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  padic numbers and applications
 Geneva  TBA
 Heidelberg  Primes in Arithmetic Progressions
 Milan  Modular arithmetic and publickey 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: padic 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 HasseMinkowski theorem states for example that a quadratic equation $ax^2+bxy+cy^2=0$ with rational coefficients has a nonzero rational solution $(x,y)$ if and only if it has nonzero real solutions and nonzero 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’ padic 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:
1012 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: TBADescription: TBA
Project Description: TBA
Time Frame:
1012 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 ContiProject 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 wellknown Riemann $\zeta$function. The proof gives a first example of how analytic objects can make a meaningful appearance in the solution of numbertheoretic 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:
1012 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 numbertheoretic information from the study of $L$functions.
Milan
Mentor: Ottavio Rizzo
Contact person: Ottavio Rizzo
Topic: Cryptography
Title: Modular arithmetic and public key cryptographyDescription:
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 visualizationDescription:

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:
1012 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 functionalsDescription: 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 nonempty 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. BlaschkeLebesgue 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: 810 weeks.
Prerequisite: Elementary geometry.
Workload: Meetings every week, independent work online, group work.

Warsaw
Mentor: Witold Bednorz
Contact person: Witold Bednorz
Topics: Matching theoremsDescription: The problem we solve in this project is how to best match two
groups of elements. A wellknown 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 hypercube 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.uniheidelberg.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)
 Charles University (Prague)