En savoir plus
This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Table des matières
Part I: Toric Geometry and Logarithmic Curve Counting. - 1. Geometry of Toric Varieties. - 2. Compactifying Subvarieties of Tori. - 3. Points on the Riemann Sphere. - 4. Stable Maps and Logarithmic Stable Maps. - 5. Cheat Codes for Logarithmic GW Theory. - Part II: Hurwitz Theory. - 6. Classical Hurwitz Theory and Moduli Spaces. - 7. Tropical Hurwitz Theory. - 8. Hurwitz Numbers from Piecewise Polynomials. - Part III: Tropical Plane Curve Counting. - 9. Introduction to Plane Tropical Curve Counts. - 10. Lattice Paths and the Caporaso-Harris Formula. - 11. The Caporaso-Harris Formula for Tropical Plane Curves and Floor Diagrams.
A propos de l'auteur
¿Renzo Cavalieri completed his PhD at University of Utah in 2005 under the direction of Aaron Bertram. He was a postdoc at University of Michigan under the mentorship of Bill Fulton for the following three years. In 2008, he became faculty at Colorado State University where he is currently a professor in the department of mathematics.
Hannah Markwig completed her PhD in 2006 at the University of Kaiserslautern in Germany, advised by Andreas Gathmann. She was a Postdoc at the Institute of Mathematics and its Applications in Minneapolis and at the University of Michigan in Ann Arbor, before she started a Juniorprofessorship at the University of Göttingen in 2008. In 2011, she moved to the University of the Saarland as a Professor, and in 2016 to the University of Tübingen.
Dhruv Ranganathan completed his PhD at Yale University in 2016 under the direction of Sam Payne. He was a CLE Moore Instructor at MIT and a memberat the Institute for Advanced Study in 2017. Since 2019, he has been at the University of Cambridge, where he is currently a professor of mathematics.
The authors have worked together since 2013, on several projects related to the themes discussed in this book. They have taught several courses, including at MSRI, Stockholm, and of course in Oberwolfach. In addition to their shared love of mathematics, the authors enjoy hiking, cooking, music, and the life-altering card game known as "tichu".
Commentaire
The book gives many technical details, along with examples and exercises for readers to familiarize themselves with the techniques. After reading it, and answering the exercises, readers should have a good basis for understanding recent papers approaching enumerative geometry with logarithmic and tropical methods. (Reinier Kramer, Mathematical Reviews, April, 2025)
The book consists of three parts written by three different authors . there is a small number of cross-references within the text, the parts are largely independent and can be read separately in an arbitrary order. Each of them may serve as an excellent introduction to the corresponding topic . (Mikhail Shkolnikov, zbMATH 1544.14001, 2024)
