Analysis Seminar

Speaker: Hadi Seyedinejad (Western)

"Irreducibility in real algebraic geometry (Part I)"

Time: 15:30

Room: MC 108

The notion of irreducibility in the conventional Zariski topology is too coarse for real algebraic sets. For example, the hyperbola xy=1 is an irreducible algebraic set which is not even connected in the Euclidean topology. Cartan umbrella is another irreducible algebraic set, which is connected, but decomposes into the union of a 'sheet' and a 'stick.' Inspired by Nash and his notion of 'sheets,' one might require then to distinguish, as called by Kurdyka, the 'rigid pieces' of a real algebraic set. We will review different approaches to defining irreducibility and irreducible components in real algebraic geometry, in which more 'good' functions than just polynomials should be considered. We may work with the ring of Nash functions, continuous rational functions, or, most notably, arc-analytic functions. Nash functions are able to detect two components for the hyperbola xy=1. Continuous rational functions are able to detect a sheet and a stick component for Cartan umbrella. But we find examples in which only the phenomenon called 'arc-symmetricity' can among the others realize a decomposition.

Speaker's web page: http://www.math.uwo.ca/index.php/profile/59/

Noncommutative Geometry

Speaker: Bogdan Nica (McGill University)

"Hyperbolic groups and Noncommutative Geometry"

Time: 12:30

Room: MC 107

C*-algebras associated to groups provide some of the most interesting and most important examples in Noncommutative Geometry. In this respect, hyperbolic groups have, time and again, proved to be particularly exciting. I will discuss three vignettes illustrating this idea.

Homotopy Theory

Speaker: James Richardson (Western)

"Presentable infinity categories"

Time: 13:00

Room: MC 107

I will introduce presentable quasicategories and discuss some of their properties. I will then discuss their relationship with combinatorial model categories.

Colloquium

Speaker: Pinaki Mondal (School of Mathematics, Physics and Technology at The College of the Bahamas)

"Milnor number, intersection multiplicity and number of zeroes of systems of polynomials"

Time: 15:30

Room: MC 107

We talk about two of the original problems that shaped the theory of Newton polyhedra: the problem of computing the Milnor number of the singularity at the origin of a generic polynomial, and computing the number of zeroes of generic polynomials. The former was addressed by Kushnirenko, who gave a beautiful formula in terms of Newton diagrams in a special case.Bernstein (following work of Kushnirenko) solved completely the latter problem for the case of (C^*)^n. In the case of C^n the problem was partially solved following the work of Khovanskii, Huber-Sturmfels, and many others. We give complete solution to both these problems. A common theme in our solution to both problems is the computation of intersection multiplicity at the origin of the hypersurfaces determined by n generic polynomials.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Random Matrix Theory (IV)"

Time: 12:30

Room: MC 106