### 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.

**1. Reconstruction of 3D chromosome structure**

HiC matrices record genome-wide contact frequencies among genomic loci. The goal of this research topic is to learn how the 3D chromosome structure is modeled from the HiC data. This project involves theoretical study as well as computations using semidefinite programming. Contact persons are Kaie Kubjas and Annachiara Korchmaros.**2. The Euclidean Distance from a real algebraic variety**

Consider a set X and a data point u in a real Euclidean space V. A natural problem is finding the closest point to u on the set X. This leads to the study of the critical points of the Euclidean Distance (abbreviated ED) function from X to u. When X has additional structure, for example when it is defined by polynomials, the number of (complex) critical points on X of the ED function between u and X is invariant and is called ED degree of X. On the contrary, the number of real critical points on X depends by the particular choice of u. Averaging over all u with respect to a measure on V leads to the (real) notion of average ED degree of X. We would like to investigate how the choice of data point u affects the nature of real critical points, namely the number of real local minima, maxima and saddle points of the ED function between u and X, starting from the case of plane curves. Concrete examples will be studied using mathematical software as Macaulay2, Maple or Mathematica. Contact persons are Kaie Kubjas and Luca Sodomaco.

The completion of the rational numbers with respect to the ordinary absolute value gives us the familiar field of real numbers, but choosing a prime number p and completing the rationals with respect to the p-adic absolute value results in something different, called the field of p-adic numbers. The purpose is to familiarise oneself with the concept of p-adic numbers, study their basic features, and learn about some algebraic properties such as, for example, Hensel’s lemma. Prerequisites are basic analysis, algebra, and number theory. Contact persons are Camilla Hollanti and Louna Seppälä. **4. Classical or quantum-entanglement-aided secure and private computation**Study the classical private information retrieval from distributed storage systems or secure distributed matrix multiplication. Extend this to entanglement-aided scenarios, and analyze the resulting schemes. It is also possible to leave out the quantum part and only concentrate on classical problems. Prerequisites: matrix and linear algebra is necessary, coding theory and quantum physics are useful but not necessary. Contact persons are Camilla Hollanti, Jie Li, and Matteo Allaix.

6. The polyhedral geometry of betti tables

**7. 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.

8. Perfect graphs and associated polytopes

9. Configurations of cycle codes

10. 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.**11. 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.

12. Image deblurring by numerical linear algebra

When taking a photograph with a camera, ideally we would like the image to be a faithful reconstruction of what happened in reality. However, in practice, images may be *blurry. *This can be due to several circumstances: for example, motion of the subject or camera out of focus. The goal of this thesis is to explore techniques to *deblur *an image. Crucial tools are a mathematical modelling of the blurring process and appropriate techniques, based on linear algebra, to recover mathematically a maximal amount of information about the "real" picture. Contact person: Prof. Vanni Noferini.**13. 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

15. Efficient quadratures for polygons

**17. Statistical learning of labeled networks**The goal is to investigate how accurately the community structure and statistical model parameters of a node-labeled random graph can be learned based on several graph samples corresponding to a time-varying stochastic model. The job requires a strong background in probability theory and stochastic processes, together with some familiarity with discrete mathematics and programming. Contact person: Prof Lasse Leskelä

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, representation theory, 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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