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29 Noncommutative Geometry
Noncommutative Geometry Speaker: Jason Haradyn (Western) "NCG Learning Seminar: Spin Geometry (1)" Time: 14:30 Room: MC 107 In the first of a series of talks, I would like to introduce the notions of a Clifford algebra of a vector space V over R and of a spin structure on a Riemannian manifold. I will discuss when a Riemannian manifold does in fact carry a spin structure, thus allowing it to admit spinors. This is not always possible because there may be topological obstructions on the manifold that inhibit it from carrying such a structure. Nevertheless, spin manifolds are useful for determining whether or not an orientable Riemannian manifold admits spinors. Once this is in place, we will look at the Dirac operator associated to a spin module and some of its properties, including how it operates on sections of the spinor bundle.
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30 Analysis Seminar
Analysis Seminar Speaker: Masoud Khalkhali (Western) "A new evaluation of zeta values at even integers, II" Time: 15:30 Room: MC 108 TBA Graduate Seminar
Graduate Seminar Speaker: Piers Lawrence (UWO Applied Math) "Mandelbrot Polynomials and Matrices" Time: 17:00 Room: MC 108 We explore a family of polynomials whose roots are related to the Mandelbrot set. The roots correspond to the k-periodic points of the iteration defining the Mandelbrot set. The Mandelbrot polynomials are defined by p0(ζ)=0 and pj+1(ζ)=ζp2j(ζ)+1. These polynomials give rise to a novel family of recursively constructed zero-one matrices whose eigenvalues are the roots of pk(ζ). The LU decomposition of the resolvent of these matrices is highly structured, and one linear solve can be done in O(n) operations. Krylov based eigenvalue solvers can then be used to compute the eigenvalues of these matrices in an efficient manner. |
31 Noncommutative Geometry
Noncommutative Geometry Speaker: Asghar Ghorbanpour (Western) "NCG Learning Seminar: Spin Groups and their Representation Theory" Time: 14:30 Room: MC 107 The spin group, Spin(n), for n>2, can be defined as the universal covering group of SO(n). They can be explicitly constructed as a subgroup of the group of invertible elements of the Clifford algebra. One can easily see that any irreducible SO(n) representation gives an irreducible representation of Spin(n), however, some irreducible Spin(n) representations cannot be constructed in this way. The main goal of this talk is to construct such representations using the representation theory of Clifford algebras. |
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2 Algebra Seminar
Algebra Seminar Speaker: Stefan Tohaneanu (Western) "From Spline Approximation to Roth's Equation via Schur Functors" Time: 14:30 Room: MC 108 Let Δ be a triangulation of a topological open disk in the real plane. Let r and d be two positive integers. On this region one defines a piecewise Cr function, such that on each triangle the function is given by a polynomial in two variables of degree ≤d. The set of these functions forms a finite dimensional vector space, and one of the major questions in Approximation Theory is to find the dimension of this space. It was conjectured that for d≥2r+1, this dimension is given by a precise formula that depends on the combinatorial information of the simplicial complex Δ, and on the local geometric data. The conjecture is very difficult, and trying to prove it for the simplest nontrivial example has been a challenge for about 10 years. Jan Minac and myself answered this question by the means of Commutative Algebra, showing also that a direct approach to solve this conjecture for this particular example leads to difficult questions in Matrix Theory, such as the LU-decomposition of an invertible matrix. In this talk I am presenting an overview of these problems. The talk is accessible to graduate students. |
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