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:

Other books or lecture notes:

Survey or expository papers:

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