algebra

AES Encryption Surrounds Us; We Surround AES Encryption

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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

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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

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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

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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

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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

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

Tensor Product

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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

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

Local Rings: Krull-Remak-Schmidt Theorem II

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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

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

Local Rings: Krull-Remak-Schmidt Theorem

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

The Torsion Free Covers II

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Canan Özeren, Dokuz Eylül University. Date: 10th 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:  We will continue to talk about the existence and uniqueness (up to isomorphism) torsion-free covers of modules over an integral domain (see [1]). We show that the classical definition of torsion-free cover and the definition of an F-cover, where F is the class of the torsion-free modules, coincide.

References

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

The Torsion Free Covers

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Canan Özeren, Dokuz Eylül University. Date: 03rd 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:  We talk about the existence and uniqueness (up to isomorphism) torsion-free covers of modules over an integral domain (see [1]). We show that the classical definition of torsion-free cover and the definition of an F-cover, where F is the class of the torsion-free modules, coincide.

References

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