Dive into the foundations of Quantum Toric Geometry in this comprehensive lecture, the first of a mini-course series. Explore the groundbreaking work of Katzarkov, Lupercio, Meersseman, and Verjovsky as they generalize toric geometry to include irrational fans corresponding to non-commutative spaces called quantum toric varieties. Discover how these quantum toric varieties relate to traditional toric varieties, drawing parallels with the relationship between quantum tori and conventional tori. Delve into key topics including stacks, non-commutative spaces, the quantum torus, and quantum toric varieties. Gain insights from speaker Ernesto Lupercio of the Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN). Examine the structure of compact toric varieties, explore non-commutative spaces through Gelfand duality, and investigate the Kronecker foliation. Study crucial concepts such as groupoids, Lie groupoids, étale groupoids, Morita equivalence, and stacks associated with foliations. Enhance your understanding of toric geometry and prepare for advanced exploration in subsequent lectures of this mini-course.
An Introductory Mini Course into Quantum Toric Geometry Lecture I - Day 1
Overview
Syllabus
Intro
Recall the basic structure of a (compact) toric variety (over C)
In general...
What are non-commutative spaces? (Gelfand duality)
Usual torus vs. Non-commutative torus aka quantum torus.
Constructing the Kronecker foliation.
Exercise.
We need Groupoids, objects that generalize groups actions (groups).
Associativity...
Internal facts... (Logic).
Natural transformations.
Group Actions produce Groupoids
Lie Groupoids
Étale Groupoids.
Morita equivalence.
Stacks associated to foliations.
Morita equivalence of algebras.
Taught by
IMSA
