Georgia Institute of Technology College of Sciences

For the classical actions of the linear algebraic groups in characteristic zero (special linear, orthogonal, symplectic) we calculate the arithmetic rank of Hilbert's nullcone ideals via a general theorem on pure subrings. Then, in positive charactersitic, we discuss topological methods that lead to an inequality between arithmetic rank of the nullcone ideal and the dimension of the invariant ring, that can be used to replace the argument of the pure subring (which is false in positive characteristic). In the outline of his argument we spend most of the time over the complex numbers, for better comfort. This is a report on a paper with J Jeffries, V Pandey and A K Singh.

In this talk, I will show how tools from VC-dimension theory can be used to address questions from incidence geometry, enumerating intersection graphs, and graph drawing.

The bi-adjacency matrix of an Erdős–Rényi random bipartite graph with bounded aspect ratio is a rectangular random matrix with Bernoulli entries. Depending on the sparsity parameter $p$, its spectral behavior may either resemble that of a classical Wishart matrix or depart from this universal regime. In this talk, we study the extreme singular values at the critical density $np=c\log n$. We present the first quantitative characterization of the emergence of outlier singular values outside the Marčenko–Pastur law and determine their precise locations as functions of the largest and smallest degree vertices in the underlying random graph, which can be seen as an analogue of the Bai–Yin theorem in the sparse setting. These results uncover a clear mechanism by which combinatorial structures in sparse graphs generate spectral outliers. Joint work with Ioana Dumitriu, Haixiao Wang and Zhichao Wang.

Dalton, Etnyre, and Traynor classified Legendrian cable links when the companion knot is both uniformly thick and Legendrian simple, and Etnyre, Min, and Chakraborty classified all cable knots of uniformly thick knots. Using convex surfaces, we build on these results to classify cable links of knots in $(S^3, \xi_\text{std})$ that are uniformly thick but not Legendrian simple, and address new questions that arise from their nonsimplicity. This is joint work with Rima Chatterjee, John Etnyre, and Hyunki Min.