evaluated on some dataset of choice and the methods shall also be compared to multilayer perceptron (MLP) networks to understand the pros and cons of using residual neural networks on data in comparison to using the more common MLP networks.

We seek 1-2 students for this project. The students are expected to have a strong mathematical background. Some experience with TensorFlow or PyTorch is meriting.

Further information

Contact: Stig Larsson (contact)

## Nasal airflow modeling

Nasal breathing can be obstructed in various ways including anatomical alterations post trauma or by swollen tissue due to inflammation for example. How such geometrical changes affects the nasal breathing, and thus our sleep quality among other things, can be illustrated by measuring the mean flow velocity as a function of time over a breathing cycle. This is a collaboration master thesis project between Chalmers and Sahlgrenska where the clinical contact person is MD Johan Hellgren johan.hellgren@orlss.gu.se

Further information

Contact: Torbjörn Lundh (contact)

## Modelling tumour growth using Diffusion Tensor Imaging

The growth of tumours depends on the anatomical structure of the tissue in which the tumour resides. In the case of brain tumours it is possible to map the structure using data from Diffusion Tensor Imaging (DTI), which can be used as a means of estimating the ability of cancer cells to move through the tissue. The aim of this project is to use publicly available DTI-data to parametrise an individual-based model of brain tumour growth, which will be compared to growth patterns obtained in mice.

Further information

Contact: Philip Gerlee (contact)

## Spatial game theory

Classical game theory has in recent years been extended in order to account for the spatial location of the agents involved in the game. The population structures considered in spatial game theory have almost exclusively been discrete (e.g. network models), but in most applications of the theory individuals are located in continuous space, e.g. cells in a tissue or individuals living in different locations. The aim of this project is to study a game where the players are located in Euclidean space and the strength of interactions depend on the distance between players. Using simulations and possibly theoretical tools the goal is to establish conditions for dominance of one strategy over the other, which will generalise the concept of Nash equilibrium in classical game theory.

Further information

Contact: Philip Gerlee (contact)