I will discuss some recent work on cut-based methods for graph-based semi-supervised learning. Classic methods such as Laplace learning are known to be degenerate in low label rate regimes and are dependent on carefully chosen heuristics to map their continuous-valued solutions to discrete labels. By considering a spectral relaxation of a graph cut problem, we formulate graph-based semi-supervised learning as the minimization of a quadratic over a Stiefel manifold. We develop sequential subspace methods to recover critical points, at which the associated multiplier matrix meets a certain upper bound condition. These critical points enjoy global optimality in certain special cases, and can be searched for more efficiently by SSM compared to alternative methods. Next, to address the difficulty of mapping between continuous-valued solutions and discrete labels, I will introduce an “exact” non-convex relaxation of the cardinality constrained minimum cut problem with supervision and our algorithm based on ADMM to solve it. This method significantly outperforms the state of the art across various label-rate and imbalanced class regimes. Our work builds on earlier results by Hager and Krylyuk (SIAM Discrete Math 1999) on graph partitioning by continuous optimization and Hager (Siam Optimization 2001) on sequential subspace algorithms for quadratic minimization over the sphere and Calder, Cook, Thorpe, Slepcev (ICML, 2020) on graph-based semi-supervised learning. Applications to classification of kNN, citation, and large product graphs at low label rates and imbalanced class and label regimes will be discussed.

Quasicrystals have exotic spectra that are challenging to understand and are the basis of several longstanding problems in spectral analysis. There is also significant excitement about utilising these exotic spectra for wave control applications. In particular, the ability to support many large spectral gaps and exhibit some reported robustness properties (possibly with topological origins) has led several groups to work on enlarging the metamaterial design space beyond just periodic geometries, into the realm of quasicrystals. The first part of this talk will focus on our recent efforts to develop efficient methods for predicting the main spectral gaps in a quasiperiodic waveguide. A common approach is to approximate the spectrum of a quasicrystal with a periodic approximation, known as a supercell. For the specific case of one-dimensional waveguides based on generalised Fibonacci tilings, we have proved that supercell approximations give accurate predictions of the main spectral gaps. This analysis is based on characterising the growth of the underlying recursion relation. We refer to these main gaps as “super band gaps” and have analytically proved their existence in a class of one-dimensional wave systems. The second part of the talk will present recent work to develop applications of the exotic spectral properties of quasicrystals to wave energy harvesting. We have shown that the rainbow trapping phenomenon of graded metamaterials can be combined with the fractal spectra of quasiperiodic waveguides to give a metamaterial that performs fractal rainbow trapping. This is achieved through a graded cut-and-project algorithm that yields a projected geometry for which the effective projection angle (and corresponding local band gap structure) is graded along its length, leading to broadband `fractal’ rainbow trapping. We have demonstrated this principle by designing and building an acoustic waveguide.

Given a finite locally free commutative group scheme G over some base scheme S one can consider the corresponding higher classifying stacks B^nG=K(G,n); these are algebro-geometric versions of the corresponding Eilenberg-Maclane spaces. I will talk about how given a reasonable cohomology theory RG_? (e.g. “?” could be structure sheaf cohomology, singular, etale, de Rham, or prismatic cohomology) one can compute the cohomology of K(G,n) in a uniform fashion. More precisely, one can construct a canonical filtration on RG_?(K(G,n)), whose associated graded is the free divided power algebra on D_?(G)[-n], where D_?(G) is a certain 2-term complex which we call the “Dieudonne module” associated to RG_?. Moreover, if multiplication by 2 on RG_? is invertible then this filtration typically splits, giving an explicit formula for RG_?(K(G,n)) as an E_{n-1}-algebra. This is joint work with Shizhang Li and Shubhodip Mondal.