Blog Archives

Topic Archive: ultrapowers

Model theory seminarFriday, October 10, 201410:45 amGC5382Note New Room

Timothy Gendron

Modular Invariant of Quantum Tori

Instituto de Matematicas, Universidad Nacional Autonoma de Mexico

The modular invariant jqt of quantum tori is defined as a discontinuous, PGL(2,Z)-invariant multi-valued map of the reals R. For θ ∈ Q, jqt(θ) = ∞ and for quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest that jqt(θ) is a finite set. In the case of the golden mean φ, we produce explicit formulas for the experimental supremum and infimum of jqt(φ) involving weighted generalizations of the Rogers–Ramanujan functions. Finally, we define a universal modular invariant as a continuous and single-valued map of “ultrasolenoids” (quotients of sheaves of ultrapowers over Stone spaces) from which jqt as well as the classical modular invariant of elliptic curves may be recovered as subquotients.

Karel Hrbacek
City College of New York, CUNY
Professor Hrbacek undertakes research in the area of mathematical logic, particularly in set theory, with a focus on non-standard analysis and nonstandard set theory. He wrote in co-authorship with Thomas Jech the highly regarded book Introduction to Set Theory.
University of Toronto
Alex Rennet is a postdoc in the Mathematics department at the University of Toronto working under the supervision of Bill Weiss. His research focus right now is in o-minimality and in particular, ultraproducts of o-minimal structures. He received his Ph.D. in 2012 at the University of California at Berkeley, under the supervision of Thomas Scanlon.