We will plan out speakers/topics for the remainder of the semester.
organizational meeting
Mar. 31, 2022 2:30pm (Math 2…
Functional Analysis
Levi Lorenzo (CU Boulder)
X
We set out with the intention that KK sets (which we now know have the structure of abelian groups) would be Hom sets in our new category. Accordingly, we need a way to compose an element of KK(A,B) with one of KK(B,C) to produce an element of KK(A,C). In this talk, we see how perhaps the most natural guesses for how to do so fail and introduce the Connes-Skandalis Connection which enables us to define composition.
The Connes-Skandalis connection
Mar. 31, 2022 3pm (Math350)
Math Phys
Anton Kapustin (Caltech)
X
M. Berry showed how to attach a line bundle and a connection on it to a family of quantum Hamiltonians with a non-degenerate ground state, under the assumption that the Hilbert space is finite-dimensional. The first Chern class of this line bundle is a topological invariant of the family. It is far from obvious if this construction can be generalized to quantum many-body Hamiltonians. Indeed, naive generalizations fail because ground states of different Hamiltonians typically correspond to inequivalent representations of the algebra of observables. Nevertheless, it is possible to construct such invariants by making use of a certain differential graded Lie algebra (DGLA) attached to a quantum lattice system. For example, it turns out that to any family of gapped Hamiltonians on a 1d lattice one can attach a “quantized” degree-3 cohomology class on the parameter space. In this talk I will outline a construction of this DGLA as well as the construction of higher Berry classes. The talk is based on a work in progress with Nikita Sopenko.
The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356
Higher Berry classes for many-body quantum lattice systems Sponsored by the Meyer Fund