### Department of Mathematics and Systems Analysis

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Here is a list of research topics for the summer trainee positions at Department of Mathematics and Systems Analysis in 2021. Indicate at least one topic in your application. However, it is recommendable to give a priority list of several topics.

Algebra and discrete mathematics

1. Algebraic geometry and data science

High-dimensional data often belongs to a toplogical space whose intrinsic dimension is lower than the number of coordinates of the data. It is of interest to estimate the intrinsic dimension of the topological space. Algebraic varieties are solution sets of polynomial equations. In this internship, you will study the intrinsic dimension for finite data sets sampled from algebraic varieties. The main goal is to study how different intrinsic dimension estimation methods depend on sampling. Prerequisites: First course in probability and statistics, Metric spaces (Euclidean spaces), recommended Abstract algebra. Contact person: Kaie Kubjas.

**2. Secure and private computation and applications to eHealthcare**The student will work in the ANTA group and take part in a project that studies distributed computation, in particular in the context of secure and private data retrieval and matrix multiplication. The project also involves healthcare related applications, and the ultimate goal is to apply the above methods to real-life data, after some pre-processing. The application part will depend on the current stage of the project and will be specified later. Prerequisites: matrix algebra, linear algebra, and abstract algebra courses with a good grade and good programming skills. Courses and expertise related to data processing and more advanced studies in algebra and discrete mathematics are considered an asset. The position best suits a master's student, but advanced and mature BSc students can also be considered. There might be an option to continue the work in collaboration with our industrial partner after the training period. Contact person: Camilla Hollanti

3. Polynomials of graphs without certain cycles or holes

This is a project in graph theory. For graphs without cycles (or holes) of length *k* mod *m* several interesting results have been proved the last years. In particular for 1 mod 2 and 0 mod 3. There are several polynomials one may construct from graphs: Independence, Chromatic, Tutte, and so on. In this project, polynomials of graphs without certain cycles or holes are explored. There are both computational and theoretical approaches. Projects suitable for all levels available. Contact person: Alexander Engström**4. Computational Combinatorics**Computational combinatorics studies abstract mathematical structures (e.g. graphs, groups) by the help of computation. The tools are an exciting mix of mathematical theory (algebra), efficient computation methods (algorithms) and high-performance computing (supercomputers). The intern will study a selected problem in computational combinatorics, implement an algorithm and perform the computations. The problem is selected according to the intern's skills and interests, for example, from additive number theory or graph enumeration (see e.g. https://en.wikipedia.org/wiki/Additive_number_theory, https://en.wikipedia.org/wiki/Graph_enumeration, arXiv:1711.08812, and arXiv:1804.03679). Prerequisites: Good programming skills (e.g. depth-first search); basic understanding of algebra, including permutation groups (from MS-C1081 Abstract algebra or otherwise). Contact person: Jukka Kohonen

**5. Nonlinear PDEs**We are recruiting one or more students to work on projects related to nonlinear partial differential equations. The projects are related to the parabolic p-Laplace equation, the porous medium equation and the total variation flow. There are many challenging research topics for bachelor, diploma and doctoral theses. It is also possible to take courses as a self-study package and to participate in international summer schools. Please contact Juha Kinnunen for more information. See also the NPDE group.

6. Mathematical analysis

We are recruiting one or more students on projects related to mathematical analysis (and PDEs). The research topics can be related to e.g. analysis on metric spaces, geometric measure theory, harmonic analysis or Sobolev- or BMO-type function spaces. Prerequisites: Euclidean spaces. If you are interested, please come and discuss about more spesific topics. Contact person: Prof. Riikka Korte. See also the NPDE group.

**7. Quaternion-based tools for numerical linear algebra**The polar decomposition of a complex number can be extended to square matrices: we can always write A=UH where U is unitary and H is Hermitian positive semidefinite. If A is invertible, the decomposition is even unique. In graphics, polar decompositions of 3x3 real matrices are important: whenever a 3D animation is displayed in a web browser, an underlying computation of (possibly many) polar decompositions is happening. The currently fastest algorithm for this task was developed in 2016 by N. Higham and V. Noferini, and it is based on a connection with quaternions. Quaternions are a non-commutative number systems, discovered in 1843 by R. Hamilton, that extends complex numbers.

The goal of this project is to study this connection and explore possible further computational applications. Depending on the student's skills, this may even lead to research level questions; for this reason the project is particularly suitable to strong students with a desire to pursue a future career in applied mathematics research. Contact person: Vanni Noferini

8. Optimal experimental design for X-ray imaging

**9. Automatic evaluation of jaw surgery**Certain operations in maxillofacial surgery are carried out to modify the anatomy and function of the lower jaw to correct abnormalities. Typically, the jaw bone is split, and the parts are rejoined to a different position using titanium screws and plates. Patient image material has been developed by 3D Magnetic Resonance Imaging and lateral X-ray radiograms both before and after the operation. The outcome of the operation can be evaluated in a number of ways. The aim of this work is to develop an image processing application for automatically evaluating the translation distance and rotation angle of the jaw tip, relative to jaw joint position, as a result of such surgery. The application can developed in MATLAB environment, using the image processing toolbox and optimisation. Contact person: Jarmo Malinen

12. Random graphs and network statistics

We are recruiting one or more research trainees to work on projects related to random graphs and network statistics. The projects are related to the analysis of statistical pairwise interaction models relevant to community detection and spread of epidemics. There are many challenging research topics for BSc, MSc, and DSc research. Courses MS-E1600 and MS-E1603 together with basic programming skills form an ideal background for the position. It is also possible to participate in the activities of an international research team working on the statistical modelling of COVID-19. Please contact Lasse Leskelä for more information.**13.** **Mathematical physics**

We are seeking to recruit one research intern in mathematical physics for the summer 2021. Possible topics include lattice models of statistical physics, constructive quantum field theory, and representation theory. Contact person: Kalle Kytölä

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