## Data Augmentation for AI using Möbius Mappings

You will further investigate the so called M-admissible class of Möbius transformation introduced in the preprint linked below, both from a theoretical complex analysis view point, but also from the application of image augmentation in AI. Knowledge of complex analysis and automatic image classification is necessary.

Further information

Contact: Torbjörn Lundh (contact)

## A game theoretic agent based approach to medal races in sailing

This project is based on a fleet race simulation master thesis by Sebastian Berg, 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 master project will use the above described system as a starting point and will then expand taking into consideration the complex pay off of the standing (ranking) of the top 10 boats prior to the “Medal Race” i.e. the final race of the series. A medal race is the final race and the points count double to clearly define the gold, silver and bronze medal winners. Here the tactics will be a mix between fleet race and match racing. Co-supervisor of this project will be Dr Laura Marimon Giovannetti, that will help with insights on the sailing background.

Further information

Contact: Torbjörn Lundh (contact)

## Investigations of methods for robust quantification and estimation surface geometries of modelled tubular anatomic struc

The purpose of this thesis work is to: investigate parameters, put conditions on modelling quality of complex surfaces of tubular anatomic structures found in the human body. The insights are drawn from studying previous work in the field and analyzing current workflows; starting from the semi-manual methods for example in the software SimVascular2 and then a Lagrangian surface description for quantification using MATLAB. To make sure that the workflow is aligned and identifying bottlenecks for further improvement of accuracy is crucial for progressions. The long-term aim is to improve care and life of patients, and non-invasive quantification methods plays an important role in this. Supervisor: Johan Bondesson, Examiner: Torbjörn Lundh

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)

## 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)

## Analysis of length scale and interfacial curvature of phase-separating polymer films

Two-phase polymer films are commonly used as pharmaceutical coatings for controlled drug release applications. In an ongoing PhD project, we characterize the impact of e. g. weight fractions of the polymers and processing conditions on the structure that is formed through phase separation, typically through the spinodal decomposition mechanism. The structures are imaged using confocal laser scanning microscopy (CLSM). The project involves image processing for quantitative analysis of the length scale of the structure as a function of time and for the shape and curvature of the interfacial surface. This analysis entails using multiple image processing tools such as filtering techniques and Fourier (FFT) domain analysis for wavelength/frequency analysis. The proposed MSc project is focused on spatial statistics and image analysis. The aim is to explore image analysis methods for analysis of length scale and interfacial curvature of phase-separating polymer films, using existing experimental data as well as simulated data. Concretely, to implement methods for synthetic image data generation using Gaussian random fields and other simulation techniques, to investigate both FFT- and non-FFT-based methods for length scale/wavelength analysis (the latter could be e.g. covariance functions, chord length distributions), curvature estimation methods, and relating local curvature to local length scale. The project will be supervised by Magnus Röding at RISE. You can contact him by email magnus.roding@ri.se.

Further information

Contact: Aila Särkkä (contact)