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 $\mathbb{R}$ 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.

TBA

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 $p_0(\zeta)=0$ and $p_{j+1}(\zeta)=\zeta p^2_{j}(\zeta)+1$. These polynomials give rise to a novel family of recursively constructed zero-one matrices whose eigenvalues are the roots of $p_k(\zeta)$. 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.

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.

Let $\Delta$ 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 $C^r$ function, such that on each triangle the function is given by a polynomial in two variables of degree $\leq d$. 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 $d\geq 2r+1$, this dimension is given by a precise formula that depends on the combinatorial information of the simplicial complex $\Delta$, 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.