# Summer trainee positions 2024

**The call closed on 25.01.2024. Thanks everybody who applied!**

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Below is a list of research topics for the summer trainee positions at Department of Mathematics and Systems Analysis in 2024. Indicate at least one topic in your application. However, it is recommended that you give a priority list of several topics. Further instructions on submitting an application can be found at

https://www.aalto.fi/en/open-positions/summer-employee-positions-2024-at-the-department-of-mathematics-and-systems-analysis

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 recruiting one student in applied algebra and algebraic geometry. Possible topics are on the intersection of algebra/algebraic geometry and combinatorics, or interactions of algebra/algebraic geometry with statistics, machine learning, optimization and biology. 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. Further courses on Galois theory and algebraic number theory are an asset (note that we are lecturing a course on algebraic number theory on period IV, and another one on Galois representations on period V). Contact person: Camilla Hollanti, camilla.hollanti(a)aalto.fi

**3.** We are recruiting one student in combinatorics and graph theory. The student will study extremal problems, and in particular how these can be approached via the so-called container method. The idea of this technique is that a wide variety of combinatorial problems can be formulated in terms of independent sets in so-called hypergraphs. We can then define “containers”, such that every independent set is in at least one of said containers. The goal of the internship would be to review the method, including some of the results that it can prove. There is also a possibility to work on open problems where the method may be of use. The project will be advised by Patricija Sapokaite (patricija.sapokaite(a)aalto.fi) and Ragnar Freij-Hollanti (ragnar.freij(a)aalto.fi)

**4. **We are recruiting research interns in one or more of the following areas: algebraic geometry (moduli spaces, enumerative geometry), representation theory (of algebraic groups, Lie algebras, Hecke algebras, etc.), knot invariants (polynomials, homological invariants), algebraic combinatorics (symmetric functions). For more information, please contact Oscar Kivinen.

Analysis

**5. **We are recruiting at least seven students to work on projects related to mathematical analysis (including analysis of metric measure spaces, harmonic analysis, partial differential equations, time-frequency analysis and topology). There are suitable topics both for Bachelor and Master’s thesis projects. Please contact Pekka Alestalo, Tuomas Hytönen, Juha Kinnunen, Björn Ivarsson, Riikka Korte, Sari Rogovin or Ville Turunen, for more information about possible projects.

**6. **We are recruiting *one student* to work on a project (Master's level) 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.

*Alternatively*, we are recruiting *one student* on a project (Bachelor's level) related to existence and uniqueness theory of ordinary differential systems, long-time behavior and stability for dynamical systems, Lyapunov exponents and chaos, or dynamic optimization. Please contact Jonas Tölle for more information.

Maths & Arts

**7. **We are recruiting students to work in areas related to geometric analysis. The approach of this research can be purely theoretical, towards some application or visualisation, artistic, based on crafting or some other interesting viewpoint. Possible topics include low dimensional topology and geometry, models of geometry, dynamics, fractals, matrix groups and algebras, algebraic topology, mathematics of origami, classification of symmetries in different geometries, Kleinian groups, orbifolds. Contact person: Kirsi Peltonen

Mathematical Physics

**8. **We are seeking to recruit research interns in areas of mathematics related to mathematical physics. Possible topics include random geometry (SLE theory, Liouville quantum gravity, related areas), discrete probabilistic models of statistical physics, aspects of constructive quantum field theory or axiomatic conformal field theory, and representation theory. For more information, please contact Kalle Kytölä (kalle.kytola "at" aalto.fi) or Eveliina Peltola (eveliina.peltola "at" aalto.fi).

Numerical Analysis

**9. **We are recruiting summer trainees in inverse problems, which is an active and expanding 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 or slight mismodelling of the studied phenomenon 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 optimal experiemental design of measurements for inverse problems and (ii) inverse problems for imprecisely known forward models. Contact person: Nuutti Hyvönen

**10.** 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

**11.** Sobolev orthogonal polynomials. Sobolev orthogonal polynomials are polynomials orthogonal with respect to a Sobolev inner product, an inner product in which derivatives of the polynomials appear. They satisfy a long recurrence relation that can be represented by a Hessenberg matrix. The problem of generating a finite sequence of Sobolev orthogonal polynomials can be reformulated as a matrix problem, that is, a Hessenberg inverse eigenvalue problem, where the Hessenberg matrix of recurrences is generated from certain known spectral information. Contact person: Harri Hakula

**12.** Deterministic stochastic PDEs. There are many subtopics within the broad scope of numerical methods for SPDEs. The topic can be adjusted based on the level and interest of the student, e.g., linear algebra, quadrature rules, or probabilistic questions. Contact person: Harri Hakula

**13.** Suppose that A and B are two square integer matrices. We say that they are similar over Z if there is a third integer matrix C,whose determinant is +1 or -1, such that AC=CB. And we say that they are similar over Q if there is an invertible rational matrix X such that AX=XB. Clearly, similarity over Z implies similarity over Q, but the opposite is not true. While similarity over Q is very well understood and can be reduced to the study of the Frobenius canonical form, the situation is much more subtle for similarity over Z. The goal of the project is to study similarity over Z, reviewing scattered results on it that exist in the literature and summarising their content in a written report. The project lies at the intersection of matrix theory, algebra (especially ring theory and module theory), and number theory. Contact person: Vanni Noferini

**14. **Cameras and machine vision are used in many engineering and industrial applications. In this topic, we study mathematical models of cameras and investigate combining video from multiple overlapping sources that are not assumed to be precisely positioned nor synchronized. Contact person: Antti Hannukainen

Operations Research and Systems Analysis

More information here.

Statistics and Mathematical Data Science

**15. **Research intern to work on an applied project that is related to detecting accidents or other outlying (non-recurring) events on highways that might require emergency response and hence, fast allocation of communication resources to first-responder teams. Basic knowledge of probability theory, statistics, and Python programming are required for this position. Contact: Natalia Vesselinova (natalia.vesselinova@aalto.fi) and Pauliina Ilmonen (pauliina.ilmonen(a)aalto.fi)

**16. **Research intern to prepare new exercise problems for the lecture course MS-C2128 Prediction and Time Series Analysis and/or for the lecture course MS-E2112 Multivariate Statistical Analysis. The work is done under the guidance of Pauliina Ilmonen. This position is suitable for someone who has already written their Bachelor’s thesis. Grade 5 from one of these courses is required for this position. Contact: Pauliina Ilmonen (pauliina.ilmonen(a)aalto.fi)

**17. **Research intern in mathematical statistics. Possible topics include topics related to functional data analysis and/or extreme value theory. Interest in studying the mathematics behind statistical methods is required for this position. Contact: Pauliina Ilmonen (pauliina.ilmonen(a)aalto.fi)

**18. **Research intern to work on the statistical analysis of clustering methods for node-attributed and high-order network data. Strong command of probability theory and linear algebra are expected. Contact: Lasse Leskelä.

**19.** A research intern for two tasks:

**a)** Translate into Finnish the exercises of the course Statistical Inference (MS-C1620).

**b)** Collection of time series data and time permitting tentative modelling of them. Time series for the date of breakage of ice of Aura River (Aurajoki), Porvoo River (Porvoonjoki), and Kokemäenjoki River (Kokemäenjoki) have been published for years 1740-1839, 1771-1839, and 1800-1849 respectively. The task is to continue these time series to the present day. They are interesting because of their very long time span. The ultimate idea is to inspect climate change with these time series.

The work is done under the guidance of Pekka Pere. Contact: Pekka Pere (pekka.j.pere "at" aalto.fi).

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