 Department of Mathematics and Systems Analysis

# Summer trainee positions 2020

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.

## Algebra and discrete mathematics

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.

##### 3. Introduction to p-adic numbers

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.

### 5. 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.

6. 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.

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
A simple graph is said to be perfect if its chromatic number equals the size of the largest clique, and the same is true for any induced subgraph. Intuitively, this means the only obstructions to colouring the nodes of the graph (such that neighbouring nodes get different colours) are local. Perfect graphs are of great importance in algebraic and algorithmic graph theory alike, and their structural properties are precisely described thy the strong perfect graph theorem. In this work, we will look closer at one of the proofs of the weak perfect graph theorem, which says that a graph is perfect if and only if its complement graph is. The proof goes through studying certain high-dimensional polytopes associated to the graphs, and we will study these polytopes in detail for special classes of perfect graphs, in the light of some open conjectures in polytope theory. This project can be carried out with no other background than basic math courses, but courses in optimization and/or graph theory are a merit. Contact person is Ragnar Freij-Hollanti.

9. Configurations of cycle codes
Cycle codes are linear binary codes whose codewords correspond to the cycles in a simple graph. They also have interesting dual codes, coming from the cuts in the same graph. After an introduction to linear coding theory, we compare this duality to other dualities on graphs, coming from algebra, geometry and combinatorics. We will also explore how to go in the opposite direction, and reconstruct the graph from combinatorial properties of the cycle code. This project can be carried out with no other background than basic math courses, but courses in linear algebra and/or graph theory are a merit. Contact person is Ragnar Freij-Hollanti.

## Analysis and nonlinear partial differential equations

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.

## Applied mathematics and mechanics

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

14. Randomized Linear Algebra
Randomized linear algebra has exited the pure research phase and is now provided for instance in the NAG libraries. In this study, iterative solution of eigenproblems via randomizedlinear algebra is examined numerically and in special cases where some extra information such as the exact spectrum is available, also theoretically. Contact person: Senior Univ. Lect. Harri Hakula

Numerical solution of partial differential equations often requires integration of weighted inner products of polynomials or piecewise polynomial approximations of functions. In recent years there has been a lot of interest in polygonal, possibly curvilinear, discretisations of domains as opposed to simple triangulations. There has been very little attention paid to quadrature design, however. Here, the emphasis in on planar configurations and mapped Gaussian quadratures. Contact person: Senior Univ. Lect. Harri Hakula

16. Optimal experimental design for an inverse heat conduction problem
Inverse problems constitute an active and expanding research field of mathematics and its applications. A fundamental feature of inverse problems is that they are ill-posed: a small amount of noise in the measured data may cause arbitrarily large errors in the estimates for the parameters of interest. This project considers a simple but notoriously ill-posed inverse heat conduction problem. To be more precise, the aim is to determine the temperature distribution in a given domain at a given time based on measurements at a later time. The emphasis is on the Bayesian formulation of the inverse problem and, in particular, on optimal experimental design: how to choose optimal positions for the temperature sensors inside the domain of interest based on the measurement model and prior information on the unknown? Contact person: Prof. Nuutti Hyvönen

## Probability, statistics, and mathematical physics

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

18. Mathematical physics
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