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.
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
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.
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.
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.
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.
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.
