Geometry and Topology

Speaker: Parker Lowrey (Western)

"Descent for derived categories "

Time: 15:30

Room: MC 107

Classically, in algebraic geometry, descent for quasi-coherent sheaves is valid for maps between schemes satisfying certain flatness and finiteness conditions. With the machinery developed in stable infinity categories, one can extend decent to the stable infinity category $QCoh(X)$ (whose homotopy category is the derived category of the scheme $X$) and to maps that are not necessarily flat. We will give some examples and discuss plausible conditions on maps to satisfy descent.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Real Submanifolds in a Complex Space I"

Time: 15:30

Room: MC 107

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

Colloquium

Speaker: Rick Jardine (Western)

"Path categories and calculations"

Time: 15:30

Room: MC 107

The path category P(X) of a space X is an invariant which is defined much like the fundamental groupoid, except that directions of paths are not formally reversed. This construction has applications in theoretical computer science, where it gives, in principle, a description of execution paths in geometric models for the behaviour of parallel processing systems. Path categories have resisted calculational analysis until just recently, in part because standard homotopy theoric methods are not applicable.

We now know that the path category P(K) of a finite simplicial complex K can be computed by an algorithmic method which is based on the existence of a finite 2-categorical resolution.

Methods of implementation and applications of this result will be discussed.

Algebra Seminar

Speaker: Andrey Minchenko (Western)

"Enhanced Dynkin diagrams"

Time: 14:30

Room: MC 107

The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. We introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow one to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R.