Engin Mermut

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.

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.

Cihan Sahillioğulları and Sedef Taşkın, Dokuz Eylül University.
Date: 10th and 17th of January, 2018, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract for the 1st talk by Cihan Sahillioğulları:
In this talk, we introduce the classification of irreducible tame representations of infinite symmetric group. First of all, we present the group we work on. Later, we talk about Olshanski’s semigroups and spherical representations of infinite bisymmetric group.
Abstract for the 2nd talk by Sedef Taşkın:
In this talk, we present an introduction to the classification of all irreducible unitary representations of the infinite symmetric group. First the infinite symmetric group will be introduced. Then we mention representations of semigroups with involution. After this, we use representations of semigorups to classify all irreducible unitary representations of the infinite symmetric group.

Meltem Güllüsaç, Dokuz Eylül University.
Date: 20th of December, 2017, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: After a review of prime modules, prime ideals and indecomposable injectives, we will prove Matlis’s Theorem. See Chapter 3 of the monograph [1].
References
[1] Lam, T. Y. Lectures on Modules and Rings. Springer, 1999.

Meltem Güllüsaç, Dokuz Eylül University.
Date: 22nd and 29th of November, 6th and 13th of December, 2017, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: We shall make an introduction to mininjective rings. See the second chapter of the book by Nicholson and Yousif.
References
[1] Nicholson, W.K. and Yousif, M.F. Quasi-Frobenius Rings. Cambridge University Press, 2003.

Noyan Er, Dokuz Eylül University.
Date: 25th of October, 1st, 8th and 15th of November, 2017, Wednesday. Time: 09:30-10:30
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: This is the first of a series of talks on QF-rings and related questions.

Sinem Benli, İzmir Institute of Technology.
Date: 18th of October, 2017, Wednesday. Time: 09:30-10:30
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In this talk, we shall give a brief summary about the motivating ideas for commutative almost perfect domains. After that, we can mention the generalization of this class of rings to noncommutative setting considered by Facchini and Parolin.
References:
[1] Facchini, A. and Parolin, C. Rings Whose Proper Factors are Right Perfect. Colloquium Mathematicae, 122, 191-202, 2011.
[2] Benli, S. Almost Perfect Rings. M.Sc. Thesis, Dokuz Eylül University, The Graduate School of Natural and Applied Sciences. İzmir/TURKEY, 2015.

Salahattin Özdemir, Dokuz Eylül University.
Date: 27th of September, 4th and 11th of October, 2017, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: We will prove the well-known result by Benson and Goodearl about flat modules over an arbitrary ring R: If a flat R-module M sits in a short exact sequence 0 → M → P → M → 0 with P projective, then M is projective. In other words, every flat periodic R-module M (of period 1) is projective. We will then talk about some of the recent generalizations of this result.

Sinem Benli, İzmir Institute of Technology.
Date: 20th of September, 2017, Wednesday. Time: 09:30-12:00
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: Firstly, we introduce the notion of primitive rings. After giving some examples and mentioning the properties of this class of rings, we shall prove the famous Jacobson Density Theorem which gives the structure of the primitive rings. Finally, we give a different proof of the fundamental Wedderburn’s structure theorem that characterizes the finite dimensional simple algebras.
References:
[1] Matej Brešar, Introduction to Noncommutative Algebra, Springer, 2014.
[2] Benson Farb & R. Keith Dennis, Noncommutative Algebra, Springer, 1991.

Meltem Güllüsaç and Hikmet Burak Özcan, Dokuz Eylül University.
Date: 9th, 16th and 23rd of August and 13th of September, 2017, Wednesday, Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: After a review of tensor product of vector spaces and tensor product of algebras over a field, we shall prove two classical results for finite-dimensional central simple algebras over a field: the Skolem-Noether Theorem and the Double Centralizer Theorem. See Chapter 4 of the book [1] by Matej Brešar. All the algebras we mention are algebras over a field. Multiplication algebra of an algebra is a useful tool, see for example the article [2] by Brešar with some results for zero product determined algebras. The fact that every linear operator on a finite dimensional central simple algebra A belongs to the multiplication algebra of A is used to give the following particular case of the Skolem-Noether theorem: every automorphism of a finite dimensional central simple algebra A is inner. Indeed a somewhat more general result is proved in [2] for the linear operators f and g on A that satisfy f(x)g(y)=0 whenever xy=0. In the recent preprint [3] by Brešar et. al., they investigate the unital algebras S such that for every finite-dimensional central simple algebra R, every homomorphism from R to the tensor product of the algebras R and S extends to an inner automorphism of this tensor product algebra. They call such algebras Skolem-Noether algebras. By the classical Skolem-Noether Theorem, every finite-dimensional central simple algebra is a Skolem-Noether algebra and in this work, they show that  various classical and important families of algebras like semilocal (in particular artinian and finite-dimensional) algebras, unique factorization domains, free algebras, etc. are Skolem-Noether algebras.
References
[1] Matej Brešar, Introduction to Noncommutative Algebras, Springer, 2014.
[2] Matej Brešar, Multiplication algebra and maps determined by zero products, Linear and Multilinear Algebra, 60:7, 763-768, 2012.
[3] Matej Brešar, Christoph Hanselka , Igor Klep and Jurij Volčič, Skolem-Noether Algebras, preprint, arXiv.org > math > arXiv:1706.08976, 2017.