Bachelor in Mathematics Student Task 2025

We group students from the universities of the 4EU+ alliance to work on joint projects!

Each group of students is composed of one student per university. Each group of students has one mentor. Each mentor is responsible for his/her own students and for his/her group. Don't hesitate to contact the coordinator or the various contact persons below for more information! For Copenhagen students, this can take the form of a PUK project.

Starting date: February 3, 2025 (blok 3 for Copenhagen).
Final week: between April 4 and April 22, depending on time constraints

  • Copenhagen - p-adic numbers and applications
  • Geneva - TBA
  • Heidelberg - Primes in Arithmetic Progressions
  • Milan - Modular arithmetic and public-key cryptography
  • Paris - Dynamical systems and visualization
  • Prague - Inequalities for geometric functionals
  • Warsaw - Matching theorems

    Copenhagen

    Mentor: Adrien Morin
    Contact person: Adel Betina and Fabien Pazuki
    Topic: p-adic numbers and applications

    Project Description:

    The field of $p$-adic numbers $\mathbb{Q}_p$ together with its ring of integers $\mathbb{Z}_p$ are fundamental analysis tools of modern number theory. They provide an alternative, for every prime $p$, to the field of real numbers. The Hasse-Minkowski theorem states for example that a quadratic equation $ax^2+bxy+cy^2=0$ with rational coefficients has a non-zero rational solution $(x,y)$ if and only if it has non-zero real solutions and non-zero solutions in $\mathbb{Q}_p$ for all primes $p$. One of the central properties of the $p$-adic numbers is Hensel's lemma, which allows us to lift a factorization modulo a prime number $p$ of a polynomial over the integers to a factorization modulo any power of $p$, and hence to a factorization over the $p$-adic integers.
    Here is an outline of the project:
    1) Construction of $p$-adic numbers, arithmetic in $\mathbb{Z}_p$.
    2) Hensel's lemma.
    3) Analysis on $\mathbb{Q}_p$: power series and Weierstrass’ p-adic preparation theorem
    4) Chosen topic(s) among the following: some applications to Diophantine equations; Monsky’s theorem on equitriangulations of squares; towards analytic geometry: Tate algebras.

    Time Frame:
    10-12 weeks in total, submission deadlines after agreement between the participants and the coordinator.

    Target group: Students from year 2 and 3 (bachelor mathematics).

    Learning Outcomes: Learning new methods in analysis and number theory.

    Workload: Meetings every week, independent work online, group work.

    Evaluation: Students receive a Danish grade (12, 10, 7, 4, 02, 00, -3). If you receive a numerical grade for this project with your local university, some conversion schemes will be applied. If you only receive a pass/fail, then all grades except 00 and -3 mean a pass.


    Geneva

    Mentor: Giovanna Di Marzo
    Contact person: Giovanna Di Marzo
    Topic: TBA
    Lecturer: TBA

    Description: TBA

    Project Description: TBA

    Time Frame:
    10-12 weeks in total, submission deadlines after agreement between the participants and the coordinator.

    Prerequisites: TBA

    Learning Outcomes: TBA

    Heidelberg

    Mentor: Andrea Conti
    Contact person: Andrea Conti and Michael Winckler
    Topic: Primes in Arithmetic Progressions
    Lecturer: Andrea Conti

    Project Description:

    Let $m$ and $k$ be two coprime positive integers. Does the arithmetic progression $\{ma+k\}_{a\in\mathbb{N}}$ contain any prime number? A classical theorem of Dirichlet shows that the answer is yes for any choice of $m$ and $k$, and that such a progression actually contains infinitely many primes. The classical proof of this theorem goes via the study of the properties of a complex analytic object, the $L$-function attached to a Dirichlet character; it is a generalization of the well-known Riemann $\zeta$-function. The proof gives a first example of how analytic objects can make a meaningful appearance in the solution of number-theoretic problems.

    Roughly speaking, the students are expected to:

    -- understand how to attach an $L$-function to a Dirichlet character, and how to prove some of its basic properties;
    -- show how the behaviour of the $L$-function is related to the distribution of prime numbers among the residue classes modulo $m$, described in terms of analytic density.

    Very motivated students can go deeper towards the modern theory of $L$-functions: there are analogues of Dirichlet's theorem for arbitrary number fields, together with finer (and harder) results on prime densities in number fields.

    Time Frame:
    10-12 weeks in total, submission deadlines after agreement between the participants and the coordinator.

    Prerequisites: TBA

    Learning Outcomes: Learning a first example of how to extract number-theoretic information from the study of $L$-functions.

    • Milan

      Mentor: Ottavio Rizzo
      Contact person: Ottavio Rizzo
      Topic: Cryptography
      Title: Modular arithmetic and public key cryptography

      Description:

      We will study applications to cryptography of arithmetic properties of congruences.
      Modular arithmetic is the main ingredient for public key cryptography, the technique that allows people to have a private conversation in a public space without the need to share in advance a secret key. We will again begin with discovering what is public key cryptography and how modular arithmetic plays its role in the game, to later focus on how to efficiently compute the greatest common divisor of two integers and how to (inefficiently) solve the discrete logarithm problem.

