The C*-algebra of equivariant Hamiltonians over point patterns

Abstract: Consider an extended (Delone) point pattern in the d-dimensional Euclidean space such that each point hosts N degrees of freedom. In many practical applications, ranging from quantum materials to meta-materials, one is interested in the collective dynamics of the degrees of freedom hosted by the pattern. As we shall see, the generators of any pattern-equivariant dynamics belong to a specific C*-algebra, which in general takes the form of a groupoid algebra and, in more manageable cases, of crossed products with discrete groups. The non-commutative geometry program for the aperiodic patterns consists in computing the C*-algebra, its K-theory and cyclic co-homology, as well as establishing index theorems for the K-theory and cyclic co-homology pairings. In these seminars I will describe several interesting cases where this program has been carried almost entirely. I have a large number of numerical simulations, which I will try to use throughout to exemplify the power of these methods.

Mathematics

Audience: researchers in the topic

Noncommutative Geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click sju.webex.com/meet/nikolaei (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.