Cambridge

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.

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.

In 1984 Cannon showed that cocompact discrete hyperbolic groups have finitely many cone types. In this talk, I will demonstrate how this result can be extended to non-positively curved k-fold triangle groups. I will further show how this implies that such groups have an automatic structure and how we can use this information to construct top dimensional l^2 cycles.

I will discuss some recent progress on the Donaldson Segal programme, and in particular how calibrated cycles (coassociative submanifolds, special Lagrangians) arise from the large mass limit of $G_2$ and Calabi Yau monopoles.

The question of profinite rigidity asks whether the isomorphism type of a group Γ can be recovered entirely from its finite quotients. In this talk, I will introduce the study of profinite rigidity in a different setting: the category of modules over a Noetherian domain Λ. I will explore properties of Λ-modules that can be detected in finite quotients and present two profinite rigidity theorems: one for free Λ-modules under a weak homological assumption on Λ, and another for all Λ-modules in the case when Λ is a Dedekind domain. Returning to groups, I will explain how these algebraic results yield new answers to profinite rigidity for certain classes of solvable groups. Time permitting, I will conclude with a sketch of future directions and ongoing collaborations that push these ideas further.

Due to connections to flat space holography, Carrollian geometry, physics and fluid dynamics have received an explosion of interest over the last two decades. In the Carrollian limit of vanishing speed of light c, relativistic fluids reduce to a set of PDEs called the Carrollian fluid equations. Although in general these equations are not well understood, and their PDE theory does not appear to have been studied, in dimensions 1+1 it turns out that there is a duality with the Galilean compressible Euler equations in 1+1 dimensions inherited from the isomorphism of the Carrollian (c to 0) and Galilean (c to infinity) contractions of the Poincar\'e algebra. Under this duality time and space are interchanged, leading to different dynamics in evolution. I will discuss recent work with N. Athanasiou (Thessaloniki), M. Petropoulos (Paris) and S. Schulz (Pisa) in which we establish the first rigorous PDE results for these equations by introducing a notion of Carrollian isentropy and studying the equations using Lax’s method and compensated compactness. In particular, I will explain that there is global existence in rough norms but finite-time blow-up in smoother norms.