Bogazici University

Summer School on Effective Geometry and Arithmetic of Curves

Feza Gürsey Center for Physics and Mathematics

and

Istanbul Center for Mathematical Sciences

4-8 June 2018.

Program

Click here for the program of the summer school

Information

The mathematical school will focus on effective and explicit methods in the theory of algebraic curves and applications to coding theory and cryptography. An important aspect of the school will be tutorials accompanying each of the lecture series. In these tutorials participants will have the opportunity to put into practice the theory from the lectures and experiment with the introduced object using various computer algebra systems. Depending on the background of the students, they might be given small research problems on which they can work together during the week. Active participation and interaction will be a key aspect of the mathematical school. This school also serves the purpose of preparing students for a sequence of conferences and workshops to be organized in the near future.

Lecturers

Nurdagül Anbar - RICAM, Linz

Algebraic function fields and applications to cryptography and coding theory

Anne Frühbis-Krüger - Leibniz Universität Hannover

Algorithmic Aspects of Desingularization of 2-dimensional schemes

Christophe Ritzenthaler - University Rennes 1

Models of curves of low genus and number of points over finite fields

Description of lectures

Nurdagül Anbar

Algebraic function fields and applications to cryptography and coding theory

-Algebraic function fields and their extensions

-Congruence function fields, rational places, The Hasse-Weil bound, maximal curves, Serre-bound, Ihara bound, Drinfeld-Vladut bound

-Towers of function fields

-Examples of exceptional function fields (Hermition, Ree, Suzuki, GK, etc.)

-Goppa codes, almost perfect sequences, low-discrepancy sequences

Anne Frühbis-Krüger

Algorithmic Aspects of Desingularization of 2-dimensional schemes

Resolution of singularities is a well-known theoretical and practical tool in algebraic geometry over fields of characteristic zero and for surfaces also in arbitrary characteristic. For surfaces in mixed characteristic, it is by far less used, as there are a number of additional algorithmic issues to consider. In this mini-course we shall first explore the tasks and challenges of algorithmic desingularization in general and explicitly provide basic algorithmic tools like blow-ups. In the second lecture we will focus on the desingulariziation invariant which controls the whole resolution process and on algorithmically determining the locus of maximal order. The last of the three lectures will be devoted to a variant of the algorithm of Cossart-Jannsen-Saito for resolving 2-dimensional schemes in arbitrary and mixed characteristic.

The mini-course is designed for PhD-students and advanced Master-students, so there will be an emphasis on instructive examples and explicit algorithmic techniques.

Christophe Ritzenthaler

Models of curves of low genus and number of points over finite fields

- Review of some basic properties: function fields, divisors, Riemann-Roch, Riemann-Hurwitz

- The canonical map and hyperelliptic curves/non hyperelliptic curves

- Equation of curves of genus up to 5

- Weil conjectures for curves with the proof of Stepanov/Bombieri

- Jacobians: application to cryptography and construction of curves with many points

Practical information

We have financial support available especially for young participants to partially/totally cover accommodation and travel expenses. People from developing countries are especially invited to apply.

In order to attend the summer school, please send a short CV (containing mathematical background and name of one/two reference(s)). Indicate if you request financial support for travel and/or accommodation. Deadline for applications will be 15th of May.

alp.bassa (a) gmail.com

ritzenthalerchristophe (a) gmail.com

Female mathematicians are particularly encouraged to apply.

Organizers

Alp Bassa, Ekin Özman and Christophe Ritzenthaler

boun campusfrance CIMPA Feza Gursey

IMBM TMD-MADrennes