**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.

# Month: November 2018

## Representation Theory of Artin Algebras

**Zübeyir Türkoğlu**, Dokuz Eylül University.

**Date:** 28th of November, 2018, Wednesday, **Time:** 09:30 – 12:00.

**Place:** Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

**Abstract:** In this seminar, we will talk about the transpose Tr. In general, Tr does not induce a functor from the mod(Λ) to the mod(Λ^{op}). But we do get a duality by replacing mod(Λ) with an appropriate category.

## Representation Theory of Artin Algebras

Meltem GÜLLÜSAÇ, Dokuz Eylül University.

**Date:** 14th of November, 2018, Wednesday, **Time:** 09:30 – 12:00.

**Place:** Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

**Abstract:** In this seminar, we will talk about a duality between finitely generated Λ-modules and finitely generated Λ^{op}-modules over an artin R-algebra Λ and we will prove the existence of enough injectives for finitely generated Λ-modules.

## Congruent Number Problem

**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

**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

**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**.

## Representation Theory of Artin Algebras

**Zübeyir Türkoğlu**, Dokuz Eylül University.

**Date:** 7th of November, 2018, Wednesday, **Time:** 09:30 – 12:00.

**Place:** Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

**Abstract:** In this seminar, we will talk about injective envelopes of finitely generated Λ-modules.