Math 6422 (Algebraic Geometry II)
Spring 2024
Tropical Algebraic Geometry
Instructor: Josephine Yu
Class time and location: Tuesdays and Thursdays, 3:30-4:45 pm, in Skiles 171
Piazza: https://piazza.com/gatech/spring2024/math6422a/home
Prerequisites: Graduate Algebra I (Math 6121). Linear algebra, rings, and fields. Only minimal background in algebraic geometry is assumed. Knowledge of commutative algebra and polyhedral geometry is helpful (see below for recommended background reading).
Textbook: Introduction to Tropical Geometry by Diane Maclagan and Bernd Sturmfels. Electronic copy available via GT library
Description: This course will be an introduction to tropical algebraic geometry, focusing on tropicalization of algebraic varieties over algebraically closed fields. The core of the course will be Chapters 2 and 3 of the textbook, on fields and valuations, basics of algebraic varieties and polyhedral geometry, Gröbner complexes, tropical varieties, the Fundamental Theorem, and the Structure Theorem.
As time permits, and depending on the interest of students, we may cover additional chapters and topics such as matroids and Grassmannians, tropical convexity and tropical linear algebra, tropical curves and divisors, tropicalization of real semialgebraic sets, and applications to computational algebra and optimization.
Background reading:
- Ideals, Varieties, and Algorithms : An Introduction to Computational Algebraic Geometry and Commutative Algebra, by David Cox, John Little, and Donal O’Shea (PDF available via GT library)
- Lectures on Polytopes, by Günter Ziegler (PDF available via GT library)
Other books or lecture notes:
- Essentials of Tropical Combinatorics, by Michael Joswig (PDF available via GT library)
- Tropical Geometry, by Grigory Mikhalkin and Jonannes Rau
- Tropical and Logarithmic Methods in Enumerative Geometry, by Renzo Cavalieri, Hannah Markwig & Dhruv Ranganathan
- Brief Introduction to Tropical Geometry by Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, and Kris Shaw
Survey or expository papers:
- A bit of tropical geometry, by Erwan Brugallé and Kris Shaw
- Moduli Spaces of Curves: Classical and Tropical, by Melody Chan
- Tropical Algebraic Geometry, by Andreas Gathmann
- Tropical Geometry Forwards and Backwards, by Dhruv Ranganathan
- Tropical Combinatorics, by Felipe Rincón, Ngoc Tran, and Josephine Yu
- Tropical Mathematics, by David Speyer and Bernd Sturmfels
- Convex and Combinatorial Tropical Geometry, by Josephine Yu
Schedule:
- Jan 9. 1.1, 1.4. tropical arithmetic, amoebas, log limit sets, newton polygons
- Jan 11. Attend Sam Payne’s colloquium talk
- Jan 16. 1.3, 1.7. plane curves, curve counting
- Jan 18. 1.5. implicitization
- Jan 23. 2.1. fields and valuations
- Jan 25. 2.2. varieties (HW1 due)
- Jan 30. 2.2. varieties
- Feb 1. 2.3 polyhedral geometry
- Feb 6. 2.4 Gröbner bases
- Feb 8. 2.5 Gröbner complexes (HW 2 due)
- Feb 13. 2.6 tropical bases
- Feb 15. 3.1 hypersurfaces (Project proposals due)
- Feb 20. 3.1 Kapranov’s theorem
- Feb 22. 3.2 fundamental theorem (HW 3 due)
- Feb 27. 3.3 structure theorem
- Feb 29. 3.4 multiplicities and balancing
- Mar 5. 3.5 connectivity
- Mar 7. 3.5 connectivity
- Mar 12. 3.6 stable intersections
- Mar 14. polytope algebra, reality (HW 4 due)
- Mar 19. spring break
- Mar 21. spring break
- Mar 26. 4.1 hyperplane arrangements
- Mar 28. 4.2 matroids
- Apr 2. 4.3 Grassmannians
- Apr 4. Real / positive Grassmannians (HW 5 due)
- Apr 9. Project reports due (on Canvas). JY out of town. Guest lecture.
- Apr 11. student presentations
- Apr 16. student presentations
- Apr 18. student presentations
- Apr 23. student presentations (last day of classes)
- Apr 24. Referee reports due
- Apr 30. Final revisions due
Student Papers
- Griffin Edwards. Extending Lafforgue’s Theorem of Rigid Matroids.
- William Frendreiss. Complexity Results using Tropical Geometry.
- Ruiqi (Rickey) Huang. Tropical Discriminants.
- Daniel Hwang. The Ehrhart Theory of Polytropes.
- Lewis McGoldrick. A Note on Tropical Geometry overNon-Archimedean Fields.
- Akash Narayanan. Hurwitz Numbers and Tropical Covers.
- Ari Pomeranz. Tropical Discriminants Paper.
- Noah Solomon. Tropical Geometry in Matroids.
- Matthew Sumanen. j-invariant for Tropical Elliptic Curves.
- Juliet Whidden. Combinatorial Interpretations of Min-Algebra.
- Jianuo Zhou and Jiaheng Li. Syzygies and Toric Ideals.