In this reading course we will study "embedded" tropical varieties: given a subvariety of the algebraic n-dimensional torus over a non-Archimedean valued field, we will define a polyhedral complex in Rn called its tropicalization. Topics include: structure of tropical varieties, the fundamental theorem of tropical geometry, matroids, tropical linear spaces and the tropical Grassmannian. We will follow the introductory textbook of Diane Maclagan and Bernd Sturmfels ([MS15]) for most of this topics, although certain papers will be also used as references. Handwritten Lecture notes from my Spring 2017 Topics course for most of the topics we will cover are available on the course's webpage.
For an overview of the topics covered in each Chapter, you can consult the following recordings of "Twelve Lectures on Tropical Geometry" delivered by Bernd Sturmfels in Summer 2020 at MPI-Leipzig.
Prerequisites: Some experience with undergraduate algebraic geometry will suffice (i.e. relation between ideals and varieties at the level of the book "Ideals, Varieties and Algorithms" (by Cox-Little-O'Shea). An enjoyment of combinatorics, especially polyhedra and graph theory, will be helpful throughout.
Tentative Schedule
The course is problem based and will cover Chapters 1-4 of [MS15]. We will meet every week and each student will present solutions to the assigned problem sets. Problems should be a roadmap to learn the material in each chapter of the book. During the problem sessions, I will provide some technical background and comment on findings that were inspired by the concrete examples given in the problem sets.
We will devote one or two problem-sessions to discuss each chapter, according to the following schedule:
- Chapter 1: Fridays, January 28, February 4 and 11.
- Chapter 2: Fridays, February 18, 25, March 4.
- Chapter 3: Monday, March 7, Friday March 25 and Monday April 4.
- Chapter 4: Fridays, April 8, 15 and Thursday April 21 (3-5pm).
Mathematical Software
Mathematical software can be very useful to study tropical varieties and polyhedral. Here are some useful packages:- Gfan.
- The tropical.lib package in Singular.
- The Tropical package for tropical computations in Macaulay 2.
- Polymake.
- Sage.
Problem Sets
There will be 4 problem sets to guide our learning and our weekly discussions. Group work is encouraged!- Problem Set 1: The tropical semifield and tropical plane curves. This set includes material in Chapter 1 of [MS15].
May the source be with you.
Plain text files for Problem 3:- GFAN Input file of the 3x3 determinant with the generators of the symmetric group. To run it on the terminal, use the following command "gfan_tropicaltraverse --symmmetry <InputFILEName >OutputFileName".
- GFAN Output file of all cones in the tropical 3x3 determinant under the action of S3×S3.
- GFAN Input file of the 5 points associated to a dense plane quartic with the generators of the symmetric group.
- GFAN Output file of all cones in the secondary fan, with orbit decomposition under the action of the symmetric group on 3 letters
May the source be with you. - Problem Set 2 : Gröbner bases over valued fields, Gröbner complex, Minkowski sums. This set corresponds to material in Chapter 2 of [MS15].
May the source be with you. Source for picture. - Problem Set 3: Tropical hypersurfaces, tropical varieties, recession fans and stable intersections. (Check out Gfan, the tropical.lib package in Singular and the tropical package in Macaulay 2). This set contains material from Chapter 3 of [MS15].
May the source be with you. Source for picture.
Plain text files and singular output for Problem 2(i):
- GFAN Input file. To run it on the terminal, use the following command "gfan_tropicalintersection --tplane <InputFILEName >OutputFileName".
- GFAN Output file of all cones.
- SINGULAR Input file. To run it on a terminal, use Singular.
- Drawing produced using SINGLAR. The program also creates a TeX file.
- GFAN Input file. To run it on the terminal, use the following command "gfan_tropicalstartingcone <InputFILEName | gfan_tropicaltraverse >OutputFileName".
- GFAN Output file of all cones.
- Problem Set 4: Matroids and tropical linear spaces. This set corresponds to material from Chapter 4 of [MS15].
May the source be with you. Source for picture.