In the first part of the 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 R^{n} called its tropicalization. Topics include: structure of tropical varieties, the fundamental theorem of tropical geometry, tropical linear spaces and the tropical Grassmannian, including matroid theory, tropical compactifications, toric degenerations and the tropical computation of plane relative Gromov-Witten invariants. 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.
In the second part of the course we will study tropicalizations from an abstract perspective and in connection with Berkovich non-Archimedean analytic spaces. We will focus on the case of abstract tropical curves, which are vertex-weighted metric graphs, and discuss their relation to algebraic and non-Archimedean analytic curves. Our goal would be to construct and study the moduli spaces of tropical curves. We will cover many cutting-edge papers in the field (see the References on the syllabus) .
Prerequisites: Some experience with algebraic geometry at the graduate level would be extremely helpful for Part 2 of the course. For Part 1, undergraduate algebraic geometry may be enough (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.
Handouts
Homework
There will be 3 homework problem sets (with due dates roughly every 2-3 weeks). These will be posted as the semester progresses and will constitute 60% of your grade.Participants are encourage to work in teams, but individual solutions must be submitted for grading and credit.
- Homework 1: due date January 30th (in class).
- Homework 2: due date March 10th (in class).
- Homework 3: due date April 7th (in class). Required submission format: pdf file produced in LaTeX.
Final Project
For the final project of this class, each student will give a 20 minute presentation in class and write a paper. This will constitute 40% of your final grade. The topic can be an exposition of a research paper related to tropical geometry. Only one student can work on a given paper, and projects will be assigned as requested. All students are strongly encourage to think about further questions related to the paper they are presenting.Suggested topics and references for final project.
Presentation Topics:
- Junjie Chen: Specialization of linear systems from curves to graphs (Matthew Baker),
- Evan Nash: Tropical Intersection Theory from Toric Varieties (Eric Katz),
- Aniket Shah: Piecewise polynomials, Minkowski weights, and localization on toric varieties (Eric Katz and Sam Payne),
- Jun Wang: The tropicalization of the moduli space of curves (Dan Abramovich, Lucia Caporaso and Sam Payne),
- Yuancheng Xie: Tropical curves, their Jacobians and theta functions (Grigory Mikhalkin and Ilia Zharkov),
- Yu Zhang: Linear Systems on Tropical Curves (Christian Haase, Gregg Musiker and Josephine Yu).
Schedule of talks: Each talk should be 20 minutes long (followed by a 5 minutes question period)
- Monday April 10th: Moduli of curves and specialization lemma (Jun Wang and Junjie Chen).
- Wednesday April 12th: Polytopal connections: linear systems on curves, and theta functions. (Yu Zhang and Yuancheng Xie).
- Friday April 14th: Toric connections: intersection theory, Minkowski weights (Evan Nash and Aniket Shah).
Term paper due date: April 24th at 9:10am.
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 cover each class, see the corresponding handwritten notes in the section entitled Lectures.
Week | Topics |
1 | Introduction and first examples; fields with valuations; polyhedral complexes. |
2 | Tropical hypersurfaces; Fundamental Theorem of tropical geometry and Structure Theorem for hypersurfaces. |
3 | Gröbner basis over valued fields; initial ideals of homogeneous ideals. |
4 | The Gröbner complex; tropical basis, Fundamental Theorem in any codimension. |
5 | Basics on Matroids; Grassmannians and its matroid stratification. |
6 | Tropical Grassmannians; Tropical linear spaces, Bergman fans of matroids. |
7 | Multiplicity and balancing; Connectivity; Structure Theorem of tropical varieties. |
8 | Abstract tropical varieties; Tropical linear spaces as abstract tropical varieties; Chow varieties. |
9 | Toric degenerations and Chow polytopes; Introduction to Toric varieties; Tropical Compactifications. |
10 | Geometric Tropicalization; Berkovich analytic spaces I. |
11 | Berkovich analytic spaces II; limits of tropicalizations. |
12 | The moduli space of tropical curves. |
13 | student presentations |
14 | Mikhalkin's Correspondence Theorem; Gromov-Witten invariants of elliptic curves and their tropical analogs; tropical geometry of genus 2 curves. |
15 | tropical geometry of genus 2 curves |
Lectures
- Lecture 1 (The tropical semiring, plane curves, tropical Bézout's theorem), January 9, 2017.
