# Bachelor in Mathematics Student Task

We group students from the six 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 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.

Time frame: 8 weeks.

Starting date: 26 April 2021.
Weeks 1-7 is on Zoom. Week 8 is the week where each group meets physically with the mentor, at the mentor's university, to hold the final seminar and discover the 4EU+ partner. The precise dates and ways to organize the weeks will be discussed with the mentors to help the students (in particular to avoid eventual conflicts with exams), we will be as flexible as possible.

Outcome: Students will work together on the topic of their choice (see below), and prepare a written report in English, a short oral presentation in English and a wiki-page in the official language of their university.

## Contacts

• Charles University (Prague)
Zbynek Pawlas (email: pawlas"at" karlin.mff.cuni.cz)
• Heidelberg University
Michael J Winckler (email: Michael.Winckler "at" iwr.uni-heidelberg.de)
• Sorbonne University
Pierre Charollois (email: pierre.charollois "at" imj-prg.fr)
• University of Copenhagen
Lars Kühne (email: lk "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)

## Topics

"Isoperimetric inequality" with Zbynek Pawlas in Prag. We will study the isoperimetric inequality, from proof to applications, and structure this information for presentation on a wiki page.

"Study of the series with general term $1/(an+bm)^k(cn+dm)^p$, with a,b,c,d fixed integers" with Pierre Charollois in Paris: We will study the arithmetic properties of these general series.

"Numerical quadrature rules - theory and implementation" with Michael J Winckler in Heidelberg: Numerical rules to calculate the area below a curve are very old. Discrete methods (like the Simpson's rule) were already well known several hundred years ago. However, these rules are not magic formulas but they are based on clever combinations of interpolation and integration. In the
BMST group we will analyze several of these rule sets, find their common construction plan and note their differences. We finally structure this information for presentation on a wiki page.

"p-adic numbers" with Lars Kühne in Copenhagen: We recall how are the real numbers are constructed from Q (Cauchy sequences!) and study p-adic absolute values, p-adic numbers, basic structure of the p-adic numbers, Hensel's Lemma, the (p-1)-th roots of unity, and elementary analysis in Q_p.

"Limit theorems by the Stein's approach" with Witold Bednorz in Warsaw: We will study limit theorems in probability, for example for Poisson type distributions.
Sources:
a) Probability Surveys Vol. 8 (2011) 210–293 Fundamentals of Stein’s method ∗ Nathan Ross
b) Stein-Chen method for Poisson approximation. ST414 (TERM 2, 2013{2014) Parthea Dey University of Warwick.

"Modular Arithmetic and public key cryptography" with Ottavio Rizzo in Milan: We will study applications to cryptography of arithmetic properties of congruences.