BMST 2027 4EU+

We group bachelor 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 8, 2027 (blok 3 for Copenhagen).
Final week: between April 11 and April 25, depending on time constraints

  • Copenhagen - p-adic numbers and applications
  • Geneva - Data-driven behavioural modeling for digital twins
  • Heidelberg - The congruent number problem and the arithmetic of elliptic curves
  • Milan - Modular arithmetic and public key cryptography
  • Paris Assas - Mathematics at the interface with the humanities and social sciences
  • Paris Sorbonne - Expander graphs
  • Prague - Inequalities in convex geometry  
  • Warsaw - Log concave distributions

    Copenhagen

    Mentor: Patricio Perez Pina
    Contact person: 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:

  • Construction of $p$-adic numbers, arithmetic in $\mathbb{Z}_p$.

  • Hensel's lemma.

  • Analysis on $\mathbb{Q}_p$: power series and Weierstrass' $p$-adic preparation theorem.

  • Chosen topic(s). (e.g., Local-Global principle, Skolem-Mahler-Lech theorem, 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.

    Prerequisites: Linear algebra, a first course in groups and rings, some topology (metric spaces suffice). Some knowledge of algebraic number theory allows for a broader choice of projects at the end of the block.

    Target group: Students from year 3 (students from year 2 are welcome, if they are ready to catch up on prerequisites).

    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: Data-driven behavioural modeling for digital twins
    Lecturer: Giovanna Di Marzo, Antonin Sedoh

    Description:

    Digital Twins (DTs) are increasingly used to simulate, monitor, and optimise complex systems. They can also be used for supporting decision-making for crisis management (e.g. natural disasters). DT come in various flavours from
    level 1: static twin (with no data), to
    level 2: functional twin (with dynamic behaviour) to
    level 3: real-time (with real-time data) up to
    level 4: intelligent one (taking actions).

    This project tackles DT of level 3 (with real-time data and dynamic behaviour). It leverages advanced data analysis and Machine Learning (ML) or AI-Based techniques to model (or complement) the dynamic behavior of a Digital Twin (DT).  The project will use real or simulated heterogeneous data, most likely heterogeneous time-series (e.g. energy consumption, weather information, social media posts, geo-spatial data, etc.).

    Case studies domains and examples: 
    Modelling connected objects behaviour
    Predictive maintenance in smart manufacturing
    Urban mobility and infrastructure maintenance
    Energy grid management
    Healthcare and patient monitoring
    Autonomous vehicles
    Trading portfolio
    Disaster response and emergence services
    Pandemic management
    Supply chain resilience in case of geopolitical crises

    The project will take place in several steps:

    1.     Case study selection and data identification

    2.     Data acquisition and pre-processing 

    3.     Behaviour modeling with ML

    4.     Evaluation and validation

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

    Prerequisites: Basics of programming.

    Target group: Bsc Math or Computer Science (2nd, 3rd year).

    Learning Outcomes: Developing AI-based services, heterogeneous data analysis and aggregations, time-series analysis, digital twin modeling.

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

    Evaluation: The evaluation is based on the work done (steps 1 to 4 above), a written report and a presentation including a demonstration.

    Heidelberg

    Mentor: Andrea Conti
    Contact person: Andrea Conti, Gebhard Böckle, and Michael Winckler
    Topic: The congruent number problem and the arithmetic of elliptic curves
    Lecturer: Andrea Conti

    Project Description:

    A natural number is said to be congruent if it is the area of a right
    triangle with sides of rational length. The first few congruent numbers are 5,6,7,13,14,15,20,21.
    While it is easy to produce an infinity of congruent numbers by looking at Pythagorean triples, it
    is very hard to determine whether a given number is congruent. This problem can be rephrased
    in terms of a question about the rational points of a certain “elliptic curve”. Elliptic curves play
    a central role in modern number theory, since their structure is rich with arithmetic information.
    Among their main features, they can be equipped with a group law, i.e. there is a way to add
    two rational points of an elliptic curve that makes the set of such points into a group.
    The goal of the project is to study some of the basic properties of elliptic curves, and show
    that a natural number $n$ is congruent if and only if the elliptic curve defined by the equation
    $y^2 = x^3 − n^2x$ has a rational point of infinite order. Depending on time and the students’
    motivation, we can go further and look at Tunnell’s criterion for determining whether a number
    is congruent assuming the celebrated Birch and Swinnerton-Dyer conjecture.
    The main reference for the project will be Koblitz’s book “Introduction to Elliptic Curves
    and Modular Forms”. Many extra references on elliptic curves can be consulted, including
    Silverman’s classic “The Arithmetic of Elliptic Curves”.

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

    Prerequisites: Basic algebra and analysis. If possible, some basics of complex analysis. If not,
    they can be covered during the seminar.

    Learning Outcomes: Learning an example of how to reinterpret a diophantine problem as a
    question about the rational points of an elliptic curve. Learning some basics of the arithmetic
    of elliptic curves.

    • 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 begin with discovering what is public key cryptography and how modular arithmetic plays its role in the game, introduce some simple public key cryptosystem (Diffie-Hellman Key Exchange, RSA, El Gamal), try to understand why they should be secure, and finally focus on digital signatures.