      Prerequisite is the first year of algebra (groups, rings, fields, modular arithmetic); some programming experience and familiarity with LaTeX is good but not strictly necessary.

      Paris

      Mentor: Frédéric Le Roux, Antonin Guilloux
      Contact person: Antonin Guilloux
      Topic: Dynamical systems and visualization

      Description:

    • The qualitative studies of transformations of the disk or the torus leads to a series of notions linked to the theory of Dynamical Systems. Some of these notions (e.g. rotation numbers and rotation sets, fractals...), some proofs of results can be visualized using a computer.

      These visualizations may be used to explain the theory to fellow students and as such are very gratifying. The process of building these visualizations requires a deep understanding of the theory. It is often a very interesting angle from which studying maths.

      The goal of this project is to present a few of these possible visualizations and build them.

      Time Frame:
      10-12 weeks in total, submission deadlines after agreement between the participants and the coordinator.

      Prerequisite: Not much, a bit of combinatorics, and calculus. And a lot of curiosity!

      Prague

      Mentor: Pawlas Zbynek
      Contact person: Pawlas Zbynek
      Topic: Inequalities for geometric functionals

      Description: There are numerous inequalities involving the fundamental
      functionals of geometric objects. These geometric inequalities may
      pertain to size functionals (lengths, areas, volumes, etc.), as well as
      those related to shape (angles, number of vertices, etc.). We primarily
      focus on planar sets, often restricting our attention to convex bodies
      (compact convex sets with non-empty interiors). For these sets, the
      functionals of interest are perimeter, area, circumradius, inradius,
      diameter, and minimal width. In the previous rounds of the BMST project,
      we explored inequalities that involve two functionals. These
      inequalities help solve optimization problems where one functional is
      fixed while another is to be maximized (or minimized). Examples of such
      inequalities and the questions they adress include:
      1. isoperimetric inequality - among all sets with a fixed perimeter,
      which has the largest area,
      2. isodiametric inequalities -  among all sets with a fixed diameter,
      which has the largest area, and which has the largest perimeter,
      3. Blaschke-Lebesgue inequality - among all convex bodies with a fixed
      constant width, which has the smallest area,
      4. Jung's inequality - among all convex bodies with a fixed
      circumradius, which has the smallest diameter,
      5. Pál's inequality - among all convex bodies with a fixed inradius,
      which has the largest width.
      In each of these scenarios, it is possible to restrict to some specific
      classes of sets, such as polygons.

      The goal of this BMST 2025 project is to investigate a broader range of
      geometric inequalities. Specifically, we will study problems where two
      functionals are fixed, and a third is to be optimized. For instance, one 
      problem might involve finding the convex body with the largest area
      given a fixed diameter and circumradius. Each student will be assigned
      one or more problems to investigate. Students will elaborate on
      solutions. The results will be compiled into a joint report.

      Time frame: 8-10 weeks.

      Prerequisite: Elementary geometry.

      Workload: Meetings every week, independent work online, group work.

    • Warsaw

      Mentor: Witold Bednorz
      Contact person: Witold Bednorz
      Topics: Matching theorems

      Description: The problem we solve in this project is how to best match two
      groups of elements. A well-known result in this context is Hall's
      Marriage Theorem, in which one tries to match boys and girls in such a
      way that the number of pairs that are formed is maximized. In a more
      complicated situation, one has a hyper-cube with a number of uniformly
      distributed points, and then independently and randomly selects the same
      number of new points. The problem is to match the points in two groups
      so that the sum of the distances between the pairs is minimal. There are
      many deep results related to this question, and we are going to examine
      some of them.

      Requirements: Mathematics, probability theory.

      Goals:
      Study results around matching theorems.

      Contacts

      • Charles University (Prague)
        Zbynek Pawlas (email: pawlas "at" karlin.mff.cuni.cz)
      • Geneva University
        Giovanna Di Marzo Serugendo (email: Giovanna.DiMarzo "at" unige.ch)
      • Heidelberg University
        Michael J Winckler (email: Michael.Winckler "at" iwr.uni-heidelberg.de) 
        Andrea Conti (email: contiand "at" gmail.com)
      • Sorbonne University
        Antonin Guilloux (email: antonin.guilloux "at" imj.prg.fr)
      • University of Copenhagen
        Adrien Morin (email: admo "at" math.ku.dk)
      • University of Milan
        Ottavio Rizzo (email: ottavio.rizzo "at" unimi.it)
      • University of Warsaw
        Witold Bednorz (email: wbednorz "at" mimuw.edu.pl) 

      Coordination: Fabien Pazuki (email: fpazuki "at" math.ku.dk)