** Tuesday and Thursday 1:30 - 2:45pm in G117 Tompkins Hall**

**Instructor:** Cynthia Vinzant (3260 SAS, email)

**Offce Hours:** Tuesday 3-4pm, Wednesday 1-2pm or by appointment

** Optional Textbook:** D. Maclagan and B. Sturmfels, *Introduction to Tropical
Geometry*, Graduate Studies in Mathematics, 161. American
Mathematical Society, 2015. (online draft)

**Syllabus**

**Description:** Tropical geometry is the study of certain
combinatorial shadows of solutions to systems of polynomial equations.
It is based on tropical algebra, where the sum of two numbers is their
minimum and the product is their sum. This turns polynomials into
piecewise-linear functions, and their zero sets into polyhedral
complexes. These combinatorial structures retain a surprising amount
of information about their classical counterparts. This course will
introduce and survey some topics in tropical geometry, including
Puiseux series and valuations, Grobner complexes, tropical varieties,
hyperplane arrangements and matroids, tropical convexity, connections
with toric varieties, Bernstein's theorem, and Viro's patchworking.
Grades will be based on occasional homework assignments and a final
project.

** Homework**

Homework 1 (.pdf .tex) due Thursday, January 28

Homework 2 (.pdf .tex) due Thursday, February 11

Homework 3 (.pdf .tex) due Thursday, February 25

Homework 4 (.pdf .tex) due Tuesday, April 5

**Final Project Information**

Project choice by March 4

Presentations April 12, 14, 19, 21

Final paper due April 29, 5pm

**Presentation Schedule**

(April 12) Faye Pasley: The Tropical Commuting Variety

(April 12) Georgy Scholten: The ultimate rank of tropical matrices

(April 14) Dan Bernstein: Subdominant Matroid Ultrametrics

(April 14) Chris Paquette: The Newton polytope of the implicit equation

(April 19) Naomi Boulware: Product Mix Auctions and Tropical Geometry

(April 19) Mike Weselcouch: Discrete concavity and the half-plane property

(April 21) Alex Hazeltine: Bitangents of tropical plane quartic curves

(April 21) Caprice Stanley: Tropical Curves in Sandpiles

** Extra reading **

What is an
Amoeba? by Oleg Viro

** Software **

Singular online

Gfan

polymake