L3

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

Motivated by the problem of reconstructing the electron density of a molecule from pulsed X-ray diffraction images (about 10e+9 per reconstruction), we develop a framework for analyzing the convergence to invariant measures of random fixed point iterations built from mappings that, while expansive, nevertheless possess attractive fixed points.  Building on techniques that we have established for determining rates of convergence of numerical methods for inconsistent nonconvex
feasibility, we lift the relevant regularities to the setting of probability spaces to arrive at a convergence analysis for noncontractive Markov operators.  This approach has many other applications, for instance the analysis of distributed randomized algorithms.
We illustrate the approach on the problem of solving linear systems with finite precision arithmetic.

 

Noether’s theorem plays a fundamental role in modern physics by relating symmetries of a Lagrangian to conserved quantities of the Euler-Lagrange equations. In ideal fluid dynamics, the theorem relates the particle labeling symmetry to a Kelvin circulation law. Circulation is conserved for incompressible flows and, otherwise, is generated by advected variables through the momentum map due to a broken symmetry. We will introduce variational principles for fluid dynamics that constrain advection to be the sum of a smooth and geometric rough-in-time vector field. The corresponding rough Euler-Poincare equations satisfy a Kelvin circulation theorem and lead to a natural framework to develop parsimonious non-Markovian parameterizations of subgrid-scale dynamics.

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open subset of algebraically smooth points, through which every invariant morphism factors uniquely. Some applications to quotients by connected groups will be discussed.
 

Combining the classical theory of optimal transport with modern operator splitting techniques, I will present a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media,materials science, and biological swarming. Using the JKO scheme, along with the Benamou-Brenier dynamical characterization of the Wasserstein distance, we reduce computing the solution of these evolutionary PDEs to solving a sequence of fully discrete minimization problems, with strictly convex objective function and linear constraint. We compute the minimizer of these fully discrete problems by applying a recent, provably convergent primal dual splitting scheme for three operators. By leveraging the PDE’s underlying variational structure, ourmethod overcomes traditional stability issues arising from the strong nonlinearity and degeneracy, and it is also naturally positivity preserving and entropy decreasing. Furthermore, by transforming the traditional linear equality constraint, as has appeared in previous work, into a linear inequality constraint, our method converges in fewer iterations without sacrificing any accuracy. We prove that minimizers of the fully discrete problem converge to minimizers of the continuum JKO problem as the discretization is refined, and in the process, we recover convergence results for existing numerical methods for computing Wasserstein geodesics. Simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our numerical method will be shown.

Directed graphs are a model for various phenomena in the
sciences. In topological data analysis particularly the advent of
applying topological tools to networks of brain neurons has spawned
interest in constructing topological spaces out of digraphs, developing
computational tools for obtaining topological information, and using
these to understand networks. At the end of the day, (homological)
computations of the spaces reveal something about the geometric
realisation, thereby losing the directionality information.

However, digraphs can also be associated with path algebras. We can now
consider applying Hochschild homology to extract information, hopefully
obtaining something more refined in terms of the combinatorics of the
directed edges and paths in the digraph. Unfortunately, Hochschild
homology tends to vanish beyond degree 1. We can overcome this by
considering different higher paths of simplices, and thus introduce
Hochschild homology of digraphs in higher degrees. Moreover, this
procedure gives an implementable persistence pipeline for network
analysis. This is a joint work with Luigi Caputi.

In this talk, we will present a new class of invariants of multi-parameter persistence modules : \emph{projected barcodes}. Relying on Grothendieck's six operations for sheaves, projected barcodes are defined as derived pushforwards of persistence modules onto $\R$ (which can be seen as sheaves on a vector space in a precise sense). We will prove that the well-known fibered barcode is a particular instance of projected barcodes. Moreover, our construction is able to distinguish persistence modules that have the same fibered barcodes but are not isomorphic. We will present a systematic study of the stability of projected barcodes. Given F a subset of the 1-Lipschitz functions, this leads us to define a new class of well-behaved distances between persistence modules, the  F-Integral Sheaf Metrics (F-ISM), as the supremum over p in F of the bottleneck distance of the projected barcodes by p of two persistence modules. 

In the case where M is the collection in all degrees of the sublevel-sets persistence modules of a function f : X -> R^n, we prove that the projected barcode of M by a linear map p : R^n \to R is nothing but the collection of sublevel-sets barcodes of the post-composition of f by p. In particular, it can be computed using already existing softwares, without having to compute entirely M. We also provide an explicit formula for the gradient with respect to p of the bottleneck distance between projected barcodes, allowing to use a gradient ascent scheme of approximation for the linear ISM. This is joint work with François Petit.

Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this talk, I will address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples—taken as a mean persistence measure computed from the subsamples—is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

This is joint work with Yueqi Cao (Imperial College London).

We consider, in a framework of iterated random conformal maps, a two-parameteraggregation model of Hastings-Levitov type, in which the size and intensity of new particles are each chosen to vary as a power of the density of harmonic measure. Then we consider a limit
in which the overall intensity of particles become large, while the particles themselves become
small. For a certain `sub-critical' range of parameter values, we can show a law of large numbers and fluctuation central limit theorem. The admissible range of parameters includes an off-lattice version of the Eden model, for which we can show that disk-shaped clusters are stable.
Many open problem remain, not least because the limit PDE does not yet have a satisfactory mathematical theory.

This is joint work with Vittoria Silvestri and Amanda Turner.