Abstract: 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.

Abstract: 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.