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 loosely follow the introductory textbook of Diane Maclagan and Bernd Sturmfels ([MS15]) for most of this topics, although certain papers will also be 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.
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 following is the schedule of topics that we plan to cover each week (it is subject to change). For a list of topics discussed during each meeting, see the section entitled Lectures.
| Week | Topics |
| 0 | Organizational meeting - Distribution of topics |
| 1 | Basics on Polyhedral Geometry - polyhedral complexes - Newton Subdivisions. |
| 2 | Basics on valued fields - Tropical hypersurfaces - tropical plane curves |
| 3 | Problem sets 1 and 2 |
| 4 | Gröbner basis over valued fields; initial ideals of homogeneous ideals. Flatness |
| 5 | The Gröbner complex: main properties and computations in Gfan. |
| 6 | Tropical varieties via tropical basis - Fundamental Theorem of tropical geometry |
| 7 | Structure Theorem (multiplicities and balancing) - Bieri-Groves' Theorem |
| 8 | Problem sets 2 and 3 |
| 9 | Basics on Matroids: bases, independent sets, rank, the lattice of flats, matroid polytopes. |
| 10 | Grassmannians and their Plücker embedding - matroid stratification of Grassmannians |
| 11 | Tropical Grassmannians - Main example: the space of phylogenetic trees |
| 12 | Tropical linear spaces I: constant coefficient case. Bergman fans of matroids. |
| 13 | Tropical linear spaces II: arbitrary coefficients- Valuated matroids - Dressians. |
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, valuations and polyhedra.
- Problem Set 2: Minkowski sums, plane tropical interpolation and tropical hypersurfaces. (Check out Gfan and the tropical.lib package in Singular)
- Problem Set 3: Gröbner bases over valued fields, the Grö,bner complex and tropical varieties.
- Problem Set 4: Matroids and tropical linear spaces.
Lectures
- Lecture 1 (The tropical semiring, plane curves, tropical Bézout's theorem), January 15, 2019.
- Lecture 2 (Basics on polyhedral geometry and polyhedral complexes, Newton polytopes, Regular subdivisions of polyhedra, Newton subdivision of Laurent polynomials), January 25, 2019.
- Lecture 3 (Fields with valuations, examples: p-adics, Puiseux series, generalized Puiseux series; Newton's method for solving equations over Puiseux series), January 29, 2019.
- Lecture 4 (Tropical hypersurfaces and the Fundamental Theorem of tropical geometry. Main example: tropical plane curves), February 1, 2019.
- Lecture 5 (Gröbner bases over valued fields: multiplicativity property for initial forms of Laurent polynomials, basics on traditional Gröner theory for monomial term orders), February 12, 2019.
- Lecture 6 (Gröbner bases over valued fields II: homogeneity is preserved, behavior of initial forms of polynomials under small perturbations, Hilbert functions and dimensions are preserved under initial degenerations), February 15, 2019.
- Lecture 7 (The Gröbner complex of a homogeneous ideal), February 19, 2019.
- Lecture 8 (Tropical bases, definition of tropical varieties, the Fundamental Theorem of Tropical Geometry. Computing tropical varieties with mathematical software (e.g. Gfan.)), March 1, 2019.
- Lecture 9 (Structure theorem of tropical varieties: balancing, connectivity, Gröbner structure of Trop(X). Tropical multiplicities from geometry and the hypersurface case; Bieri-Groves theorem), March 5, 2019.
- Lecture 10 (Basics on matroids (I): rank, bases, circuits, examples: graphical matroids, uniform matroid, vectors in a linear space (realizability.)), March 26, 2019.
- Lecture 11 (Matroid basics: loops, coloops, lattice of flats of a matroid. non-realizable matroids: Fano and non-Pappus matroid. Discussion of examples from Problem Set 4.), April 2, 2019.
- Lecture 12 (The Grassmannian of d-planes in n-space and its Plücker embedding, matroid stratification and its properties), April 5, 2019.
- Lecture 13 (The Tropical Grassmannian and the space of phylogenetic trees; the Pl&uulm;cker relations and the four-point conditions), April 9, 2019.
- Lecture 14 (Tropical Linear spaces I: constant coefficients; Bergman fans of matroids; ordered complex of the lattice of flats of a matroid), April 19, 2019.
- Lecture 15 (Tropical Linear spaces II: arbitrary coefficients; valuated matroids, local tropical linear spaces, the Dressian DrM of a matroid M as the moduli space of valuated matroids with underlying matroid M; structure theorem of tropical linear spaces), April 23, 2019.