Geometry and Topology

Speaker: Priyavrat Deshpande (Western)

"Arrangements of Submanifolds"

Time: 15:30

Room: MC 108

A real arrangement of hyperplanes is a collection of finitely many hyperplanes in a real vector space. It is known that the combinatorics of the intersections of these hyperplanes contains substantial information about the topology of the complement of the hyperplanes in the real as well as complexified space. For example, the cohomology of the complexified complement can be expressed in terms of the intersection lattice associated with the arrangement. The face poset of an arrangement defines a simplicial complex (the Salvetti complex) which has the homotopy type of this complement.

Noncommutative Geometry

Speaker: Ali Motadelro (Western)

"Metric aspects of noncommutative geometry II"

Time: 14:00

Room: MC 106

Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

Pizza Seminar

Speaker: Farzad Fathizadeh (Western)

"What does the spectral theorem say?"

Time: 17:00

Room: MC 107

The Spectral Theorem, and the closely related Spectral Multiplicity Theory is a gem of modern mathematics. It is about the structure, and complete classification, up to unitary equivalence, of normal operators on a Hilbert space. This theorem is the generalization of the theorem in linear algebra which says that every normal, in particular selfadjoint, matrix is unitarily equivalent to a diagonal matrix; or, in simple terms, is diagonalizable in an orthonormal basis. The extension of this result to infinite dimensions is by no means obvious and involves many new subtle phenomena that have no analogue in finite dimensions. The final result has many applications to pure and applied mathematics, mathematical physics, and quantum mechanics. In this talk, a proof of the spectral theorem for Hermitian operators on a Hilbert space will be outlined and some applications will be discussed. This talk should be accessible to undergraduate students.

Noncommutative Geometry

Speaker: Mehdi Mousavi (Western)

"Equivariant de Rham cohomology II"

Time: 14:00

Room: MC 106

Stacks Seminar

Speaker: Jeffrey Morton (Western)

"Topological and Smooth Groupoids"

Time: 13:30

Room: MC 107

In this talk, I will give some basic definitions and facts about topological and Lie groupoids. I will describe some examples involving group actions and configuration spaces of geometric structures. I will outline some work analogs of standard constructions in differential geometry, such as differential forms, in this context. I will also describe how these same examples can be described in terms of other kinds of structures: stacks (specifically, topological and differentiable stacks), and $C*$-algebras associated to the groupoids. This talk will be a fairly introductory presentation of the subjects of this seminar.

Colloquium

Speaker: Edward Bierstone (Toronto)

"Resolution except for minimal singularities"

Time: 15:30

Room: MC 108

The subject of the talk is resolution of singularities in algebraic or analytic geometry. Resolution of singularities leads to a space with only normal crossings singularities (i.e., transverse self-intersections). It therefore makes sense to consider normal crossings singularities acceptable from the start, and try to resolve singularities except for normal crossings. We will discuss the following question (a variant of a problem of Janos Kollar). Can we find the smallest class of singularities S with the following properties: (1) S includes all normal-crossings singularities; (2) every variety or space X admits a proper mapping f: X' --> X such that X' has only singularities in S, and f is an isomorphism over the locus of points of X having only singularities in S? No technical background will be assumed.

Stable Homotopy

Speaker: Enxin Wu (Western)

"Self-maps and Periodicity for Modules over the Steenrod Algebra"

Time: 10:30

Room: MC 108

Algebra Seminar

Speaker: Eric Schost (Western)

"Computing roadmaps"

Time: 14:30

Room: MC 108

Consider the following questions (coming for instance from motion planning problems): given two points on a real algebraic set $S$, do they belong to the same connected component? If so, how can we connect them?

Canny introduced "roadmaps" as a way to reduce such problems to computations with curves only. Given $s$ polynomial equations with rational coefficients, of degree $d$ in $n$ variables, Canny's algorithm, and its generalizations by Basu, Pollack and Roy, have a cost polynomial in $(s D^n)^n$.

This is depressingly high; as a result, none of these algorithms is practical for realistic instances. Indeed, one would rather expect a cost polynomial in $s D^n$. I will present ongoing work with Mohab Safey El Din toward this goal.