The moduli space of cubic surfaces and its compactifications are classical and date back to the mid-nineteenth century. While recent progress has been made in describing compactifications of moduli spaces for fully marked cubic surfaces using Kollár-Shepherd-Barron-Alexeev (KSBA) stable pairs with uniform weights, this talk explores an extension into asymmetric weights.
In particular, we investigate the KSBA compactification of moduli spaces of cubic surfaces with a single marked line by considering an asymmetric weight on one line and a uniform weight on the remaining 26 lines. We provide an explicit wall-and-chamber decomposition of the weight domain, giving 20 distinct chambers. These chambers yield new KSBA coarse moduli spaces that reveal how interactions between the marked line and cubic surface singularities give new wall crossings. We also give explicit descriptions of these weighted stable pairs parameterized by the moduli spaces in each chamber.
I will discuss a famous 40-year-old conjecture from string theory known as the S-duality modularity conjecture. It predicts that a certain partition function encoding the count of stable solutions to the partial differential equations describing D-brane interactions, supported on complex surfaces deforming inside a Calabi-Yau threefolds is given by a modular form. Depending on how these surfaces deform in the ambient Calabi-Yau threefold, one obtains different counting problems and correspondingly different versions of the S-duality conjecture. I will explain an algebro-geometric reformulation of this problem and survey a series of results obtained with collaborators over the past 15 years toward proving the conjecture in various geometric settings. Finally, I will describe ongoing work on the most difficult version of the conjecture, which involves tools such as Tyurin degeneration, derived intersection theory, and the categorification of Donaldson-Thomas invariants.
Rational neural networks are feedforward neural networks with a rational activation function. These networks found their applications in approximating the solutions of PDE, as they are able to learn the poles of meromorphic functions. In this talk, we are going to consider the simplest rational activation function, sigma = 1 / x, and study the geometry of family such architectures. We will show that the closure of all possible shallow (one hidden layer) networks is an algebraic variety, which called a neurovariety.
The moduli space of pointed rational curves has a natural action of the symmetric group permuting the marked points. In this talk, I will present combinatorial and recursive formulas for the induced representation on the cohomology of the moduli space. These formulas are derived from wall crossings of birational models, governed by Hassett’s theory of weighted stable curves and Choi-Kiem’s theory of delta-stability of quasimaps. These results allow us to investigate positivity and log-concavity of the representation. Based on joint works with Jinwon Choi and Young-Hoon Kiem.
A theory of heights of rational points on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about arithmetic counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over function fields inspired by twisted stable maps of Abramovich and Vistoli. For some well-behaved class of stacks, we obtain moduli spaces of points of fixed height whose geometry controls the number of rational points on the stack. I will outline an approach for more general stacks which is closely related to the geometry of the moduli space of vector bundles on a curve. This is based on joint work with Park and Satriano.
