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Tropical and Logarithmic Methods in Enumerative Geometry de Renzo Cavalieri

Tropical and Logarithmic Methods in Enumerative Geometry: Oberwolfach Seminars, cartea 52

Autor Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan
en Limba Engleză Paperback – 2 oct 2023
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.
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Specificații

ISBN-13: 9783031394003
ISBN-10: 3031394003
Pagini: 159
Ilustrații: XIV, 159 p. 45 illus., 11 illus. in color.
Dimensiuni: 168 x 240 mm
Greutate: 0.29 kg
Ediția:1st ed. 2023
Editura: Springer International Publishing
Colecția Birkhäuser
Seria Oberwolfach Seminars

Locul publicării:Cham, Switzerland

Cuprins

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.

Notă biografică

​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”.

Textul de pe ultima copertă

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.

Caracteristici

Provides a review and introduction of each of these 3 fields with a particular eye to the interactions Provides hands-on exercises, plenty of examples and pictures The interplay of tropical, logarithmic and enumerative geometry presented is original