Algebra and Lie research groups | Network of Ontario Lie Theorists | Seminars | Algebra seminar | Lie seminar |
Perspectives in representation theory |
MAT 2742 | MAT 1302 | Sessional Dates |
| Student | Year | Project |
| Stéphane Guérin | 2009 | Quivers and three-dimensional Lie algebras |
| Andrew Sirjoosingh | 2008 | An elementary proof of Gabriel's theorem |
| Student | Graduated | Thesis |
| Joel Lemay (M.Sc.) | 2011 | Valued graphs and the representation theory of Lie algebras |
| Caroline El-Chaâr (M.Sc.) | 2010 | The Onsager algebra |
I often have an opening for a postdoctoral fellow. Applicants should be recent Ph.D.s (or those expecting to receive their Ph.D. before the start of the appointment) working in a field closely related to my research areas. Those interested should submit an application to the department as well as contact me personally by email.
I am currently accepting graduate students. There are many intriguing problems in geometric, combinatorial, and categorical representation theory accessible at the master's and Ph.D. level (see below for potential projects). Students interested in working in this area should have completed courses in abstract algebra (uOttawa courses MAT3141 and MAT3143 or their equivalents). Descriptions of these uOttawa courses can be found here.
Each summer, I often have an opening for an undergraduate student as part of the NSERC USRA program. For students not already familiar with the theory of Lie algebras, the project would consist of independent reading in this area together with writing up solutions to one or more problems. For students with some background in the area, more advanced reading would be included in addition to the written work. Applicants should have a background in linear algebra (MAT 1341, 2141) and group theory (MAT 2143). Students interested in a project to take place in the summer should contact me in December or early January (the application deadline is typically February 1 each year).
Selected students receive a one-time $1,000 award and devote, during one academic term, at least 50 hours to a research project conducted by the faculty sponsor.
Lie groups and Lie algebras are indispensable tools in modern mathematics and mathematical physics. Lie groups are mathematical objects that have both geometric and algebraic structure. In particular, they are simultaneously differentiable manifolds and groups. They are a mathematically precise way of studying the continuous symmetry of mathematical and physical structures. For example, in quantum physics, particles correspond to representations of Lie groups. There are many potential projects in Lie theory for undergraduate and graduate students in both classical Lie theory as well as connections between Lie theory and other subjects (see below).
A quiver is a directed graph consisting of vertices and arrows, and so the study of quivers is naturally related to graph theory. However, there are many surprising connections to other areas of mathematics, including geometry (e.g. the McKay correspondence) and Lie theory. The starting point of the connection between quivers and Lie theory is Gabriel's Theorem, which states a beautiful relationship between "representations" of quivers and "root systems" of Lie algebras. This connection has flourished into an extensive theory including the subjects of Hall algebras and quiver varieties. Projects in this area can involve many different fields including graph theory, Lie theory, representation theory, and algebraic geometry.
In general terms, geometric representation theory involves reformulating often classical results in algebra in new geometric terms. For example, one realizes a representation of a group or Lie algebra (this is a vector space on which a group or Lie algebra acts – think of a group of rotations of a plane or 3-dimensional space) as the homology of some topological space. Such realizations can yield new insight because they allow one to use geometric tools in the study of representation theory as well as representation theoretic tools in the study of geometry and topology.
Category theory is a mathematical formalism that allows one to organize and study in an abstract way structure that is common to many different subjects. For instance, one can study properties of maps between objects and prove results that apply to maps between sets or groups. The advantage is that one does not need to reprove the results in each setting (in this case, for sets and for groups) because one has proven them in the more general setting of categories.
Recently, there has been considerable interest in using categories in a different way. This involves a process called categorification. Seeing as this field is relatively young, there are many good projects at the undergraduate and graduate level. These projects can involve connections to geometry, algebra, topology, and knot theory.