Ultrafilters and Some of Their Applications

Haydar Göral, İzmir Institute of Technology . Date: 26th of April, 2024, Friday, Time: 13.00 – 14.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Class B255

Abstract: In this talk, we first define ultrafilters on a non-empty set. Then, we see that there is a natural topology on the set of ultrafilters on the positive integers, and in fact this is the Stone-Cech compactification of the positive integers. Moreover, this compact space has an associative binary operation extending the addition of the positive integers. Finally, we will show how this topological semigroup is crucial for proving Ramsey theoretical results from combinatorics.

A Countable Prime Avoidance Theorem and Its Generalization to Prime Submodules

M. R. Pournaki, Sharif University of Technology. Date: 29th of March, 2024, Friday, Time: 13.00 – 14.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: One of the fundamental cornerstones of commutative ring theory is the prime avoidance theorem, which states that, if \frak{p_1},\ldots,\frak{p_n} are prime ideals of R and \frak{a} is an ideal of R such that \frak{a}\subseteq {\bigcup_{i=1}^n} \frak{p_i}, then \frak{a}\subseteq \frak{p_i} for some 1\leq i\leq n. In this talk, we give a proof for the countable version of this theorem due to Sharp and Vamos. The proof uses the celebrated Baire’s category theorem in metric spaces. We then discuss its generalization to the prime submodules.

A Glimpse to the Stanley Depth: A Geometric Invariant for Modules

M. R. Pournaki, Sharif University of Technology. Date: 22nd of March, 2024, Friday, Time: 13.00 – 14.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: Stanley depth is a geometric invariant of a module which has some common properties with the homological depth invariant. In this talk, we first review briefly the basic concepts of Stanley depth and then discuss some of the recent developments in the theory.

 

 

 

A Glimpse to Unit Graphs: The Graphs Arising From Rings

M. R. Pournaki, Sharif University of Technology. Date: 15th of March, 2024, Friday, Time: 13.00 – 14.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: The ring \mathbb{Z}_2\times\mathbb{Z}_2, having only one unit, cannot be generated by its units. It turns out, in the general theory of rings, that this is essentially the only example. In this talk, we give an elementary proof of “A finite commutative ring with nonzero identity is generated by its units if and only if it cannot have \mathbb{Z}_2\times\mathbb{Z}_2 as a quotient.” The proof uses graph theory and pointed out to the story of how unit graphs have been arisen. At the end, we discuss some of the recent developments in the theory.

When Three Subjects of Mathematics Meet Each Other

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.

AES Encryption Surrounds Us; We Surround AES Encryption

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

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

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

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

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.