Tropical Geometry Seminar
Mathematics Department
University of Georgia
Fall 2009 - Spring 2010
Tuesdays 3:30 - 4:30, Boyd Grad Studies room 410
Organizers: Angela Gibney and Clint McCrory
Useful links:
Wikipedia - tropical geometry
A list of introductory and expository references is included.
Bernd Sturmfels' course on tropical geometry, Spring 2009
Current draft of the Maclagan-Sturmfels book: Introduction to Tropical Geometry (updated November 4, 2009)
The plan of our seminar is to work through these notes in the spring semester.
MSRI tropical geometry program, Fall 2009
Notes and videos of the lectures of all three MSRI workshops in the program are available from MSRI streaming video lectures:
- Introductory Workshop: Tropical Geometry (8/24-8/28)
- Tropical Geometry in Combinatorics and Algebra (10/12-10/16)
- Tropical Structures in Geometry and Physics (11/30-12/4),
Schedule of lectures:
Tuesday, April 27- no meeting (department meeting)
Tuesday, April 20 - Josephine Yu (Georgia Tech), "Linear systems on tropical curves"
A tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as an embedded polyhedral complex. We also show that curves for which the canonical divisor is not very ample are hyperelliptic. This is joint work with Christian Haase and Gregg Musiker.
Tuesday, April 13 - Benjamin Nill, "Tropical varieties"
Maclagan-Sturmfels, chapter 3, section 1
Tuesday, April 6 - no meeting (Cantrell Lectures)
Tuesday, March 30 - Benjamin Nill, "Tropical bases"
Maclagan-Sturmfels, chapter 2, section 5
Tuesday, March 23 - no meeting
Tuesday, March 16 - Benjamin Nill, "Tropical bases"
Maclagan-Sturmfels, chapter 2, section 5
Tuesday, March 9 - no meeting, spring break.
Tuesday, March 2 - cancelled for snow!
Tuesday, February 23 - Mike Usher,"The Gröbner complex"
Maclagan-Sturmfels, chapter 2, section 4
Tuesday, February 16 - Mike Usher, "Gröbner bases"
Maclagan-Sturmfels, chapter 2, section 4
Tuesday, February 9 - Clint McCrory, "Polyhedral geometry"
Maclagan-Sturmfels, chapter 2, section 3
Tuesday, February 2 - Danny Krashen, "Fields and valuations; algebraic geometry"
Maclagan-Sturmfels chapter 2, sections 1and 2.
Tuesday, January 26 - Danny Krashen, "Fields and valuations"
Maclagan-Sturmfels chapter 2, section 1: Definition of valuations, the field of Puiseux series.
Tuesday, January 19 - meeting cancelled
Tuesday, January 12 - Danny Krashen, "Introduction to tropical geometry"
Topics from chapter 1 of the Maclagan-Sturmfels book.
Tuesday, December 1 - Mike Usher and Robert Varley, "Amoebas and moment maps"
Tuesday, November 17 - Angela Gibney, "The tropical Grassmannian," part 2.
Tuesday, November 3 - Angela Gibney, "The tropical Grassmannian"
Tuesday, October 13 - Angela Gibney, "Embedding $\bar{M}_{0,n}$ into a toric variety using tropical geometry"
Tuesday, September 15 - Clint McCrory, "Amoebas of plane curves"
Legendre transform revisited. Multiplicities of tropical curves and the balancing condition. Classical amoebas (Gelfand-Kapranov-Zelevinsky) - definition and examples.
Tuesday, September 8 - Clint McCrory, "Introduction to tropical plane curves"
The tropicalization of a complex curve X = { f(z,w) = 0 } is a planar graph Γ that's a "combinatorial shadow" of X. Definition and simple examples of tropicalization; relation to Legendre transform.