The classification of mathematical objects is an important problem in many branches of mathematics, notably also in algebraic geometry. Typically, these classification cannot be achieved by (finite) lists. Instead one builds a new algebraic variety whose points correspond to then isomorphism class of the objects to be classified — a moduli space. This leads to the questions: how does one construct such classifying spaces and what can one say about the geometry of these moduli spaces? I will discuss these question in exemplary cases (old and new).
Classification in algebraic geometry and the geometry of moduli spaces
Feb. 21, 2023 3:30pm (MATH 3…
Topology
Daniel Spiegel (CU Boulder)
X
The study of parametrized many-body gapped Hamiltonians and their topological phases represents a fast-growing frontier at the intersection of mathematical physics and condensed matter physics. In the limit of infinite system size, observable quantities are represented by elements of a uniformly hyperfinite C*-algebra A and the object of study is a continuous function into the space P(A) of pure states of A. With a continuous family of pure states, questions arise of how to do classical C*-algebraic maneuvers like the GNS construction and Kadison transitivity theorem in a way that depends continuously on the input data. I will present new theorems pertaining to both the norm and weak* topologies on P(A) to address these questions and others that have arisen in the study of parametrized many-body quantum systems.
Topological Results on Pure State Space Inspired by Parametrized Quantum Phases
If you have a non-invertible unary operation on a finite set, a natural thing to do with it is to create a compositional power which satisfies . Using this compositional power of , you can produce a simpler clone on a smaller set which satisfies every height-one identity which is satisfied by the original clone. By repeating this process, we can eventually reduce questions about height-one identities in finite algebras to the case of idempotent algebras. It turns out that one can do something similar with two-variable operations: starting from any idempotent binary operation on a finite set, there is a (nontrivial) functorial construction of a two-variable operation in the clone generated by which satisfies the identities . Such operations were used extensively by Andrei Bulatov in his proof of the CSP Dichotomy Conjecture, and I call them ``partial semilattice operations''. I will describe this functorial construction and show how we can make use of such operations to simplify the behavior of binary absorption in finite algebras while preserving many types of height-one identities.
Classification in algebraic geometry and the geometry of moduli spaces