arithmetic

The Analytic Continuation of the Riemann Zeta function

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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. 

Arithmetica İzmir 2

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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

Rational Points on Curves

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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.

Probabilistic Methods in Number Theory

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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.

Knotted Strings in the plane

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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 20th 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.