Enumerative Geometry and the Quantum Torus

Explore enumerative geometry and the quantum torus in this comprehensive lecture by Michael Polyak from the Israel Institute of Technology. Delve into tropical geometry as a piecewise linear approach to algebraic geometry, examining how tropical curves serve as planar metric graphs with specific balancing, slope rationality, and integrality requirements. Investigate a generalization of tropical curves that removes slope rationality and integrality constraints. Learn about classical enumerative problems in algebraic geometry, focusing on counting complex or real rational curves through point collections in toric varieties. Discover how this counting procedure relates to cycle construction on rigid pseudotropical curve moduli, and explore the connection between these cycles and Lie algebras. Examine the relationship between complex and real curve counting and the quantum torus Lie algebra. Delve into more complex counting invariants, such as Gromov-Witten descendants, and their connection to the super-Lie structure on the quantum torus. The lecture covers topics including an introduction to enumerative and tropical geometry, tropical curves as weighted balanced graphs, the correspondence theorem, refined tropical counting, pseudotropical curves, weights and enumeration, rigid marked curves, and evaluation maps and compactification.