Year: 2018

Genus of complex projective algebraic curves

by

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.

Annihilators of Cartier Quotients

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Mehmet Yeşil, The University of Sheffield.
Date: 12th of December, 2018, Thursday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: Let R be a commutative Noetherian ring of prime characteristic p, M be an R-module and e be a positive integer. Let f:R→R be the Frobenius homomorphism given by f(r)=r^p for all r in R whose e-th iteration is denoted by f^{e}. An e-th Cartier map on M is an additive map C:M→M such that rC(m)=C(r^{p^{e}}m) for all r in R and m in M. An R-module is called a Cartier module if it is equipped with a Cartier map. In the case that the Frobenius homomorphism is finite and M is a finitely generated R-module equipped with a surjective Cartier map, it is proved by M. Blickle and G. Böckle in [1] that the set of annihilators of Cartier quotients of M is a finite set of radical ideals consisting of intersections of the finitely many primes in it. In these talks, I will consider the case that R is a finite dimensional polynomial ring over a field of prime characteristic p, and I take a computational view of this finiteness result and drop the finiteness condition on the Frobenius homomorphism to give an alternative proof to the result.
References:
[1] M. Blickle and G. Böckle. Cartier modules: finiteness results.

Representation Theory of Artin Algebras

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 5th of December, 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 continue to 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.

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.

Representation Theory of Artin Algebras

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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

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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

by

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.