Stability Conditions for Algebraic Curves
Directed by Dr. Sebastian Bozlee
If one starts with a smooth algebraic curve (something like (x^2 + y^2 – 1)x = 1) and continuously changes its equations, it eventually breaks into pieces that are not smooth (something like (x^2 + y^2 – 1)x = 0). How exactly it breaks down is not unique: for a given degeneration there are many choices of singular limit. For example, some limits might only have “nodes” (xy = 0) while others might have “cusps” (y^2 = x^3). Algebraic geometers are interested in finding coherent choices of singular limits called “stability conditions”: each one gives rise to a “moduli space” of curves which helps us understand how algebraic curves are related to each other.
We will not use algebraic geometry to approach this problem. Instead we will approach it by studying combinatorial data on piecewise linear “tropical curves.”
One kind of combinatorial datum is that of an “extremal assignment.” These date back to 2010, but their combinatorics are only well-studied in genus zero. A more recent kind of combinatorial datum is that of a “universal mesa,” which are only well-understood in genus one.
We will start by learning some background on tropical curves. Once we’re comfortable with basic examples, we will search for more families of extremal assignments and universal mesas in a two pronged fashion.
The first prong involves creating and extending code to programmatically search for extremal assignments and universal mesas. These are large searches and will benefit from some clever coding.
The second prong involves developing theory to understand the outputs of our programs and to discover new families of extremal assignments and universal mesas.
Not required, but desirable: some abstract algebra (finitely generated, torsion-free abelian groups), some experience with basic combinatorial notions and graphs, and some programming skills (especially Python and/or C/C++).