Representation Theory - Summer SchoolGulbenkian Foundation, LISBON — 15 to 19 JULY 2013
sketch for the panel Começar by José de Almada Negreiros - image reproduced with kind permission of the Gulbenkian Foundation ProgramThis school comprises three 5-lecture courses (in English) aimed at 1st- or 2nd-year undergraduate students of mathematics, complemented by problem sessions.CoursesRepresentations of finite groupsby Eric SOMMERS (University of Massachusetts)Abstract: Representation theory is the study of how algebraic objects act on vector spaces. In this course the algebraic object is a finite group and a representation means a coherent way of associating a matrix with each element of the group so that the collection of matrices reflects the structure of the group. The subject began with the work of Frobenius in the 1890's. This course will provide an introduction to finite group representations, with a focus on the properties of the character table of a group. The pre-requisites are linear algebra and some exposure to group theory. Reference: Chapter 2 of Michael Artin's Algebra textbook provides a concise survey of group theory and chapters 1,3 and 4 cover most of the needed linear algebra. Representations of quiversby Pavel ETINGOF (MIT)Abstract: A quiver Q is just a set of dots (or vertices) connected by arrows, i.e., an oriented graph. A dimension vector d for Q is an assignment of a non-negative integer d(i) to every vertex i of Q. A representation of Q of dimension d is a collection of matrices A(i,j,b) with d(j) rows and d(i) columns defined for each arrow b going from i to j (with entries in some field k, for example, real or complex numbers). An equivalence of representations transforms A(i,j,b) to S(j)-1A(i,j,b)S(i) where S(i) are some invertible square matrices of size d(i). Q is said to be of finite type if for every d, it has a finite number of equivalence classes of representations, which is independent of the field k. 40 years ago, P. Gabriel showed that connected quivers of finite type are exactly the simply laced Dynkin diagrams (for any orientation of edges). Dynkin diagrams is a remarkable class of graphs related to many different mathematical objects, such as Platonic solids, reflection groups, Lie algebras and Lie groups, simple singularities, etc. We will prove Gabriel's remarkable theorem and explore some of these connections. This course will assume a strong background in linear algebra and some basic familiarity with abstract algebra. The free online book Linear Algebra as an introduction to Abstract Mathematics by I. Lankham, B. Nachtergaele and A. Schilling covers all the necessary Linear Algebra background. Continuous symmetry and Lie algebrasby Peter TRAPA (University of Utah)Abstract: The study of continuous symmetry (as opposed to discrete symmetry) was undertaken by Sophus Lie in the latter part of the 19th centuty. Remarkably, such symmetries are controlled to a large extent by "infinitesimal" algebraic structures nowadays called Lie algebras. This course will be devoted to the fundamental properties of Lie algebras and their connections to symmetry. The pre-requisites for this course are the same as for the other two courses. ScheduleThe opening of the school will be in Auditório 3 at 9.30 on Monday with the presence of
Friday, July 19th, there will be a dinner followed by a gathering where participants will be able to talk to the lecturers about their careers in a relaxed setting. The dinner will take place at 8pm at Hotel Açores Lisboa, a five minute walk from the Foundations' headquarters. Mini-conferenceOn Saturday, July 20th, there will be student mini-conference at the Gulbenkian Foundation which will include a talk by Prof. Etingof.All school participants are invited to attend this mini-conference. Schedule of the mini-conference of July 20th 2013 AssistantsPedro Vieira () and Ricardo Campos ()VenueThe school takes place in the headquarters of the Gulbenkian Foundation (Avenida de Berna, Lisbon), with lectures and problem sessions in Sala 1 (note the change), with exception of the opening which will take place in Auditório 3. |
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