**Place: **Dokuz Eylül University, Mathematics Department, B256

**Date:** 10 May 2019

Our workshop is supported by TMD (MAD). We are grateful to them for this support.

Deadline for application is 2 May 2019

Application form click

**Invited speakers**

Ali Ulaş Özgür Kişisel, Middle East Technical University

Ayberk Zeytin, Galatasaray University

Ekin Özman, Boğaziçi University

Yıldırım Akbal, Atılım University

**Program**

9:15—9:30: Opening

9:30—10:45: Ali Ulaş Özgür Kişisel

10:45—11:15: Coffee break

11:15—12:30: Ekin Özman

12:30—14:30: Öğle arası

14:30—15:45: Yıldırım Akbal

15:45—16:15: Coffee break

16:15—17:30: Ayberk Zeytin

**Ali Ulaş Özgür Kişisel**, Middle East Technical University

**Title:** Line Arrangements Over Different Base Fields

**Abstract**: There are various obstructions regarding the existence of line arrangements in the projective plane over a given base field. In this talk, some of these obstructions and how they depend on the chosen base field will be explained.

**Ekin Özman,** Boğaziçi University

**Title:** Modularity, rational points and Diophantine Equations

**Abstract:** Understanding solutions of Diophantine equations over rationals or more generally over any number field is one of the main problems of number theory. One of the most spectacular recent achievement in this area is the proof of Fermat’s last theorem by Wiles. By the help of the modular techniques used in this proof and its generalizations it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve or rational points on its twists play a central role in this approach. In this talk, I will summarize the modular method and mention some recent results about points on modular curves. This is joint work with Samir Siksek.

**Yıldırım Akbal,** Atılım University

**Title:** Waring’s Problem, Exponential Sums and Vinogradov’s Mean Value Theorem

**Abstract:** Having introduced Hardy&Littlewood Circle method, we will jump to Waring’s Problem: representability of a large integer as the sum of s kth powers of positive integers, which was the main motivation of Vinogradov to study a system equations (called Vinogradov’s system). Next we move on Vinogradov’s mean value theorem: a non-trivial upper-bound on the number of solutions to Vinogradov’s system, and then mention the milestone contributions of Vinogradov, Wooley and Bourgain (rip) et al.

Last but not least, some applications of Vinogradov’s mean value theorem on exponential sums will be given.

**Ayberk Zeytin,** Galatasaray University

**Title:** Arithmetic of Subgroups of PSL2(Z)

**Abstract:** The purpose of the talk is to introduce certain arithmetic questions from a combinatorial viewpoint. The fundamental object is the category of subgroups of the modular group and its generalizations. I will try to present the different nature of arithmetic of subgroups of finite and infinite index and their relationship to classical problems. I plan to formulate specific questions at the very end of the presentation and, if time permits, our contribution to both worlds.

This is partly joint with M. Uludag