Professor Po-Ling Loh will talk about; 'Private estimation in stochastic block models'

We study the problem of private estimation for stochastic block models, where the observation comes in the form of an undirected graph, and the goal is to partition the nodes into unknown, underlying communities. We consider a notion of differential privacy known as node differential privacy, meaning that two graphs are treated as neighbors if one can be transformed into the other by changing the edges connected to exactly one node. The goal is to develop algorithms with optimal misclassification error rates, subject to a certain level of differential privacy.

We present several algorithms based on private eigenvector extraction, private low-rank matrix estimation, and private SDP optimization. A key contribution of our work is a method for converting a procedure which is differentially private and has low statistical error on degree-bounded graphs to one that is differentially private on arbitrary graph inputs, while maintaining good accuracy (with high probability) on typical inputs. This is achieved by considering a certain smooth version of a map from the space of all undirected graphs to the space of bounded-degree graphs, which can be appropriately leveraged for privacy. We discuss the relative advantages of the algorithms we introduce and also provide some lower-bounds for the performance of any private community estimation algorithm.

This is joint work with Laurentiu Marchis, Ethan D'souza, and Tomas Flidr.

Symplectic capacities are symplectic invariants that measure the “size” of symplectic manifolds and are designed to capture phenomena of symplectic rigidity.

In this talk, I will focus on symplectic capacities of fiberwise convex domains in cotangent bundles. This setting provides a natural link to the systolic geometry of the base manifold. I will survey current results and discuss the variety of techniques used to compute symplectic capacities, ranging from billiard dynamics to pseudoholomorphic curves and symplectic homology. I will illustrate these techniques using disk tangent bundles of ellipsoids as an example.

Professor Colbrook is going to talk about: 'A Computational Framework for Infinite-Dimensional Nonlinear Spectral Problems' 

Nonlinear spectral problems -- where the spectral parameter enters operator families nonlinearly -- arise in many areas of analysis and applications, yet a systematic computational theory in infinite dimensions remains incomplete. In this talk, I present a unified framework based on a solve-then-discretise philosophy (familiar, for example, from Chebfun!), ensuring that truncation preserves convergence. The setting accommodates unbounded operators, including differential operators with spectral-parameter-dependent boundary conditions. 
In the first part, I introduce a provably convergent method for computing spectra and pseudospectra under the minimal assumption of gap-metric continuity of operator graphs -- the weakest natural setting in which the resolvent norm remains continuous. 
In the second part, I develop a contour-based framework for discrete spectra of holomorphic operator families, with a complete analysis of stability, convergence, and randomised sketching based on Gaussian probes. This perspective unifies and extends many existing contour integral methods. Examples throughout highlight practical effectiveness and subtle phenomena unique to infinite dimensions, including the perhaps unexpected sensitivity to probe selection when seeking to avoid spectral pollution.

I will discuss an ongoing joint work with Luigi Appolloni and Andrea Malchiodi concerning the above objects. Minimal surfaces are critical points of the area functional, which is analytic in this case, so we should expect critical points (minimal surfaces) to be either isolated or to belong to smooth nearby minimal foliations. On the other hand, the flat plane of multiplicity two in $\mathbb{R}^3$ can be (in compact regions) approximated by a blown-down catenoid, which will converge back to the plane with multiplicity two in the limit. Hence a plane of multiplicity two cannot be thought of as being isolated, or belonging solely to a smooth family, because there are “nearby” minimal surfaces of distinct topology weakly converging to it. We will nevertheless prove that, when the ambient manifold is closed and analytic, this type of local degeneration is impossible amongst closed and embedded minimal surfaces of bounded topology: such surfaces, even with multiplicity are either isolated or belong to smooth families of nearby minimal surfaces.