algebra

Biquandle colorings of knots and knotoids

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Neslihan Gügümcü, Izmir University of Economics.
Date: 24th of October, 2018, Thursday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: A (classical) knot is basically a loop in 3-dimensional space with a possible entanglement. Understanding the entanglement type of a given knot; to distinguish it from other knots or to see if its entanglement can be resolved and the knot can be turned into just a simple loop, lies as a central problem of knot theory. Knot invariants are tools used for solving this problem. Recently, Turaev introduced knotoids which are a natural extension of knots giving rise to generalizations of many knot invariants and also many new concepts.
In this talk I firstly introduce basic notions of classical knot and knotoid theory. Then I present some generalized algebraic structures such as quandles and biquandles. Finally I show how to color a knot/knotoid diagram by using a biquandle, and how to derive invariants for them by this coloring.
This is a joint work with Sam Nelson at Claremont McKenna College, USA.

PS Rings

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date:  26th of September, 2018, Wednesday
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In this seminar, we will talk about rings with projective socle. These rings are called PS rings. The following characterization of PS rings were given by W. K. Nicholson and J. F.Watters “R is a PS ring if and only if the right annihilator of every left maximal ideal is of the form fR where f is  an idempotent in R if and only if R has a faithful module with projective socle.”

Cotorsion Pairs in Categories of Quiver Representations and Maximal Equivalence between Categories of Complexes

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Sinem Odabaşı, Institute of Physics and Mathematics, Science Faculty, The Universidad Austral de Chile (UACh).
Date: 26th of July, 2018, Thursday. Time: 10:00 – 12:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: The talk is divided into two parts:
1) Completness of the induced cotorsion pairs in categories of representations of a quiver. (arXiv:1711.00559)
The concept of representations of a quiver can be traced back to [Gab72]. Now, it is known that the theory of representations of a quiver plays a crucial role in several branches of mathematics, such as Lie Algebras, quantum groups, etc. In this talk, for a given quiver Q and an abelian category C we will be interested in certain (relative) homological aspects of the category Rep(Q,C) of C-valued representations of Q. Namely, given a cotorsion pair (A,B) in an abelian category C, which satisfies certain mild conditions, there are several ways to lift that cotorsion pair to a cotorsion pair in Rep(Q,C), see [HJ16]. In [Question 7.7, HJ16], the authors proposed the following question: Is it true that if the cotorsion pair (A,B) is complete, then so are these induced cotorsion pairs in Rep(Q,C)? In this talk, we show that the answer to this question is affirmative under certain conditions. In addition, we provide a quiver that does not satisfy these conditions, but gives an affirmative answer to the aforementioned question.
2) On maximal equivalence of subcategories of chain complexes.
The notion of adjoint functors is a weaker version of equivalence of categories. So when one has an adjoint functor based on an object, it is natural to come up with the question of determining objects in such a way that its associated adjoint functor turns out to be an equivalence of categories just like it happens in the so-called Morita equivalence. Having this phenomenon, an (right) R-module P is said to be *-module if the representable functor Hom_R(P,-) induces a maximal equivalence between Mod-R and Mod-A, where A:=End_R(P), that is, an equivalence between the subcategory gen(P) of P-generated R-modules and the subcategory cogen(P^*) of P^*:= Hom_R(P,E)-cogenerated A-modules, where E is a cogenerator for Mod-R. Inspired from this, we introduce two notions of chain complexes, called *-complex and modified *-complex, each of which leads to a maximal equivalence between categories of chain complexes through the total hom complex and the modified hom-complex, respectively. In this talk, we will discuss their characterizations and their relations with (pre)silting complexes. This is a work in progress.
References:
[Gab72] Gabriel, P. (1972). Unzerlegbare Darstellungen. I, Manuscripta Math. 6, 71-103.
[HJ16] Holm, H. & Jorgensen, P. (2016). Cotorsion pairs in categories of quiver representations. Kyoto J. Math. to appear. arXiv:1604.01517v2.
[Oda17] Odabasi, S. (2017). Completeness of the induced cotorsion pairs in categories of quiver representations. Accepted in Journal of Pure and Applied Mathematics. arXiv:1711.00559v1.

Representation Theory of Artin Algebras

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date:  11th of July, 2018, Wednesday
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.

Representation Theory of Artin Algebras

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 30th of May, 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 projectivization technique of finitely generated modules over an artin algebra and we will give the proof of Krull Schmidt Theorem for finetely generated modules over an artin algebra.

Cryptology

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Gizem Derbentakar, Ege University.
Date: 23rd of May, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: We will first give some general information about cryptology. After presenting the basics, we will talk about Primality Tests. The talk is based on the book “A Course in Number Theory and Cryptography” by Neal Koblitz.
References
[1] Koblitz, N. A Course in Number Theory and Cryptography. Second edition. Springer, 1994.