Curve Counting and Tropical Geometry

Professor: Dhruv
Ranganathan (MIT)
Período:    26 e 28 / 03 / 2018   –  10:30 às 12:00 / 15:30 às 17:00
Local: Sala 349

This course will explore the role of tropical methods in curve counting, or enumerative geometry, and the related aspects of the geometry of moduli spaces. These tropical methods equate algebro-geometric counting problems with corresponding graph theoreic ones, and have their roots in mirror symmetry and degeneration methods in Gromov-Witten theory. We will develop these ideas, paying a great deal of attention to one simple example, namely counts of covers of P1 by P1 with prescribed ramification data. Using this example as a lens, we will explore connections with the geometry of Berkovich spaces and the combinatorics of logarithmic structures. As the course concludes, we will outline the theory of logarithmic stable maps, and how it can be engineered out of our simple example, revealing a powerful and general theory.