Department of Mathematics and Systems Analysis

Studies

Summer trainee positions 2019

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.

Algebra and discrete mathematics

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

A central problem in knot theory is to distinguish if two knots are equal or not. One common approach is to associate polynomials with knots, and if the polynomials are different, then so are the knots. We would like to computationally investigate polynomials of this type for knots with certain symmetries, using software as maple or mathematica. Contact persons are Alexander Engström and Florian Kohl.

11. The maths behind Bitcoin

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.

12. Introduction to modular forms in order to prove some of the Ramanujan congruences

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.

13. Quaternion algebras and wireless communications

Understand how one can design space-time block codes for wireless communications using quaternion algebras. Analyse the connection to well-rounded lattices and construct some concrete examples, e.g. the Golden code. Prerequisites: basic linear algebra course. Some number theory background is useful, but not strictly necessary. 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.

Analysis and nonlinear partial differential equations

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.

Applied mathematics and mechanics

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

19. On convergence of the Legendre expansion
Consider surface z = f(x,y) and it's Legendre expansion of degree L(p). In this study, the convergence of the partial sums in appropriate Lp-norms is explored. In particular, the convergence rates in the L infinity-norm are not know in the general case and are of interest. This is an active area of research. Contact person: Senior Univ. Lect. Harri Hakula

20. Model order reduction
Reduced order modelling is important tool in many engineering fields, such as, electrical and mechanical engineering. The aim in this work is to study application of these techniques to system of ODEs. Both numerical and theoretical study will be conducted. Contact person: Assistant Prof. Antti Hannukainen

Stochastics, statistics, and mathematical physics

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

Systems and operations research

More information here.

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