Engin Mermut

Noyan Er, Dokuz Eylül University.
Date: 27th of February, 2019, Wednesday. Time: 09:30-10:30
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: This will the the beginning of a series of sporadic talks related to some classical methods bridging Ring Theory and Representation Theory, which will feature some well known results due to Eisenbud and Griffith.

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.

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.

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.

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.

Mehmet Akif Erdal, Bilkent University.
Date: 21st of March, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: Abstract homotopy theory, also known as homotopical algebra, provides a framework that allows us to use tools from homotopy theory in various different settings, ranging from representation theory to mathematical physics. For short, it is the study of everything that is related with higher categories. This talk will be a motivational introduction to the subject. We will discuss some of the main purposes of abstract homotopy theory and state some basic definitions and examples. In particular, we will talk about definitions of categories with weak equivalences, homotopical categories and their homotopy category and we will discuss existence of the homotopy category. Later, if the time permits, we state some of the difficulties in the construction of homotopy categories and discuss why we need extra structures, such as model categories or (co)fibration categories.

Aslı Güçlükan İlhan, Dokuz Eylül University.
Date: 7th and 14th of March, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In this series of talks, we will introduce the fundamental group
and discuss some of its applications including the proof of the
fundamental theorem of algebra. In this talk, we will prove that every
group can be realized as a fundamental group. We will also discuss
the theory of covering spaces for graphs. As a consequence, we show that
every subgroup of a free group is free. If time permutes, we give a quick
introduction to basic notions of homotopy theory such as cofibrations,
fibrations and weak equivalences.

Aslı Güçlükan İlhan, Dokuz Eylül University.
Date: 28th of February, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: A small cover is a smooth closed manifold which admits a
locally standard (Z/2)^n-action whose orbit space is a simple
convex polytope. The notion of a small cover is introduced by Davis and
Januszkiewicz as a generalization of real toric manifolds. In this talk,
we first discuss the small covers over cubes whose complete
characterization is given by Choi-Masuda-Oum. Using this classification,
they also show that small covers over cubes satisfy the cohomological
rigidity problem. We also discuss the recent results obtained by
Altunbulak-Güçlükan İlhan about the number of small covers over product of
simplices.

Aslı Güçlükan İlhan, Dokuz Eylül University.
Date: 21st of February, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In this series of talks, we will introduce the fundamental group
and discuss some of its applications including the proof of the
fundamental theorem of algebra. In this talk, we will calculate the
fundamental group of a circle and give some of its applications. We also
discuss the van Kampen theorem which allows us to compute the fundamental
group of a space from the simpler ones. Then we prove that every group
can be realized as a fundamental group.