Tropical geometry is the algebraic geometry over the min-plus algebra.
It is a young subject that in recent years has both established itself
as an area of its own right and unveiled its deep connections to numerous
branches of pure and applied mathematics. From an algebraic geometric point
of view, algebraic varieties over a field with non-archimedean valuation are
replaced by polyhedral complexes, thereby retaining much of the information
about the original varieties.
This course offers an introduction to tropical geometry,
with emphasis on algebraic, computational and combinatorial aspects.
One concrete goal is to prepare the students for possible participation in
activities of the Fall 2009 research program on Tropical Geometry at MSRI
I am collaborating with Diane Maclagan on a book project
on ``Introduction to Tropical Geometry''.
A current incomplete draft of our manuscript is posted
here in pdf format. Please do e-mail us your comments.
You are strongly encouraged to browse the web for your favorite tropical papers. Obvious places to start are Wikipedia and the arXiv. Students with applied interests will enjoy reading Chapter 3 in the book by Baccelli, Cohen, Olsder and Quadrat on Synchronization and Linearity.
Schedule of lectures:
January 20: An invitation to tropical mathematics
January 22: Eigenvalues and eigenvectors
January 27: Fields and algebraic varieties
January 29: Polyhedral basics and Grobner basics
February 3: Tropical varieties
February 5: The fundamental theorem (Diane Maclagan)
February 10: Matroids and tropicalization of linear spaces
February 12: Secant varieties
February 17: Counting curves
February 19: Hurwitz numbers (Hannah Markwig)
February 24: Convexity
February 26: The rank of a matrix
March 3: Grassmannians and linear spaces
March 5: Tropical hyperplane arrangements (Federico Ardila)
March 10: SAGE Days at MSRI
March 12: Determinantal varieties
March 17: Grobner cones and mirror symmetry
March 19: Tropical discriminants
March 31: The Newton polytope of the implicit equation
April 2: Elimination and mixed fiber polytopes
April 7: Back to the basics I (Bernd)
April 9: Back to the basics II (Diane)
April 14: Noah Forman: The image of tropical matrix multiplication
April 14: Adam Boocher: Tropical lines on smooth tropical surfaces [Vigeland]
April 16: Laura Escobar: Bergman complexes, Coxeter arrangements and graph associahedra [Ardila-Reiner-Williams]
April 16: Anne Shiu: The positive Bergman complex of an oriented matroid [Ardila-Klivans-Williams]
April 21: Claudiu Raicu: Fibers of tropicalization [Payne]
April 21: Cynthia Vinzant: Logarithmic limit sets of real semi-algebraic sets [Alessandrini]
April 23: Jeff Doker: Combinatorics and genus of tropical intersections [Steffens-Theobald]
April 23: Melody Chan and Dustin Cartwright: Symmetric rank and tree rank of matrices
April 28: Alex Fink: Reparametrizations of the Grassmannian
April 28: Felipe Rincon: A matroid invariant via the K-theory of the Grassmannian [Speyer]
April 30: Shuchao Bi: Tropical and Kapranov ranks of tropical matrices [Rubei]
April 30: Morgan Brown: Compactifications of subvarieties of tori [Tevelev]
May 5: Daniel Erman and Tony Varilly: Moduli of del Pezzo surfaces [Hacking-Keel-Tevelev]
May 5: Bianca Viray: Tropical Enriques surfaces
May 7: Kirsten Freeman: Tropical hyperplane arrangements
May 7: Angelica Cueto: First steps in tropical intersection theory [Allermann-Rau]
There will be regular assignments, posted here in pdf format, during the
first eight weeks of the course.
Homework 1: due January 27. Solutions due to Alex Fink
Homework 2: due February 3. Solutions due to Shaowei Lin
Homework 3: due February 10. Solutions due to Melody Chan
Homework 4: due February 24. Solutions due to Angelica Cueto
Homework 5: due March 12. Solutions due to Kristen Freeman
Projects: Registered students will work on a course project in tropical geometry. There are two phases for your project:
In phase one, you will read a research paper in tropical geometry and write a brief report about it. In phase two, you will identify
a relevant research problem and pursue it as far as you can. You may also be asked to present your paper and/or problem in class.
Tuesday, March 17: Tell me which paper you have chosen.
Tuesday, April 7: Submit a brief written report which summarizes your paper and describes your research problem.
Tuesday, May 19: Submit the final report about your research project.
Postdocs Fall 2009: Please start interacting as soon as possible with the following terrific postdoctoral fellows:
Tropical postdocs at MSRI: Tristram Bogart, Erwan Brugalle, Ehthan Cotterill, Anders Jensen, Eric Katz, Lucia Lopez de Medrano,
Gregg Musiker, Benjamin Nill, Mounir Nisse, Lisa Nilsson, Alan Stapledon, Luis Tabera, Lauren Williams, Josephine Yu.
Berkeley postdocs with related interests: Janko Boehm, Alexander Engstrom, Valerie Hower, Christopher Manon,
Brett Parker, Philipp Rostalski, Raman Sanyal, Alexander Schoenhuth, Mauricio Velasco.