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.
Grades will be based on weekly homework sets (50%)
and the final term paper (50%).
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.
939 Evans is reserved
on Fridays, 12:00-2:00pm,
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.
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
2:10 Qiao Zhou: Toric Connections I
2:50 Justin Chen: Toric Connections II
2:10 Bryan Wang: Tropical Convexity and Tree Space
2:50 Bo Lin: Fermat-Weber Points
2:10 Ashwin Iyengar: Tropical Surfaces and Manifolds
2:50 Thomas Blomme: The Correspondence Theorem
2:10 Charlie Reid: Quartic Surface Polytopes
2:50 Liz Ferme: Hyperbolic Polynomials
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
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
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