- Lecture 2 (Fields with valuations, examples: p-adics, Puiseux series, generalized Puiseux series), January 11, 2017.
- Lecture 3 (Basics on polyhedral geometry), January 13, 2017.
- Lecture 4 (Regular subdivisions of polyhedra, Newton subdivision of Laurent polynomials, Tropical hypersurfaces), January 18, 2017.
- Lecture 5 (Tropical hypersurfaces and the Fundamental Theorem of tropical geometry), January 20, 2017.
- Lecture 6 (The Fundamental Theorem and Structure theorem for tropical hypersurfaces, duality between tropical hypersurfaces and Newton subdivision of Laurent polynomials), January 23, 2017.
- Lecture 7 (Gröbner bases over valued fields: multiplicativity property for initial forms of Laurent polynomials, basics on traditional Gröner theory for monomial term orders), January 25, 2017.
- Lecture 8 (Gröbner bases over valued fields II: homogeneity is preserved, behavior of initial forms of polynomials under small perturbations), January 27, 2017.
- Lecture 9 (Gröbner bases over valued fields III: behavior of initial ideals under small perturbations, Hilbert functions and dimensions are preserved under initial degenerations), January 30, 2017.
- Lecture 10 (The Gröbner complex of a homogeneous ideal), February 1, 2017.
- Lecture 11 (Tropical bases, definition of tropical varieties, the Fundamental Theorem of Tropical Geometry), February 3, 2017.
- Lecture 12 (Basics on matroids: rank, bases, circuits, examples: graphical matroids, uniform matroid, vectors in a linear space (realizability); non-realizable matroids: Fano matroid), February 6, 2017.
- Lecture 13 (Matroid basics: loops, coloops, lattice of flats of a matroid; the Grassmannian of d-planes in n-space and its Plücker embedding, matroid stratification and its properties), February 8, 2017.
- Lecture 14 (The Tropical Grassmannian and the space of phylogenetic trees), February 13, 2017.
- Lecture 15 (The Tropical Grassmannian II: The compact tropical Grassmannian Gr(2,n); modular interpretation of Trop Gr_{∅}(d,n)), February 15, 2017.
- Lecture 16 (Tropical Linear spaces I: constant coefficients; Bergman fans of matroids; ordered complex of the lattice of flats of a matroid; matroid polytopes), February 17, 2017.
- Lecture 17 (Tropical Linear spaces II: arbitrary coefficients; valuated matroids, local tropical linear spaces, the Dressian Dr_{M} of a matroid M as the moduli space of valuated matroids with underlying matroid M; structure theorem of tropical linear spaces), February 20, 2017.
- Lecture 18 (Structure theorem of tropical varieties: balancing, connectivity, Gröbner structure of Trop(X)), February 22, 2017.
- Lecture 19 (Tropical multiplicities from geometry and the hypersurface case; Bieri-Groves theorem), February 24, 2017.
- Lecture 20 (Recession fans and tropicalization with respect to trivial valuations; definition of abstract tropical varieties; Abstract tropical hypersurfaces are realizable via the ray-shooting algorithm), February 27, 2017.
- Lecture 21 (Tropical linear spaces are abstract tropical varieties), March 1, 2017.
- Lecture 22 (Chow varieties, Chow forms and Chow polytopes: definitions, properties and examples), March 3, 2017.
- Lecture 23 (Chow polytopes, toric degenerations, orthant shooting algorithm), March 6, 2017.
- Lecture 24 (Introduction to toric varieties and their tropicalizations, examples: A^{n} and P^{n}, affine toric varieties via rational polyhedral cones in the R-span of the cocharacter lattice of the dense torus), March 8, 2017.
- Lecture 25 (Introduction to toric varieties II: gluing affine pieces and their tropical counterparts, examples; visualization of their tropicalization in a polytope; Tropical compactifications over trivial valuation via Tevelev's theorem, examples), March 10, 2017.
- Lecture 26 (Divisorial valuations, definition, examples; tropicalization via divisorial valuations; Simple normal crossings vs. combinatorial normal crossings on divisorial boundaries; Geometric Tropicalization after Hacking-Keel-Tevelev), March 20, 2017.
- Lecture 27 (Berkovich analytification for tropical geometry: non-Archimedean absolute values on K from valuations, totally disconnected induced topology, behavior under completion; basic definition of analytification for (Spec A)^{an}s for finitely generated K-algebras, topology on analytic spaces, main example: A^{1} with trivial valuation.), March 22, 2017.
