Math 189: Tropical Geometry

Spring 2013


In this course we will study an introduction to Tropical Geometry, including tropical arithmetic and linear algebra, tropical varieties, tropicalization, and other assorted topics and applications.


Dagan Karp
Office hours: TBA, and open door.
Office Location: Olin 1267


We will use Introduction to Tropical Geometry, by Diane Maclagan and Bernd Sturmfels.


Homework and classwork will account for 70% of the course grade, with the remaining 30% determined by a final project.


Homework will be due in class on Tuesdays. Please consult the HMC mathematics homework format guidelines for helpful tips on homework submission and formatting.


Students should use LaTeX for homework, but hand drawn figures are fine.


Students who need disability-related accommodations are encouraged to discuss this with the instructor as soon as possible.

Honor Code

Though cooperation on homework assignments is encouraged, students are expected to write up their own solutions individually. That is, no copying. Comprehension is the goal, so even with cooperation, you should understand solutions well enough to write them up yourself. It is appropriate to acknowledge the assistance of others; if you work with others on a homework question, please write their names in the margin. The HMC Honor Code applies in all matters of conduct concerning this course.

Homework Assignments

All HW's refer to Introduction to Tropical Geometry, unless noted otherwise.

  • HW #1. Due Tuesday, Jan 29.

    1. Let p be a tropical polynomial in n variables. Prove that, as a function from R^n to R, p is continuous, piecewise linear with a finite number of pieces, and concave.
    2. What conditions on the coefficients of a tropical quadratic in a single variable ensure its graph has 3 pieces? Also, give an example of such and graph it.
    3. Problem 1.9.1.
    4. Problem 1.9.2.

  • HW #2. Due Tuesday, Feb 5.

    Read Section 1.3.
    Problems 1.9.3, 1.9.4, 1.9.5, 1.9.6.

  • HW #3. Due Tuesday, Feb 12.

    Read Section 1.4.
    Problems 1.9.7, 1.9.9.

  • HW #4. Due Thursday, Feb 21.

    Read Section 2.1.
    Problems 2.7.1, 2.7.2.

  • Due Thursday, Mar 7.

    Choose topic for final project.

  • HW #6. Due Thursday, Mar 14.

    Read Section 2.3.
    Problems 2.7.7, 2.7.8, 2.7.9.

  • HW #7. Due Thursday, Apr 4.

    Read Section 3.1.
    Problems 2.6.1 (a)-(f)

  • No Class Thursday, Apr 25.

  • FINAL PROJECT. Due Thursday, May 2.

    Oral Presentation: 12 minutes + Q&A = 15 min total
    Paper: No more than 5 pages. (Note that 5 LaTeX pages is much more than a 12 minute talk.)