# Summer trainee positions 2023

Below is a list of research topics for the summer trainee positions at Department of Mathematics and Systems Analysis in 2023. Indicate at least one topic in your application. However, it is recommended that you give a priority list of several topics.

If you are a student outside Finland, or an international/exchange student in Finland, please also check the Aalto Science Institute international summer research programme.

https://www.aalto.fi/en/aalto-science-institute-asci/how-to-apply-to-the-asci-international-summer-research-programme

Algebra and Discrete Mathematics

**1. **We are seeking to recruit a research intern in algebra and discrete mathematics. Possible topics include algebraic geometry and interactions of algebra and discrete mathematics with statistics, machine learning, optimization and genomics. Contact person: Kaie Kubjas, kaie.kubjas(a)aalto.fi

**2.** The Algebra, Number Theory, and Applications (ANTA) Group is seeking to hire an intern to work on a project related to number theory. The student should have taken at least Linear Algebra and Abstract Algebra to qualify for the position. Contact person: Camilla Hollanti, camilla.hollanti(a)aalto.fi

**3.** The Algebra, Number Theory, and Applications (ANTA) Group is seeking to hire an intern to work on a project related to coding theory and federated learning. The student should have taken at least Linear Algebra and Abstract Algebra to qualify for the position. Contact person: Camilla Hollanti, camilla.hollanti(a)aalto.fi

**4.** We are recruiting an intern to work on a project in combinatorics, more specifically to study so-called almost affine representations of matroids. The project is rather elementary, but the student is expected to have taken Abstract Algebra or Foundations of Discrete Mathematics. Other projects in matroid theory or graph theory are also possible, depending on the student's interests. Contact person: Ragnar Freij-Hollanti, ragnar.freij(a)aalto.fi

Analysis and Nonlinear Partial Differential Equations

**5. **We are recruiting one or more students to work on projects related to harmonic analysis and nonlinear partial differential equations. 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.** We are recruiting students to work on project related to mathematical analysis, for example on analysis of metric measure spaces, harmonic analysis or partial differential equations. Please contact Riikka Korte for more information about possible projects. (Also Pekka Alestalo provides topics on metric spaces and topology. You can use this topic number also if you are interested in working with him.)

**7.** We are recruiting one (or more) student(s) to work on a project related to nonlinear stochastic partial differential equations (nonlinear SPDEs). Nonlinear SPDEs are used to describe the influence of random fluctuations on mathematical models from fluid dynamics, material science, mathematical neuroscience, flows in porous media, molecular biology, or ecology, to name a few. The research level of the project can be adjusted to bachelor's, master's or doctoral level. Please contact Jonas Tölle for more information.

**8.** We are looking for a summer intern to learn about the notion of Fractal Uncertainty Principle (FUP), a new method in harmonic- and semiclassical analysis that in the recent years has found powerful applications in control theory of hyperbolic PDEs, gravitational waves emitted by black holes and quantum chaos. You will learn the basics of fractal geometry and Fourier analysis needed to understand FUP, and establish FUP for some new examples that are useful in quantum chaos. Contact: Tuomas Sahlsten, tuomas.sahlsten ‘at’ aalto.fi, website

Mathematical Physics

**9. **We are seeking to recruit research interns in several areas of mathematics related to mathematical physics. Possible topics include random geometry (SLE theory, Liouville quantum gravity, related areas), lattice models of statistical physics, aspects of constructive quantum field theory or axiomatic conformal field theory, topological recursion, complex geometry, and representation theory. No background in physics is required for the positions. For more information, please contact Eveliina Peltola (Eveliina.Peltola "at" aalto.fi).

**10. **We are looking for a summer intern to work on a project related to coupled chaotic systems and their quantizations. In this project you will learn on basics of chaotic dynamical systems (e.g. Arnold’s cat maps), effect of coupling to their chaos, and quantization procedures for cat maps. This project will help in understanding the mathematics of quantum many-body chaotic systems (like some models of quantum computers) in the case of simple toy models, where not many mathematically rigorous results yet exist. Contact: Tuomas Sahlsten, tuomas.sahlsten ‘at’ aalto.fi, website

Numerical Analysis and Inverse Problems

**11. **We are recruiting summer trainees in inverse problems, which is 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. For example, the reconstruction tasks of many medical imaging modalities are mathematically formulated as inverse problems. Possible topics include: (i) Bayesian experiemental design for inverse source problems such as *electroencephalogram* (EEG) and (ii) Comparison of light propagation models for *diffuse optical tomography* (DOT). Contact person: Nuutti Hyvönen

**12. ***Hyperbolic drums.* Recent developments in condensed matter and metamatrial research have led to interest in model problems in hyperbolic geometry. Hyperbolic drums are a simple extension of the standard elastic membrane problem to hyperbolic geometry, i.e., to Poincare disk. Many interesting questions arise both in analytic and numeric contexts, for instance, what are the asymptotics of the discrete spectrum as the membrane is extended toward horizon where the spectrum is continuous. This project is part of ongoing research. Contact person: Harri Hakula

**13. ***Zeros of associated Legendre polynomials.* Many problems in physics admit solutions as expansions of special functions. Transferring the problems to hyperbolic geometry leads to new expansions with special functions with less well understood properties. Of particular interest are the zeros of associated Legendre polynomials in their general setting with complex parameters. Overview of the known analytic results and robust numerical methods builds a foundation for future work on this topic and its applications. Contact person: Harri Hakula

**14. ***Quadratures on polygons.* Integrating weighted inner producs of polynomials and piecewise polynomial functions remains a challenge in high performance computing. All practical rules are composite rules of some kind, however, for any given fixed configuration there are many different choices leading to a combinatorial selection problem. Understanding the fundamentals of the computational complexity issues and devising robust designs is the goal in this project. Contact person: Harri Hakula

**15. ***Smith form of powers of a matrix.* The Smith canonical form of a matrix is an important result in matrix theory; it applies, for example, to matrices whose elements are integers or univariate polynomials, and has many applications in mathematics including for instance solving systems of linear Diophantine equations. The goal of this project is to explore how the invariant factors (diagonal elements in the Smith form) of a square matrix A relate to those of the powers of A. For a much more detailed description of the project, please check also the Aalto Science Institute call (page 36 of the linked document).* *Contact person: Vanni Noferini

Stochastics and Statistics

**16. **We are seeking to recruit research interns in mathematical statistics and probability theory. Possible topics include:

- Imbalanced classification (of text data), multivariate extreme value theory, and functional data analysis. Contact person: Pauliina Ilmonen (pauliina.ilmonen(a)aalto.fi)
- Network statistics, random graphs, and random hypergraphs. Contact person: Lasse Leskelä (lasse.leskela(a)aalto.fi)

Systems and Operations Research

More information here.

Page content by: webmaster-math [at] list [dot] aalto [dot] fi