##### Brill-Noether existence on finite graphs

The Brill-Noether Existence Theorem says that every smooth projective algebraic curve of genus g over an algebraically closed field has a divisor of degree d and rank at least r, provided that g is greater than or equal to (r+1)(g-d+r). Tropical geometry gives an intricate dictionary of analogies and implications between the algebraic geometry of divisors on algebraic curves and the combinatorial geometry of divisors on graphs and, based on this analogy, the Brill-Noether existence conjecture (posed by Matthew Baker in 2008) predicts that every graph of genus g should have a divisor of degree d and rank r. This simple combinatorial conjecture remains unproved, even in small cases. Participants in this project will learn about the theory of ranks of divisors on graphs, develop techniques for proving the conjecture in special cases by day, and search for counterexamples by night. (Although this project is inspired by classical Brill-Noether theory, no prior background in algebraic geometry is required.)

##### Tropical line arrangements

Tropical geometry studies combinatorial, piecewise linear shadows of objects from algebraic geometry, sometimes in surprising ways. For instance, the most common tropical method for studying arrangements of lines in the plane is to remove the lines and tropicalize their complement. Participants in this project will pursue a more naive approach to the tropical geometry of arrangements of lines, studying the tropicalizations of the lines themselves. Possible research directions include the combinatorial classification of tropicalizations of Fano arrangments in the projective plane over F_2((t)), tropical analogues of they Sylvester-Gallai Theorem, which says that given any configuration of at least three points in R^2, not all collinear, there is at least one line that passes through exactly two points, and the Guth-Katz Theorem, which bounds the minimal number of distinct distances between N points in the plane.

##### Cones of divisors

In our calculus classes we frequently use the geometry of curves and surfaces defined by polynomials to get insight into the solutions to given problems. In algebraic geometry we study higher dimensional generalizations of these curves and surfaces in search of geometric insight that can be applied to theoretical mathematics as well as to physics, engineering and computer science. Studying these higher dimensional analogs of curves and surfaces requires careful study of several invariants, one of which is called the cone of divisors. It turns out that elementary arguments together with some good intuition and creativity can lead to the description of some unknown cones of divisors of interesting geometric objects. This project involves extensive use of computational linear algebra, while the required basic algebraic geometry can be learned during SUMRY.

##### Orbits on modular character varieties

In this project we will investigate a connection between group theory and number theory. We will learn about an action of GL_2(Z) on the solutions to the Markoff diophantine equation x^2 + y^2 + z^2 = 3xyz over the finite field F_p, p prime. We’ll be interested as a starting point in the orbits of a single fixed element of GL_2(Z) as p varies. There are many interesting variants of this problem to be investigated numerically and theoretically.

##### Strategies in bidding games

Bidding games combine aspects of combinatorial games and economic games by having players bid for the right to move next, instead of alternating moves, while playing a traditional two player game such as tic-tac-toe, connect four, or chess. The goal, of course, is to win the game, but the optimal strategies change wildly when one player can potentially make several moves in a row. Traditional bidding games via first price auction, where both players bid and then the high bidder pays and makes a move, are relatively well studied, but even minor variations on this structure are largely unexplored. Participants in this project may explore bidding versions of multiplayer games, strategies and structure theory for bidding games with different auction structures, such as all-pay bidding, or variations in which players can bid on different packages of moves.