Geometry and Topology

Speaker: William Slofstra (Waterloo)

"Schubert varieties and inversion hyperplane arrangements"

Time: 15:30

Room: MC 107

Freeness is an interesting algebraic property of complex hyperplane arrangements. The standard examples of free arrangements are the Coxeter arrangements, which consist of the hyperplanes normal to the elements of a finite root system. It is a natural (open) question to determine when a subarrangement of a Coxeter arrangement is free. Surprisingly, for the inversion subarrangements this question seems to be closely connected to the combinatorics of Coxeter groups and Schubert varieties. I will talk about two aspects of this connection: (1) the equality between the exponents of a rationally smooth Schubert variety and the exponents of the corresponding inversion arrangement, and (2) a criterion for freeness of inversion arrangements using root-system pattern avoidance.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

We continue with: ---The index problem and characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch. Baris Ugurcan (UWO) will speak in the second part.

Colloquium

Speaker: Lance Littlejohn (Baylor University)

"Glazman-Krein-Naimark theory, left-definite theory and the square of the Legendre polynomials differential operator"

Time: 15:30

Room: MC 107

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator $A$ in $L^2(-1,1)$ having the Legendre polynomials $\{P_{n}\}_{n=0}^{\infty}$ as eigenfunctions. As a particular consequence, they explicitly determine the domain $\mathcal{D}(A^2)$ of the self-adjoint operator $A^{2}$. However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point $x=\pm1$ in $L^2(-1,1)$ meaning that $\mathcal{D}(A^2)$ should exhibit four boundary conditions. In this talk, after a gentle `crash course' on left-definite theory and the classical Glazman-Krein-Naimark (GKN) theory, we show that $\mathcal{D}(A^2)$ can, in fact, be expressed using four (separated) boundary conditions. In addition, we determine a new characterization of $\mathcal{D}(A^2)$ that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple - and natural - and are equivalent to the boundary conditions obtained from the GKN theory.

Algebra Seminar

Speaker: Andrei Minchenko (Weizmann Institute of Science)

"Simple Lie conformal algebras"

Time: 14:30

Room: MC 107

The notion of a Lie conformal algebra (LCA) comes from physics, and is related to the operator product expansion. An LCA is a module over a ring of differential operators with constant coefficients, and with a bracket which may be seen as a deformation of a Lie bracket. LCA are related to linearly compact differential Lie algebras via the so-called annihilation functor. Using this observation and Cartan's classification of linearly compact simple Lie algebras, Bakalov, D'Andrea and Kac classified finite simple LCA in 2000.

I will define the notion of LCA over a ring $R$ of differential operators with not necessarily constant coefficients, extending the known one for $R=K[x]$. I will explain why it is natural to study such an object and will suggest an approach for the classification of finite simple LCA over arbitrary differential fields.

Colloquium

Speaker: Ian Hambleton (McMaster)

"Manifolds and symmetry"

Time: 15:30

Room: MC 107

This will be a survey talk about connections between the topology of a manifold and its group of symmetries. I will illustrate this theme by discussing finite group actions on spheres and products of spheres, and infinite discrete groups acting properly discontinuously on products of spheres and Euclidean spaces.