algebra

Representation Theory of Artin Algebras

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 25th of April, 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,  firstly we will continue talk about the radicals of rings and modules. Then we will talk about the structure of projective modules over left Artinian rings.

Representation Theory of Artin Algebras

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Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 11th of April, 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 present parts of the book “Representation Theory of Artin Algebras” by Auslander, Reiten and Smalo. In the first session, we will talk about  finite length modules, Jordan Holder Theorem and the notions of right-left minimal morphisms.

Model Structures on Categories of Chain Complexes

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Mehmet Akif Erdal, Bilkent University.
Date: 28th of March, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: This is the second talk in the homotopy theory series. We will go through the definitions of a model category and compare elements of these definitions with notions of classical homotopy theory. Then, we will discuss standard (Quillen) model structures on the category of chain complexes of modules over a ring.

Basic Elements of Abstract Homotopy Theory

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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.

The Fundamental Group and Some of Its Applications, III

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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.

Small Covers over Product of Simplices

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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.