This semester we’ll have talks on two themes i. Tropical curves and their divisor theory, ii. Green’s conjecture on syzygies of canonical curves. We’ll start with the basics of tropical part (about 2-3 weeks) and then move to basics of Green’s conjecture part (2-3 weeks). We’ll then move on to more advanced aspects of both these topics.

This semester (probably the next) we plan to focus on aspects of syzygies: combinatorial, algebraic, geometric. We’ll devote the first few lectures to the basics of syzygies.

In the future, we plan to cover various aspects of the interplay between Combinatorics and Commutative Algebra-Algebraic Geometry. Some possible topics are

i. Syzygies: connections to Combinatorics, Syzygies of curves particularly Green’s conjecture and the related conjecture for Graph Curves, Boij-Soederberg theory, Syzygies in Tropical geometry.

ii. Vector Bundles with potential connections to Tropical Geometry.

iii. Dessins D’enfants:

iv. Brill-Noether theory on Algebraic curves and Tropical curves, Combinatorial Aspects of Moduli Spaces.

v. Non-archimedean Geometry in connection with Tropical Geometry, particular Berkovich Spaces,

vi. Combinatorics and Hodge theory, Vector Partition Functions.

vii. Ideals of Powers of Linear Forms.

viii. Tutte Polynomials of Graphs, Zeta Functions associated to Graphs.

ix. Syzygies in Real Algebraic Geometry.

x. Tropical Geometry in Dimensions Two and Higher.

We plan to pick a theme every semester followed by talks by local faculty and students. These talks will be introductory and will assume very little background. We will then invite speakers from outside for more specialized talks on this theme.