### Department of Mathematics and Systems Analysis

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Here is a list of the research groups and research topics available for the summer summer trainee positions at the Department of Mathematics and Systems Analysis. Please select at least one research group and at least one topic and specify them in your application. It is also possible to give a priority list of several research groups and topics in the application. The candidates will be interviewed after the deadline and the decisions will be made by 4 March 2019 at the latest.

**9. The polyhedral geometry of betti tables**

Betti tables are foundational invariants in algebraic geometry and they are situated in cones explicitly described by Boij-Söderberg theory. The goal of this project is to investigate the low-dimensional polyhedral geometry of these cones using software as polymake. Contact persons are Alexander Engström and Milo Orlich.

**10. Polynomials of knots with symmetries**

Study of basic tools such as elliptic curves and finite fields in order to understand how bitcoins (and other cryptocurrencies) work. Prerequisites: basic algebra and analysis course. Contact persons are Camilla Hollanti and Laia Amorós.

Pure maths topic. Modular forms appear in different areas of mathematics, making strange connections of parts apparently unrelated. There are many conjectures still to solve. Literature review. Prerequisites: basic algebra and analysis course. Contact persons are Camilla Hollanti and Laia Amorós.

14. Characterizing the requirements of optimal repair for MDS codes in distributed storage systems

Distributed storage systems built on a large number of unreliable storage nodes have important applications in large-scale data center settings, such as Facebook's coded Hadoop, Google Colossus, and Microsoft Azure, and in peer-to-peer storage settings, such as Ocean Store, Total Recall, and DHash++. To ensure reliability, redundancy is imperative for these systems. A popular option to introduce redundancy is by calling upon MDS codes, which provide the optimal tradeoff between fault tolerance and storage overhead. The optimal repair bandwidth of MDS codes was characterized by Dimakis et al. in 2007. The requirements of optimal repair for MDS codes were subsequently characterized in several ways such as in the matrix terminology and subspace terminology for the case of d=n-1 (connecting all the surviving nodes to repair a failed node). However, the requirements of optimal repair for MDS codes for the case of d<n-1 and/or that can optimally repair multiple nodes are not fully characterized in the literature, at least in the matrix terminology. The goal is to read some papers on MDS codes in distributed storage systems, get to know the existing methods of how to characterize the requirements of optimal repair for MDS codes in the case of d=n-1 and make a summary. Try to characterize the requirements of optimal repair for MDS codes for the case of d<n-1 and/or that can optimally repair multiple nodes, perhaps in matrix terminology. Prerequisites: Good at linear algebra, have some knowledge about finite fields. Contact persons are Camilla Hollanti and Jie Li.

**15. 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.**16. Mathematical analysis**I recruit at most students to work on topics related to mathematical analysis. Some PDE topics are also possible. The research topics for bachelor or diploma thesis can be related to e.g. analysis on metric spaces, harmonic analysis or 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.

**17. Kelvin transform and electrostatic measurements**The aim is to study, both numerically and theoretically, how the size and location of a grounded and ideally conducting inhomogeneity inside a three-dimensional ball affects boundary measurements of current and voltage on the surface of the ball. The Kelvin transform provides a mathematical tool for tackling such a problem. Contact person: Prof. Nuutti Hyvönen

**18. Bayesian Quadratures**In this study, the connection between classical and probabilistic numerical quadratures are examined both numerically and theoretically. The connection between classical numerical analysis and probabilistic numerical methods is likely to be the future direction in applied reserarch in this area. Contact person: Senior Univ. Lect. Harri Hakula

**21. Structure of clustered random graph models**

The goal is to detect structural differences between random intersection graphs and hyperbolic random graphs, using theoretical calculations and numerical simulation experiments. The job requires a good background in discrete mathematics and probability, as well as some familiarity with programming for conducting numerical experiments. Contact person: Prof. Lasse Leskelä

**22. Statistical learning of labeled networks**

The goal is to study how the community structure and parameters of a node-labeled random graph can be learned based on several graph samples corresponding to a time-varying stochastic block model. The job requires a good background in discrete mathematics and probability, as well as some familiarity with programming for conducting numerical experiments. Contact person: Prof. Lasse Leskelä**23. Representation theory and combinatorics**

We are seeking to recruit a student to work on representation theory and combinatorics. Potential topics include analysis of Monte Carlo Markov Chain for graph colorings, transfer matrix techniques for statistical physics models, and Coxeter groups. Solid basics in abstract algebra and linear algebra are required, and a knowledge of stochastic processes and as well as programming skills are considered an asset. Contact person: Kalle Kytölä**24. Topics in mathematical structures arising in quantum field theory**

We are seeking to recruit a student to work on particular mathematical structures arising in quantum field theory. Potential topics include: knot invariants, quantum groups, representations of infinite-dimensional Lie algebras, and constructive (probabilistic) quantum field theory. Solid basics are required in one or more among: (1) analysis, (2) abstract algebra, or (3) probability. Contact person: Kalle Kytölä and/or Christian Webb

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