      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 Assas

      Mentor: Lisa Morhaim
      Contact person: Lisa Morhaim
      Topic: Mathematics at the interface with the humanities and social sciences

      Description: This project will develop mathematics at the 
      interface with humanities and social sciences. It will focus mainly on 
      Economics but also Law, Geography, History, Sociology, Anthropology 
      can be discussed.
      To be as self-contained as possible while going as deep as possible,
      each topic will include developing 1) the needed mathematical 
      background,  2) modeling issues 3) the theorem(s) and proofs 4) some 
      further development and 5) further and current issues and applications 
      (including AI). The humanities and social sciences topics can embrace 
      game theory, social choice theory and voting, networks, argumentation, 
      kinship, chronologies, etc and topics in functional analysis, graph 
      theory, optimization,logic, algebra, topology and geometry are among 
      the mathematics involved.
      Works such as the ones by Kenneth Arrow, Nicolas de Condorcet, George 
      Dantzig, Gérard Debreu, Leonid Kantorovich, Tjalling Koopmans, Andreu 
      Mas-Colell, John Nash, John von Neumann, Lloyd Shapley, Herbert Simon, 
      André Weil, etc. are among the ones that may be studied.

      Based on the interests of the group, we will choose some specific topics.

    • Paris Sorbonne

      Mentor: Bram Petri, Antonin Guilloux
      Contact person: Antonin Guilloux
      Topic: Expander graphs

      Description:

    • A graph is a set of points (usually called vertices) that is connected by lines (usually called edges). This simple structure, describing the connections between some set of objects, plays a role in many different branches of mathematics, ranging from geometry and topology to group theory, analysis, probability theory and combinatorics. An "expander graph" is a sequence of finite graphs with growing numbers of vertices that are both sparse and well-connected. Here, sparseness is measured in terms of the degrees of the vertices of the graph: the number of edges incident to these vertices. We ask that the degrees of all vertices in our sequence is uniformly bounded. However, we also ask that the graphs are well connected. This, one measures with their isoperimetric constant (also called Cheeger constant) or equivalently, the spectral gap of their adjacency matrix.
      The goal of the project is to first learn the basics of the theory of expander graphs. After this we will study (near) optimal spectral expansion, so called (near) Ramanujan graphs. We will study both random consturctions and explicit constructions. There have been spectacular recent advances on this subject and we will study some of these. Afterwards, also depending on the preferences of the participants, there can also be a programming component. We will write a program that generates these graphs and will study various geometric properties of them.
      Part of the project, especially the beginning, will be based on the survey article "Expander graphs and their applications" by Hoory, Linial and Wigderson that the reader is also encouraged to consult for more information on the subject.

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

      Prerequisite: Linear algebra and basic modular arithmetic.

    • Prague

      Mentor: Pawlas Zbynek
      Contact person: Pawlas Zbynek
      Topic: Inequalities in convex geometry 

      Description: 
      There are numerous inequalities involving fundamental
      geometrical functionals of convex bodies (compact convex sets with
      non-empty interiors). In previous rounds of the BMST project, we
      investigated several classical geometric inequalities for planar convex
      bodies, including the isoperimetric inequality, isodiametric inequality,
      Blaschke-Lebesgue inequality, Brunn-Minkowski inequality, and
      Rogers-Shephard inequality.

      The goal of the BMST 2027 project is to explore selected further
      classical inequalities in convex geometry. Possible topics include
      Urysohn's inequality, Bonnesen-type inequalities, and Minkowski's
      inequalities for mixed volumes. Each student will be assigned one or
      more inequalities to investigate. The results will be compiled into a
      joint report.

    • Time frame: 8-10 weeks.

      Prerequisite: Real analysis (measure, integration), basic convex geometry.

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

    • Deliverables: Joint written report, presentation (20-30 minutes).

    • Warsaw

      Mentor: Witold Bednorz
      Contact person: Witold Bednorz
      Topics: Log-concave distributions

      Description: Log-concave distributions constitute one of the most
      important classes of distributions. Formally, a random vector X in
      d-dimensional real space is required to have a density of the form
      $\exp(-U(x))$, where U is a convex function.
      The analysis of such vectors involves studying the moments of linear
      projections of the form $<t,X>$ and subsequently analyzing norms—an
      approach that can be generalized to studying the expected value of the
      quantity $\sup_{t\in T}<t,X>$. Significant progress has recently been made
      in analyzing these types of problems through the use of tools from
      information theory and statistics. The main objective is to present new
      directions and ideas that can be further developed through research.

      Requirements: Knowledge of probability theory. I intend to provide the
      necessary probabilistic foundations for all the tools used in this
      analysis.

      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
        Andrea Conti (email: andrea.conti"at"iwr.uni-heidelberg.de)
        Gebhard Böckle (email: gebhard.boeckle"at"iwr.uni-heidelberg.de )
        Michael J Winckler (email: Michael.Winckler "at" iwr.uni-heidelberg.de) 
      • Paris Assas
        Lisa Morhaim (email: lisa.morhaim "at" assas-universite.fr)
      • Sorbonne University
        Antonin Guilloux (email: antonin.guilloux "at" imj.prg.fr)
        Bram Petri (email: bpetri"at"imj-prg.fr)
      • University of Copenhagen
        Patricio Perez Pina (email: papp "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)