Questions on Homework etc: Fridays 12:00-2:00pm (in 939 Evans)

Last Day of Class: Thursday, April 28.

Term Papers Due: Thursday, May 12.

**Description:**
Tropical geometry is algebraic geometry over the min-plus algebra.
It is a young subject that in recent years has both established itself
as an area of its own right and unveiled its deep connections to numerous
branches of pure and applied mathematics. From an algebraic geometric point
of view, algebraic varieties over a field with non-archimedean valuation are
replaced by polyhedral complexes, thereby retaining much information
about the original varieties.
This course offers a first introduction to tropical geometry.

by Diane Maclagan and Bernd Sturmfels, Graduate Studies in Mathematics, AMS, 2015.

**Grading:**
Grades will be based on weekly homework sets (50%)
and the final term paper (50%).

**Homework:**
There will be a weekly homework assignment,
to be handed in on Tuesdays.

Late homework will not be accepted.
The assignments, posted below, refer to the text book.

No homework after spring break, so you can work on the term papers.

**Tropical Fridays:**
939 Evans is reserved
on Fridays, 12:00-2:00pm,
for discussions.

This provides an informal setting to
interact and collaborate with other students.

**Final Project:** Students will write a term paper on a topic
related to the class.
Papers by two authors are as welcome as

single-authored papers.
A written proposal for your project is due on
Tuesday, March 15. You will be invited to give a

lecture in
the second half of April. The final version of your paper is due on
Thursday, May 12. Hard copies much preferred.

** DAILY SCHEDULE: **

Jan 19: 1.3 Plane Curves, 1.4 Amoebas and Their Tentacles

Jan 21: 1.5 Implicitization, 1.7 Curve Counting, 1.8 Compactifications

Jan 26: 2.3 (JK) Polyhedral Geometry, (MB) the software
polymake

Jan 28 (AS): 2.1 Fields, 2.2 Algebraic Varieties

Feb 2: 2.4 Gröbner Bases

Feb 4: 2.5 Gröbner Complexes, 2.6 Tropical Bases

Feb 9: 3.1 Hypersurfaces

Feb 11: 3.2 The Fundamental Theorem, 3.3 The Structure Theorem

Feb 16: 3.4 Multiplicities and Balancing

Feb 18: 3.5 Connectivity and Fans,
the software
Gfan

Feb 23: 4.1 Hyperplane Arrangements, 4.2 Matroids

Feb 25: 4.3 Grassmannians

Mar 1: 4.4 Linear Spaces

Mar 3: 4.5 Surfaces

Mar 8: 5.2 Tropical Convexity

Mar 10: 5.3 The Rank of a Matrix

Mar 15: 5.1 Eigenvalues and Eigenvectors

Mar 17: 3.6 Stable Intersections

Mar 29: (JK) 5.5 Monomials in Linear Forms

Mar 31: (ET) Eigenvectors of Tropical Tensors

Apr 5: 4.6 Complete Intersections

** Student presentations: **

April 7:

2:10 Qiao Zhou: Toric Connections I

2:50 Justin Chen: Toric Connections II

April 12:

2:10 Bryan Wang: Tropical Convexity and Tree Space

2:50 Bo Lin: Fermat-Weber Points

April 14:

2:10 Ashwin Iyengar: Tropical Surfaces and Manifolds

2:50 Thomas Blomme: The Correspondence Theorem

April 19:

2:10 Charlie Reid: Quartic Surface Polytopes

2:50 Liz Ferme: Hyperbolic Polynomials

April 21:

2:10 Lynn Chua: Elliptic Curves

2:50 Brandon Williams: Divisors and the Riemann-Roch Theorem

April 22 (939 Evans):

12:00 Julio Soldevilla: The Tropical Positive Grassmannian

12:40 Albert Zheng: Dynamic Programming

1:20 Eric Chen: Tropical Varieties in Representation Theory

April 28:

2:10 Madeline Brandt: Curves of Genus 2

2:50 Christopher Eur: Curves of Genus 3

April 29 (939 Evans):

12:00 Sophia Elia: Polytropes

12:40 Shiyu Li: Tropical Linear Programming

1:20 Nishant Pappireddi: Tropical Semiring and Newton Polytopes

** Homework assignments: **

due Jan 26: Chapter 1: 8,9,11,16,17,18,24,25,26,29

due Feb 2: Chapter 2: 2,3,7,8,10,11,27,28

due Feb 9: Chapter 2: 14,15,20,22,23,25,26

due Feb 16: Chapter 3: 1,2,4,9,11,13,14,16

due Feb 23: Chapter 3: 15,21,22,23,24,31,33,34

due Mar 1: Chapter 4: 2,4,7,12,13,15,17,22

due Mar 8: Chapter 4: 21,23,24,26,27,28,30

due Mar 15: Chapter 5: 3,6,7,8,9,10,14,16,17