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.

# Month: February 2018

## The Fundamental Group and Some of Its Applications, II

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.

## Invariant Basis Property and the Ideal Structure of Leavitt Path Algebras

Müge Kanuni, Düzce University.

Date: 14th 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 ring *R* is said to have Invariant Basis Number property, or more simply IBN, in case no two free left *R*-modules of different rank are isomorphic. W. G. Leavitt constructed some non-IBN algebras — what we now call Leavitt algebras — in the 1960’s. The Leavitt path algebra of a quiver with one vertex and *m* loops turns out to be Leavitt algebra *R* of type (1,*m*), that is a non-IBN algebra where *R* is isomorphic to *m*-copies of *R* as a left module and not isomorphic to *n*-copies of *R* for any 1 < *n* < *m*. On the other hand, there is an abundance of examples of Leavitt path algebras which have IBN. Moreover, we give an algorithm to decide whether a Leavitt path algebra has IBN or not. If time permits, we discuss the ideal structure of Leavitt path algebras.

## The Fundamental Group and Some of Its Applications

Aslı Güçlükan İlhan, Dokuz Eylül University.

Date: 7th 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 the first talk, after a quick

discussion of what algebraic topology is, we will give the definition and

some properties of the fundamental group. We will also discuss the picture

hanging problem as a motivation.