Mutlu Koçar, Galatasaray University

24/05/2019, Time: 10:30

Place: B256

Mutlu Koçar, Galatasaray University

24/05/2019, Time: 10:30

Place: B256

Sedef Taşkın (DEU)

03/05/2019, Time: 10:00

Place: B256

In his epoch-making memoir of 1860 Riemann showed that the key to the deeper investigation of the distribution of the primes lies in the study of zeta function. Riemann proved that the zeta function can be continued analytically over the whole plane and its only pole being a simple pole at s=1. In this talk, we first introduce the gamma function. After that we mention analytic continuation of the zeta function. Finally, we obtain its functional equation.

**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

Hikmet Burak Özcan (DEU)

Date&Time: 29/03/2019, 10:00

Place: B256

In this talk, after defining rational points on a curve we will address the problem of finding the rational points on curves. We will give a recipe in order to generate a new rational point from already known ones. After that we will introduce the notion of elliptic curves and mention the rational points on elliptic curves. Finally we will refer to the well-known results, Mordell’s Theorem and Siegel’s Theorem.

**Doga Can Sertbaş**, Cumhuriyet University

Date and Time: 22/03/2019, 10:00

Place: B256

In 1947, Erdos gave a lower bound for the diagonal Ramsey numbers R(k,k). His proof contains purely probabilistic arguments where the original problem is not related to the probability theory. This pioneering work of Erdos gave rise to a new proof technique which is so called the probabilistic method. According to this method, one just obtains the existence of a particular mathematical object in a non-constructive way. In this talk, we first introduce the Ramsey numbers and then explain the basics of the probability theory. After mentioning the fundamentals of the probabilistic method, we give several examples from the number theory. In particular using probabilistic inequalities, we show how one can prove some number theoretic results which seem completely unrelated to the probability theory.

**Neslihan Güğümcü**, Technical University of Athens

Date and Time: 15/03/2019, 10:00

Place: B256

Abstract: Planar curves have been studied since the time of Gauss. Gauss was one of the first to notice that they can be handled combinatorially by codes (named as Guass codes) that are strings of labels encoding self-intersections. Whitney classified all immersed curves up to a topological relation called regular homotopy by using the winding number of immersion maps. In the first half of the 20^{th} century Reidemeister showed that classical knot theory is equivalent to the study of immersed curves in the plane, whose self-intersections are endowed with a combinatorial structure, with an under/over-data. With this extra structure, regular homotopy needs to transforms into a richer equivalence relation generated by Reidemeister moves. Since then knot theory is a classical subject of topology, bringing us many interesting questions relating to combinatorial topology.

In this talk, we will talk about knotoids (introduced by Turaev) that provide us a new diagrammatic theory that is an extension of classical knot theory. Problem of classifying knotoids lies at the center of the theory of knotoids. We will construct a Laurent polynomial with integer coefficients for knotoids called the affine index polynomial and we will show how it contributes to the classification problem.

**Halil Oruç**, DEU**Abstract:** Vandermonde matrix and interpolation, elementary and complete symmetric functions, combinatrorial identities and Blossoms.

Date and Time: 01/03/2019, 10:00

Place: **B256, DEU mathematics department**

**Selçuk Demir**, DEU

**Abstract:** Schnirelmann Density will be defined and its relation to basis

properties of sequences of integers will be explained. Some classical results,

namley the theorems of Schnirelmann, Mann and Erdos will be discussed.

Date and Time: 22/01/2019, 10:00

Place: **B256**

**Haydar Göral**, (DEU)

**Abstract:** This talk will be on the sum of reciprocals of primes and the probability of choosing a prime number.

**Date:** 11/01/2019, **Time**: 10:15

**Place:** The room B259, Mathematics department, DEU

**2019 January 4**: Celal Cem Sarıoğlu (DEU)

**Abstract**: In this talk, we will introduce how can we compute the arithmetic and geometric genus of an irreducible projective algebraic curve and how they are related to the genus of an oriented Riemann surface.

**Time:** 10:15

**Place:** Room B259