algebra – Page 2 – Dokuz Eylül University Faculty of Science Department of Mathematics

When Three Subjects of Mathematics Meet Each Other

March 20 2024March 2 2024 by Zübeyir Türkoğlu

M. R. Pournaki, Sharif University of Technology. Date: 8th of March, 2024, Friday, Time: 13.00 – 14.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Class B255.

Abstract: Fermat’s little theorem states that if $p$ is a prime number, then $a^p \equiv a$ (mod $p$ ) holds true for any integer $a$ . One may ask what happens when $p$ is not a prime. The answer to this question seems little known to mathematicians, even to number theorists (as Dickson said in his “History of the Theory of Numbers”). In this talk, we discuss the missing result which is essentially due to Gauss and its generalizations.

Quantitative unique continuation or “If we don’t know everything, how much do we actually know”?

May 21 2024February 26 2024 by M. Pınar Eroğlu

Matthias Täufer, Analysis group, FernUniversität in Hagen, Germany Date: 1th March 2024, Friday Time: 13:00 Place: DEU, Faculty of Science, Department of Mathematics, Room B255

Abstract: Unique continuation is a basic property of many partial differential equations stating that solutions vanishing on subsets must be identically zero. In many cases one would like to have a quantitative version of that, meaning that one can bound the norm of solutions by their norm on subsets. In this talk, we review some history of quantitative unique continuation and present several results on quantitative unique continuation in unbounded domains. Based on joint works with Ivica Nakic (Zagreb), Martin Tautenhahn (Leipzig), Sedef Özcan (Dokuz Eylül), Paul Pfeiffer (Hagen), Albrecht Seelmann (Dortmund) and Ivan Veselic (Dortmund).

AES Encryption Surrounds Us; We Surround AES Encryption

January 4 2024January 4 2024 by Zübeyir Türkoğlu

Orhun Kara, İzmir Institute of Technology . Date: 31th of January, 2024, Wednesday, Time: 10.30 – 12.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract:

The NIST Advanced Encryption Standard, AES, is without doubt the most used encryption algorithm all over the world. AES supplies confidentiality in almost all the ubiquitous cryptographic protocols including Whatsapp security, TLS, WPA. In this talk, we introduce algebraic aspects and the design philosophy of AES. Moreover, we try to convince the audience that AES appears sufficiently secure to protect our data by presenting supporting security evidence and introducing some unsuccessful attack attempts.

Binary Cyclic Group Codes

December 26 2023 by Zübeyir Türkoğlu

Mustafa Kemal Turak, Dokuz Eylül University. Date: 4th of January, 2024, Thursday, Time: 11.00 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, we will first give the definition of cyclic codes. We will then show that each cyclic code can be seen as an ideal in the quotient ring $F[x]/(x^{n}-1)$ . Next, we will introduce group codes and examine the relationship between cyclic group codes and cyclic codes.

Torsion Free Covers of Line Quivers

December 5 2023December 4 2023 by Zübeyir Türkoğlu

Canan Özeren, Dokuz Eylül University. Date: 6th of December, 2023, Wednesday, Time: 13.30 – 14.30. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Torsion-free covers exist for abelian groups (see [1]). The coGalois group of automorphisms a torsion-free cover $\phi: T \rightarrow A$ of an abelian group $A$ is defined in [2] as the group of $f: T \rightarrow T$ s.t. $\phi f= \phi$ and is denoted by $G(\phi)$ . The abelian groups for which the coGalois group is trivial were characterized in [3]. The notion of coGalois group can be defined in any category where we have a covering class. In [4], coGalois groups have been studied in the category of representations of the quiver $q_2 : \cdot \rightarrow \cdot$ . We talk about the existence and uniqueness of the torsion free-cover of an object in $(q_n, Z-mod)$ (see [5]).

References:

[1] E. Enochs: Torsion-free covering modules. (1963).

[2] E. Enochs, J. R. Rozas and L. Oyonarte: Compact coGalois groups. (2000).

[3] E. Enochs and J. Rada: Abelian groups which have trivial absolute \\ coGalois group. (2005).

[4] Paul Hill, Abelian group pairs having a trivial coGalois Group. (2006).

[5] Molly Dunkum Wesley, Phd Thesis at The Graduate School University of Kentucky. (2005)

Bialgebroids and Dual Calculus

November 8 2023 by Zübeyir Türkoğlu

Keremcan Doğan , İstanbul Technical University Postdoctoral Researcher.

Date: 10th of November, 2023, Friday.

Time: 12.30 – 13.30.

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

Abstract: In this talk, we will first explain why algebroids constitute a good framework for generalizations of geometric structures suitable for string and $M$ theories. After a quick overview of their fundamental properties, we will focus on bialgebroids and algebroid calculus. Then, we will be interested in the extensions of Drinfel’d doubles using these notions and certain compatibility conditions between them. We will finish the discussion with the relation between our constructions and exceptional geometries required for string theory. If time permits, we will slightly touch upon the global picture about the formal bundle rackoids.

Tensor Product II

October 30 2023October 30 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 1st of November, 2023, Wednesday, Time: 13:30 – 14:30. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Let $R$ be a ring with identity and let $M$ and $N$ be $R$ -modules ( $M$ a right $R$ -module and $N$ a left R-module). We denote the tensor product with $M\otimes_R N$ . This week, we will continue to talk about various properties of tensor product of M and N. References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.
Bland, P. (2011). Rings and Their Modules. Berlin/New York: de Gruyter.

Tensor Product

October 20 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 25th of October, 2023, Wednesday, Time: 13:30 – 14:30. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Let $R$ be a ring with identity and let $M$ and $N$ be $R$ -modules ( $M$ a right $R$ -module and $N$ a left R-module). In this talk, we will define tensor product of M and N. We will also discuss various properties of tensor product of M and N. We will denote the tensor product with $M\otimes_R N$ .

References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.
Bland, P. (2011). Rings and Their Modules. Berlin/New York: de Gruyter.

Local Rings: Krull-Remak-Schmidt Theorem II

June 5 2023May 29 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 7th of June, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Every injective module over a noetherian ring is a direct sum of directly indecomposable submodules. The question arises as to whether and in what sense such a decomposition is uniquely determined. This question is answered by the Krull-Remak-Schmidt Theorem. The proof of the Krull-Remak-Schmidt Theorem assumes that the endomorphism rings of the direct summands are local rings. Hence we have, first of all, to introduce local rings and then to state sufficient conditions in order that the endomorphism ring of a directly indecomposable module is local. References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.

Local Rings: Krull-Remak-Schmidt Theorem

May 18 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 24th of May, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings). Abstract: Every injective module over a noetherian ring is a direct sum of directly indecomposable submodules. The question arises as to whether and in what sense such a decomposition is uniquely determined. This question is answered by the Krull-Remak-Schmidt Theorem. The proof of the Krull-Remak-Schmidt Theorem assumes that the endomorphism rings of the direct summands are local rings. Hence we have, first of all, to introduce local rings and then to state sufficient conditions in order that the endomorphism ring of a directly indecomposable module is local. References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.