(in Polish) Algebry operatorów dające się widzieć II

General data

Course ID:	1000-1M20AOW2
Erasmus code / ISCED:	11.1 The subject classification code consists of three to five digits, where the first three represent the classification of the discipline according to the Discipline code list applicable to the Socrates/Erasmus program, the fourth (usually 0) - possible further specification of discipline information, the fifth - the degree of subject determined based on the year of study for which the subject is intended. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	(unknown)
Name in Polish:	Algebry operatorów dające się widzieć II
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	(unknown)
Type of course:	elective monographs
Short description:	Graph algebras: path algebras, Leavitt path algebras, graph C-algebras, from pushouts of graphs to pullbacks of graph C-algebras.
Full description:	Graph C-algebras proved to be tremendously successful in studying the K-theory of operator algebras. They are currently at the research frontier of noncommutative topology enjoying a substantial ongoing research output. The goal of this lecture course is to explain the fundamentals of path algebras and Leavitt path algebras so as to build from scratch and in a systematic way the knowledge of graph C-algebras. The course begins with the introduction of the path algebra of a directed graph (quiver), which is defined as the linear span of all finite paths in the graph with the multiplication given by the composition of paths. Thus the number of finite paths in a graph is the dimension of its path algebra. Next, a key step is introduce the Cuntz-Krieger relations in the path algebra of the ghost extension of a graph - they define the Leavitt path algebra of the graph as the quotient of the path algebra of the extended graph by the ideal generated by the Cuntz-Krieger relations. Taking the ground field of the Leavitt path algebra of a graph to be the field of complex numbers, and defining an involution in terms of the extended graph, we obtain a complex -algebra. Now, we can define graph C-algebras as the universal enveloping C-algebras of Leavitt path algebras. Here key results to be explained concern representations on a Hilbert space and the ideal structure of graph C-algebras. The course culminates with applications in noncommutative topology. First, we prove that, by equipping graphs with Leavitt morphisms, the assignment of graph C-algebras to graphs becomes a contravariant functor into the category of C-algebras and -homomorphisms. Then we show when this contravariant functor turns pushouts of graphs into pullbacks of graph C-algebras. All this is exemplified by plethora of natural examples rooted in classical topology.
Bibliography:	1. Graph Algebras, Piotr M. Hajac, Mariusz Tobolski, arxiv 1912.05136. 2. Leavitt Path Algebras, Gene Abrams, Pere Ara, Mercedes Siles Molina. 3. Algebras and Representation Theory, Karin Erdmann, Thorsten Holm. 4. C*-algebras and Operator Theory, Gerard J. Murphy.
Learning outcomes:	Acquiring a working knowledge of graph C*-algebras allowing one to start research in this area of mathematics. Depending on the level of involvement, this course might lead either to a Master thesis or a PhD dissertation.
Assessment methods and assessment criteria:	regular attendance or an oral exam

This course is not currently offered.

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