Georgia Institute of Technology College of Sciences

This talk was cancelled due to the current status. The following is the original abstract for the talk. The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves.

A math-themed variety show including music, improv comedy, a poetry slam, juggling, a fashion show (audience members can join in)  and more, right there on the stage of the fabulous Highland Ballroom!   Tickets  are $10.00.

Cancelled due to COVID-19

The pure spherical p-spin model is a Gaussian random polynomial H of degree p on an N-dimensional sphere, with N large. The sphere is viewed as the state space of a physical system with many degrees of freedom, and the random function H is interpreted as a smooth assignment of energy to each state, i.e. as an energy landscape. 

In 2012, Auffinger, Ben Arous and Cerny used the Kac-Rice formula to count the average number of critical points of H having a given index, and with energy below a given value. This number is exponentially large in N for p > 2, and the rate of growth itself is a function of the index chosen and of the energy cutoff. This function, called the complexity, reveals interesting topological information about the landscape H: it was shown that below an energy threshold marking the bottom of the landscape, all critical points are local minima or saddles with an index not diverging with N. It was shown that these finite-index saddles have an interesting nested structure, despite their number being exponentially dominated by minima up to the energy threshold. The total complexity (considering critical points of any index) was shown to be positive at energies close to the lowest. Thus, at least from the perspective of the average number of critical points, these random landscapes are very non-convex. The high-dimensional and rugged aspects of these landscapes make them relevant to the folding of large molecules and the performance of neural nets. 

Subag made a remarkable contribution in 2017, when he used a second-moment approach to show that the total number of critical points concentrates around its mean. In light of the above, when considering critical points near the bottom of the landscape, we can view Subag's result as a statement about the concentration of the number of local minima. His result demonstrated that the typical behavior of the minima reflects their average behavior. We complete the picture for the bottom of the landscape by showing that the number of critical points of any finite index concentrates around its mean. This information is important to studying associated dynamics, for instance navigation between local minima. Joint work with Antonio Auffinger and Yi Gu at Northwestern.