Travel support grant for the program on "Interactions between topological recursion, modularity, quantum invariants and low-dimensional topology"

University Of California-Davis, Davis CA

Investigators

Abstract

This award provides travel support for U.S. based participants to the 2016 Australian program titled "Interactions between Topological Recursion, Modularity, Quantum Invariants and Low-Dimensional Topology," to be held at the newly inaugurated Australian Mathematical Research Institute in Melbourne, MATRIX@Melbourne, from November 28, 2016, to December 23, 2016. The Program starts with a week of (Australian) summer school aimed at graduate students and postdoctoral scholars, followed by two week-long international conferences featuring talks by experts from all over the world on the subjects listed in the title, and with diverse background. The focus of the Program is to find a systematic mathematical understanding of the physical idea of `quantization' applied to functions in two variables. Quantization was discovered by Planck, Einstein, Heisenberg, Schoedinger, Dirac, and other giants in the early 20th century, as a procedure to change classical Newtonian mechanics to then newly discovered quantum mechanics. One hundred years later, the same quantization plays a role of discovering new keys, or invariants, to identify finer structures of spaces surrounding us. The classical invariants are called homology and homotopy of the space, and these tell us the coarse structure. The quantum invariants tell us far more refined information of the given space. The designed interaction between U.S. researchers and Australian mathematicians is expected to fertilize the study of this exciting new area of research. At the same time, the Program aims at inspiring early career researchers, in particular, women scientists, into this field. The Program will gather leading experts from both mathematics and physics to address recent advances and explore new connections between topology, number theory, and topological recursion. Such a meeting is particularly timely as quantum invariants in low-dimensional topology are discovered, while topological recursion and quantum curves are making rapid advances. Connections between these fields are now becoming rigorous mathematical theory. On the topology side, in the past four years we have witnessed exciting activity on the connection between number theory and ideal triangulations of (mostly hyperbolic) 3-manifolds. On topological recursion side, the most notable achievement is the recent solution of the Remodeling Conjecture of toric Calabi-Yau orbifolds of dimension three. And on the quantization aspect of the interplay, Gaiotto's conjecture on quantizing Higgs bundles of Hitchin component is solved. The Program aims at studying outstanding conjectures, including (1) Hyperbolic Volume Conjecture; (2) AJ-Conjecture; and (3) Quantum Modularity Conjecture. New ideas from topological recursion and quantum curves are expected to play a key role on studying these conjectures. A significant advance on the field is expected as a result of the 4-week activity. The URL of the Program is the following. http://www.matrix-inst.org.au/events/interactions-between-topological-recursion-modularity-quantum-invariants-and-low-dimensional-topology/