Georgia Institute of Technology College of Sciences

First of two talks.

In Bernoulli percolation, we place edges independently with probability p between vertices of a lattice, and study the resulting random geometry. In the first talk, I will give a friendly introduction to the model and discuss some of the motivating questions and the history of methods in the high dimensional setting. In the second talk, I will discuss our recently developed tool, the incipient infinite cluster, and describe how we use it to derive the exact limiting distribution of distances within clusters, as well as prove sharp asymptotics for k-point connectivity functions. These results together establish several aspects of the conjecture that the scaling limit of high dimensional critical clusters is the Integrated Super-Brownian Excursion. Based on joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe.

The Conway knot C is an 11-crossing knot that bounds a locally flat disk in B^4. After many decades of unknown status, C was proven to not bound a smoothly embedded disk by Piccirillo. In this talk I will show that in fact, no sum of Conway knots bounds a smoothly embedded disk. This is joint work with Marco Golla (and independent of work of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.)

Ptychographic phase retrieval is an ill-posed nonlinear inverse problem typically formulated as large-scale nonconvex optimization. I will present stochastic multigrid methods, beginning with the known-probe case as a foundation for blind ptychography. The approach uses quadratic surrogate models that majorize the data-misfit objective, together with compatible coarse-grid models that accelerate surrogate minimization. We extend this construction to blind ptychography, where the object and probe must be recovered jointly. Geometric-mean updates coordinate changes to the object and probe within each scan, while intensity-weighted minibatch synthesis combines the resulting local updates across scan positions. I will discuss the mathematical analysis and present results using synthetic data and experimental X-ray measurements.

We are going to discuss recent developments in the study of the local asymptotic behavior of the second correlation function of characteristic polynomials for several important examples of non-Hermitian random matrices including adjacency matrices of weighted Erdos-Renyi graphs and  band matrices, where non-zero entries are present only in the narrow strip around the main diagonal. The talk is based on the joint papers with Ievgenii Afanasiev and Maria Shcherbina.