Course Announcement - Spring 2021

Math 274: Tropical Geometry

Instructor: Bernd Sturmfels

Office hours: e-mail me to set up an appointment.

Meeting times: Mondays, Wednesdays and Fridays, 9:00-9:50, on zoom

First Day: Wednesday, January 20.
Last Day: Wednesday, April 28.

Prerequisites: Basics of commutative algebra, combinatorics and algebraic geometry.

Description: Tropical geometry is algebraic geometry over the min-plus algebra. This young subject has both established itself as an area of its own right and unveiled its deep connections to numerous branches of pure and applied mathematics. One passes from algebraic geometry to combinatorics, by replacing algebraic varieties over a valued field with polyhedral complexes. This process retains much information about the original varieties. This course offers a systematic introduction to this subject.

Course text: Introduction to Tropical Geometry by Diane Maclagan and Bernd Sturmfels, Graduate Studies in Math, AMS, 2015.

Syllabus: All six chapters: 1 Tropical Islands, 2 Building Blocks, 3 Tropical Varieties, 4 Tropical Rain Forest, 5 Tropical Garden, 6 Toric Connections.

       Click here for a book review by Michael Joswig, published in Jahresbericht der DMV. You can find lots of resources on tropical geometry online.
       I recommend the books Essentials of Tropical Combinatorics by Michael Joswig and Tropical Geometry by Grigory Mikhalkin and Johannes Rau.

FORMAT: You watch one recorded lecture on or before each weekend. E-mail me your questions (if any). We discuss that lecture on Monday.
       On Wednesday there is a live lecture on other topics from the same chapter. On Friday there is a discussion of exercises from that chapter.

Homework: You solve at least six problems from the exercises in each chapter. Since there are six chapters, this makes 36 exercises.
       Collaboration welcome. Submission by e-mail. The deadlines are six Sundays. They are listed below. No late homework, please.

Grading: We will count your homework, but also other contributions. You might give a lecture or lead an exercise session. Remember: this is an advanced graduate class. You set your own goals. You decide whether you reached them. The professor and peers are your resources.

Jan 20: Basics and Welcome
Jan 22: Introductions and Software
Jan 25: Discussion of recorded lecture # 1
Jan 27: 1.5 Implicitization
Jan 29: Exercises from Chapter 1 (led by Yelena Mandelshtam)
Feb 1: Discussion of recorded lecture # 2
Feb 3: 2.3 Polyhedral Geometry
Feb 5: Exercises from Chapter 1 (led by Kyle Huang)
Feb 7: Homework due for Chapter 1
Feb 8: Discussion of recorded lecture # 3
Feb 10: 2.6 Tropical Bases
Feb 12: Exercises from Chapter 2 (led by Mara Belotti)
Feb 15: No class
Feb 17: Discussion of recorded lecture # 4
Feb 19: Exercises from Chapter 2 (led by Eunice Sukarto)
Feb 21: Homework due for Chapter 2
Feb 22: Discussion of recorded lecture # 5
Feb 24: 3.5 Connectivity and Fans
Feb 26: Exercises from Chapter 3 (led by Yassine El Maazouz)
Mar 1: Discussion of recorded lecture # 6
Mar 3: 3.6 Stable Intersection (presented by Maddie Weinstein)
Mar 5: Exercises from Chapter 3 (led by Chiara Meroni)
Mar 7: Homework due for Chapter 3
Mar 8: Discussion of recorded lecture # 7
Mar 10: 4.1 Hyperplane Arrangements (presented by Theodore Coyne)
Mar 12: Exercises from Chapter 4 (led by Mariel Supina)
Mar 15: Discussion of recorded lecture # 8
Mar 17: 4.2 Matroids (presented by Andy Zhang)
Mar 19: Exercises from Chapter 4 (led by Shelby Cox)
Mar 28: Homework due for Chapter 4
Mar 29: Discussion of recorded lecture # 9
Mar 31: 5.1 Eigenvalues and Eigenvectors (presented by Eric Jankowski)
Apr 2: Exercises from Chapter 5
Apr 5: Discussion of recorded lecture # 10
Apr 7: 5.4 Arrangements of Trees (presented by Alexander Mcdonough)
Apr 9: Exercises from Chapter 5 (led by Yulia Alexandr)
Apr 11: Homework due for Chapter 5
Apr 12: Discussion of recorded lecture # 11
Apr 14: 6.4 Tropical Compactifications (presented by Yifan Chen)
Apr 16: Exercises from Chapter 6 (led by Adam Dhillon)
Apr 19: Discussion of recorded lecture # 12
Apr 21: 6.6 Degenerations (presented by Juliette Bruce)
Apr 23: Exercises from Chapter 6 (led by Hongyi Liu)
Apr 25: Homework due for Chapter 6
Apr 26: 6.7 Intersection Theory (presented by Xianlong Ni)
Apr 28: Madeline Brandt speaks on Tropical Geometry of Curves