arithmetic

Genus of complex projective algebraic curves

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Celal Cem Sarıoğlu (DEÜ)

December 14, Time:10:15, Place: B259

Abstract: From the Geometry-Topology courses everyone knows the number of holes of an orientable compacts surface is known as a genus,and it is related with its Euler characteristic. On the other hand, in algebraic geometry, there are two related definitions of genus of an irreducible projective algebraic curve C: the arithmetic genus, and the geometric genus. If the curve C has no singular points these two concepts will coincide and also coincide with the topological definition applied to Riemann surfaces of C. 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.

Misusing elliptic curves in key agreement protocols

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Berkant Ustaoğlu (IYTE)
November 30, time: 10:15, place: B259
Abstract: Cryptography intersects various scientific fields. Achieving true security is a highly non-trivial task. One of the many reasons is easy miscommunication among the various fields. This talk will illustrate the above idea: initially we will develop a model for secure and authentic key establishment. Then employing some plain ideas we will point out how things can go wrong. Lastly, we will extend those principles to elliptic curves arguing that omitting simple for in one field details when communicating knowledge leads to major issues more so if the ingredients require in depth understanding in specialised areas.

Congruent Number Problem

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Hikmet Burak Özcan 

Date&Time: November 23, Friday,  10:15

Abstract: In this talk, we will first mention about the importance of rational points on curves and show the correspondence between Pythagorean triples and rational points on the unit circle. Secondly, we will introduce a well known arithmetic problem, congruent number problem and show that the congruent number problem is equivalent to a problem about rational squares in arithmetic progressions. Lastly, we will concentrate the relationship between the congruent number problem and rational points on the elliptic curve.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B259.

Arithmetica İzmir 1

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Date: November 16, 2018

Place: Dokuz Eylül University, Mathematics Department, B255

Invited speakers

Alp Bassa, Boğaziçi University

Ayhan Günaydın, Boğaziçi University

Şermin Çam Çelik, Özyeğin University

Doğa Can Sertbaş, Cumhuriyet University

Ayhan Günaydın, Boğaziçi Üniversitesi

Title: Polynomial Exponential Equations

Abstract: Let V be an algebraic set in C^{2n} over complex numbers. We would like to find a generic element of V of the form (x,exp(x)) where exp is the complex exponential map defined on C^n. Of course, we need certain conditions necessary for the existence of such an element. These assumptions will be explained in the beginning of the talk. Then we present a way to change this question into a question about the solutions of polynomial‐exponential equations over the complex numbers where the variables run through the rationals. After that we make a reduction to integers variables. Finally focusing on the case n=1, we explain why such a generic element should exists.
(Joint work with Paola D’Aquino, Antongiulio Fornasiero, and Giuseppina Terzo.)

Alp Bassa, Boğaziçi Üniversitesi

Title: Good Recursive Towers over Prime Fields Exist

Abstract: In the past, various methods have been employed to construct high genus curves over finite fields with many rational points. One such method is by means of explicit recursive towers and will be the emphasis of this talk. The first explicit examples were found by Garcia–Stichtenoth over quadratic finite fields in 1995. Shortly after followed the discovery of good towers over cubic finite fields in 2005 (Bezerra–Garcia-Stichtenoth) and all nonprime finite fields in 2013 (Bassa–Beelen–Garcia–Stichtenoth). The questions of finding good towers over prime fields resisted all attempts for several decades and lead to the common belief that such towers do not exist. In this talk I will try to give an overview of the landscape of explicit recursive towers and present a recently discovered tower over prime fields. This is joint work with Christophe Ritzenthaler.

Doğa Can Sertbaş, Cumhuriyet Üniversitesi

Title: Density results on Egyptian Fractions

Abstract: Any finite sum of distinct unit fractions is called an Egyptian fraction. For some fixed positive natural number k, a restricted Egyptian fraction is also defined as the sum of at most k unit fractions, where the repetition is allowed. These type of fractions have been studied extensively and there are some open problems related to them. In this talk, we introduce some of these problems and mention some arithmetic and analytic properties of restricted Egyptian fractions. In particular, we provide analytic proofs which show that restricted Egyptian fractions are not dense in the interval [0,1] in the sense of height and topology.

Şermin Çam Çelik, Özyeğin Üniversitesi

Title: Special Values of Dirichlet Series

Abstract: In this talk, we will show that real numbers can be strongly approximated by linear combinations of special values of Dirichlet series.

 

 

Introduction to Elliptic Curves

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Beste Akdoğan, Dokuz Eylül University

Date&Time: November 9, Friday,  10:15

Abstract: In this talk, we introduce the notion of elliptic curves. First, we present geometric and algebraic interpretations of elliptic curves. Later, we talk about the set of rational points on elliptic curves and mention Mordell’s theorem, which states that the set of rational points on an elliptic curve is a finitely generated abelian group. Finally,  we relate elliptic curves with L-series, and present some open questions about this subject.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B259.

Infinitude of Primes and Primes in Arithmetic Progressions

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Haydar Göral, Dokuz Eylül University

Date: 26/10/2018, Friday, Time: 10:15

Abstract: We first give two proofs of the infinitude of primes using topology and geometry. Then, we will see the connection between the Riemann zeta function and the reciprocal sum of prime numbers. Lastly, we will discuss how this idea leads us to Dirichlet’s theorem on primes in arithmetic progressions.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B259.

Around the Szemerédi Theorem 3

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Selçuk Demir, Dokuz Eylül University

Date: 19/10/2018, Friday, Time: 10:15

Abstract: This will be the third of a series of talks devoted to some topics around the Sezemerédi Theorem. We briefly recall what we did in the first two talks, and then we show the relation between the Szemerédi theorem and the Ergodic theory.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B259.

Around the Szemerédi theorem 2

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Selçuk Demir, Dokuz Eylül University

Date: 05/10/2018, Friday, Time: 10:15

Abstract: This will be the second of a series of talks devoted to some topics around the Sezemerédi Theorem. We briefly recall what we did in the first talk, and then we state the results that lead us to the Szemerédi theorem.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B259.