- Lecture 28 (Berkovich analytification for tropical geometry II: complete description of (A^{1})^{an} (points and metric) for complete non-Archimedean algebraically closed fields K via sup norms on discs; Berkovich's classification theorem: correspondence between points in (A^{1})^{an} and nested sequences of discs in K; example over p-adics), March 24, 2017.
- Lecture 29 (Berkovich analytification for tropical geometry III: Topology and metric on (A^{1})^{an}, branching at each point; Berkovich unit disc; proof of Berkovich's classification theorem for the Berkovich unit disc), March 27, 2017.
- Lecture 30 (Berkovich analytification for tropical geometry IV: proof of Berkovich's classification theorem for the Berkovich line; the unit disc as an inverse limit of finite union of segments via retraction maps; analytification of smooth and complete curves: skeletons via semistable regular models over valuation rings; the example of an elliptic curve with bad reduction; tropical varieties are shadows of analytifications), March 29, 2017.
- Lecture 31 (Berkovich analytification for tropical geometry V: the piecewise linear behavior of the tropical map from X^{an} to Trop(X,i) with explicit examples on curves; Poincarë-Lelong formula; inverse limits of tropicalizations: Payne's theorem; Hrushovski-Loeser Theorem: local contractibility of X^{an}; faithful tropicalizations with examples on how to repair non-faithful embeddings), March 31, 2017.
- Lecture 32 (Berkovich analytification for tropical geometry VI: faithful tropicalization for elliptic cubics, the negative valuation of the j-invariant as the tropical j-invariant; repairing embeddings into honeycomb form and using tropical modifications; faithfulness in hiher dimensions: skeleton norms for (A^{n})^{an}, Shilov boundary points, faithful tropicalization for the Grassmannian of planes Gr(2,n) with proof sketch; extensions of this result to subvarieties of tori and toric varieties by Gubler-Rabinoff-Werner via extended skeleta.), April 3, 2017.
- Lecture 33 (Moduli spaces of curves and their tropical analogs I: the moduli functor, representability, main example: M^{0,n} and its modular compatification: from smooth rational marked curves to stable marked curves; relation to the Grassmannian Gr(2,n) and the space of trees; Tevelev's theorem: M_{0,n} is a tropical compactification), April 5, 2017.
- Lecture 34 (Moduli spaces of curves and their tropical analogs II: boundary of M_{0,n} strafied by combinatorial types of dual graphs to stable marked curves; natural morphisms between M_{0,n}'s: forgetful and gluing morphisms; relation to strong semistable models with markings: Berkovich skeleta and extended skeleta, with no metric; other moduli spaces: M_{g,n} and M_{g} with their combinatorial statification; tropical analogs (stable tropical curves): M_{g,n}^{trop} and M_{g}^{trop}, examples in genus 2), April 7, 2017.
- Lecture 35 (Moduli of tropical curves III: combinatorial structure as stacky fans; abstract vs. parameterized tropical curves), April 17, 2017.
- Lecture 36 (Curve counting with tropical techniques: Mikhalkin's correspondence theorem, combinatorial formulas for genus, multiplicities of tropical curves; why do we need 3d+g-1 conditions to have a finite count of degree d genus g curves in P^{2}; counting curves of degree d and genus g in P^{2} via lattice path counts on dΔ_{2}; explicit calculations on the example of rational plane cubics: N_{0,3} = N_{0,3}^{trop} = 12; calculation of the number of singular plane quadrics: N_{-1,2} = N_{-1,2}^{trop} = 3), April 19, 2017.
- Lecture 37 (Tropical Geometry of Genus 2 curves; motivation: GW invariants of smooth genus 1 curves, the classical and tropical counts and their comparison; genus 2 curves as hyperelliptic covers of P^{1}; tropical hyperelliptic covers, example of degree 4), April 21, 2017.
- Lecture 38 (Tropical Geometry of Genus 2 curves II: complete description of the Berkovich skeleta (and the metric) from tropical hyperelliptic covers of rational metric trees on 6 leaves; effective characterization in terms of 7 witness cases, depending on the valuations of the branch points and their differences; tropicalization of Igusa invariants and their continuous piecewise linear behavior on M_{2}^{trop}), April 24, 2017.