University of Washington, Spring 2020

Math 563 -- Graphs and Tropical Curves

Instructor:

Lecture time and address:

Lecture notes:

I will upload my handwritten notes to a folder in Google Drive and share the link with you.

Description:

We will discuss some fundamental results on both finite and metric graphs (abstract tropical curves). We will also present a few open problems along the way, some of which might be within reach for strong undergraduate or graduate students.

Advanced topics might include: Metric Graphs and Tropical Curves, Chip-Firing Games, Abel-Jacobi and Riemann- Roch Theory on Graphs, Potential Theory, and maybe even some recent applications in Algebraic Geometry and Number Theory.

Course requirements:

Students will study and present topics that compliment the material presented in lectures, and will write a short expository article. Obviously, the final paper does not have to be expository, and you are welcome to solve some of the open problems that I propose instead.

I expect you to attend most classes and participate actively in class disccussions. If at some point you decide to stop coming to class, please drop the course.

Projects:

• Approaches to Stanley's Conjecture
  (Peter Gylys-Colwell, Amzi Jeffs)


• Ramsey Theory: Unavoidable patterns
  (Brian Nugent)


• Demonstration of Aldous-Broder and Wilson's algorithm to find a uniform spanning tree of a graph
  (Rahul Chandra)


• Electrical Networks and Random Walks
  (Grandom Greaton, Nikhil Pandya)


• The Tutte Polynomial
  (Grant Robinson)


• Bijections, Bernardi, and Burning: Oh My!
  (Cameron Wright)


• Simplicial complexes with integral Laplacian spectra
  (Lei Xue)


• Vector Bundles on Graphs and the Riemann Roch Theorem
  (Junaid Hasan, Josh Southerland)


• A Brief Introduction to Spectral Graph Theory
  (Catherine Babecki, Kevin Liu, Omid Sadeghi)


• Properties of the tau invariant of metric graphs
  (Victor Reis)

References:

Some books:

• Diestel, Reinhard -- Graph theory, Fifth, Graduate Texts in Mathematics, vol.173, Springer, Berlin, 2018. MR3822066
• Bollobás, Béla -- Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York, 1998. MR1633290
• Biggs, Norman -- Algebraic graph theory, Second, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. MR1271140
• Godsil, Chris; Royle, Gordon -- Algebraic graph theory, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, New York, 2001. MR1829620
• Stanley, Richard P. -- Combinatorics and commutative algebra. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. x+164 pp. MR1453579
• Lyons, Russell; Peres, Yuval -- Probability on trees and networks. Cambridge Series in Statistical and Probabilistic Mathematics, 42. Cambridge University Press, New York, 2016. xv+699 pp. MR3616205

Some papers:

• Crapo, Henry H. -- The Tutte polynomial. Aequationes Math. 3 (1969), 211--229. MR0262095
• Björner, Anders; Lovász, László; Shor, Peter W. -- Chip-firing games on graphs. European J. Combin. 12 (1991), no. 4, 283--291. MR1120415
• Bacher, Roland; de la Harpe, Pierre; Nagnibeda, Tatiana -- The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France 125 (1997), no. 2, 167--198. MR1478029
• Biggs, Norman -- Algebraic potential theory on graphs. Bull. London Math. Soc. 29 (1997), no. 6, 641--682. MR1468054
• Baker, Matthew; Norine, Serguei -- Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215 (2007), no. 2, 766--788. MR2355607
• Baker, Matthew; Shokrieh, Farbod -- Chip-firing games, potential theory on graphs, and spanning trees. J. Combin. Theory Ser. A 120 (2013), no. 1, 164--182. MR2971705
• Stanley, Richard P. -- Hyperplane arrangements, parking functions and tree inversions. Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 359--375, Progr. Math., 161, Birkhäuser Boston, Boston, MA, 1998. MR1627378
• Mikhalkin, Grigory; Zharkov, Ilia -- Tropical curves, their Jacobians and theta functions. Curves and abelian varieties, 203--230, Contemp. Math., 465, Amer. Math. Soc., Providence, RI, 2008. MR2457739
• Baker, Matthew; Faber, Xander -- Metrized graphs, Laplacian operators, and electrical networks. Quantum graphs and their applications, 15--33, Contemp. Math., 415, Amer. Math. Soc., Providence, RI, 2006. MR2277605
• de Jong, Robin; Shokrieh, Farbod -- Metric graphs, cross ratios, and Rayleigh's laws, 2018. Preprint available on arXiv:1810.02638.

Disability Resources and Accommodations:

Reasonable accommodations will be provided to students who have a documented disability that may affect their ability to participate in course activities or to meet course requirements. Students with disabilities are encouraged to contact the office of Disability Resources for Students (DRS).

Religious Accommodations::

Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW's policy, including more information about how to request an accommodation, is available at Religious Accommodations Policy. Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form.

Academic Integrity:

Each student in this course is expected to abide by the University of Washington's code of academic integrity. Any work submitted by a student in this course for academic credit will be the student's own work. More information about this code is available here.

Intellectual Property and Copyright:

All course materials are intellectual property belonging to the author. Students are not permitted to share any course materials without the expressed permission of the instructor.