Analysis Seminar

Speaker: Yael Karshon (University of Toronto)

"Diffeological, Frolicher, and differential spaces"

Time: 15:30

Room: MC 108

Differential calculus on Euclidean spaces can be generalized in different ways. A diffeological structure on an (arbitrary) set is given by maps from open subsets of Euclidean spaces to the set (satisfying some axioms); a differential structure is given by maps from the set to R; a Frolicher structure is given by maps from R to the set as well as maps from the set to R. We describe relations between these structures and present examples, including good subsets, bad quotients, and infinite dimensional spaces of smooth maps.

Speaker's web page: http://www.math.toronto.edu/karshon/

Dept Oral Exam

Speaker: Nadia Alluhaibi (Western)

"On Vector-Valued Automorphic Forms On Bounded Symmetric Domains "

Time: 14:00

Room: MC 107

The main points of this talk are as follows:

- constructions of vector-valued automorphic forms on bounded symmetric domains via Poincare series

- vector-valued automorphic forms associated to submanifolds of the complex unit ball - studying the behavior of asymptotics of the inner product of two Poincare series associated to submanifolds of the complex unit ball, for large weights, with examples.

Colloquium

Speaker: Blake Madill (Waterloo)

"On some techniques in modern radical theory"

Time: 15:30

Room: MC 107

The notion of a radical of a ring was first suggested by Wedderburn in 1908 and first used by Koethe in 1930. We say a class of rings C is a radical class of rings if it is homomorphically closed, the sum of all ideals in C of a ring R, denoted by C(R), is again in C, and C(R/C(R))=(0). We call C(R) the radical of R associated to the class C. The idea of radical theory is to collect ring theoretical information in C(R) and be able to then give more information about R. In this talk we will discuss some of the classical results in radical theory and describe some of modern techniques being used in current research. In particular, techniques from graded-ring theory and theoretical computer science will be discussed. This talk will be accessible to a general mathematical audience

Algebra Seminar

Speaker: Hadi Seyedinejad (Western)

"Effective extensions of the ring of polynomials in real algebraic geometry"

Time: 14:30

Room: MC 107

Some key tools, such as the Nullstellensatz, of course fail in algebraic geometry over the reals. However, having less of such machinery that works universally (as in the case over an algebraically closed field) does not make real algebraic geometry any less fascinating. One needs instead to come up with the right tools that are tailored to a phenomenon of interest. We are interested in 'irreducible components' of real algebraic sets, particularly as they relate to the sheets of Nash on such sets. As it turns out, we need to extend the classical ring of polynomials and obtain a wider class of functions to work with. We compare different approaches and present our point of focus, namely the ring of arc-analytic functions, in which even the Nullstellensatz is proven to hold.