Riemannian metrics and Laplacians defined for complex distributions on manifolds.
problem Defining metrics and Laplacians for distributions on manifolds of varying rank.
method Introduced a Riemannian metric and Laplace operator for generalised smooth distributions on manifolds.
result Essentially self-adjoint Laplacian on compact manifolds, hypoellipticity proven.
Develops a new global theory of generalised functions on manifolds.
problem Lack of a global theory for generalised functions on manifolds.
method Constructs a global theory based on smoothing operators and generalised Lie derivative.
result Generalised Lie derivative commutes with embedding of distributions.
Abstract reviews distributions and subbundles in differential geometry.
problem Understanding distributions and subbundles in differential geometry.
method Systematic review of distributions and subbundles, including sheaves and differentiability cases.
result Detailed consideration of Orbit Theorem and its applications.
A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a presc…
In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…
Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
problem Compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
method Introduced a locally convex topology, extended compactness theorem, studied pseudo-differential operators, and applied to microlocal defect measures.
result Extended microlocal defect measures and compensated compactness theorem to Sobolev wave front set spaces.
The κ-generalised distribution fits daily stock returns well.
problem Stock returns are often heavy-tailed, not normally distributed.
method Used the κ-generalised distribution with a Monte-Carlo goodness of fit test. result The κ-generalised distribution fits historic daily stock returns well for a significant proportion of analyzed stocks. Extends nonlinear theory of distributional geometry.
problem Developing a theory for nonsmooth differential geometry.
method Extending Colombeau theory to tensor fields, introducing Lie derivative and covariant derivative, defining generalised metric.
result Preserves Einstein equations and curvature of cones in nonsmooth geometry.
Local equivalence shown between specific distributions and flat Cartan distribution.
problem Establishing local equivalence between specific distributions and flat Cartan distribution.
method Change of coordinates mapping specific distributions to flat Cartan distribution.
result Local equivalence between maximally symmetric (2,3,5)-distributions and flat Cartan distribution. Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
The surgery technique of Gromov and Lawson may be used to construct families of positive scalar curvature metrics which are parameterised by Morse functions. This has played an important role in the study of the space of metrics of positive scalar curvature on a smooth manifold and its corresponding moduli spaces. In t…
Improved neural networks for relational reasoning by projecting high-dimensional data to low-dimensional manifolds.
problem Out-of-distribution generalization in complex relational reasoning tasks.
method Neuroscience-inspired inductive-biased module projecting high-dimensional object representations to low-dimensional manifolds.
result Significantly better out-of-distribution generalization performance on relational reasoning tasks.
Proves smooth solutions for generalised Monge-Ampère equations on projective manifolds.
problem Existence of smooth solutions for generalised Monge-Ampère equations on projective manifolds.
method Intersection numbers and degenerate concentration of mass result.
result Proves existence of smooth solutions for generalised Monge-Ampère equations on projective manifolds.
PAC-Bayes bound requires prior to place mass on high-performing predictors.
problem Explaining generalization in machine learning.
method Analyzing necessary conditions for PAC-Bayes bounds to provide meaningful generalization guarantees.
result Achieving a target generalisation level requires the prior to place sufficient mass on high-performing predictors.
Diffusion models adapt to data geometry through log-domain smoothing.
problem Understanding why diffusion models generalize well across diverse domains.
method Investigating the role of score matching and log-domain smoothing in diffusion models.
result Log-domain smoothing adapts the diffusion model to the data manifold.
The paper proves a mass theorem for non-spin manifolds with low regularity curvature.
problem Establishing a mass theorem for non-spin manifolds with low regularity curvature.
method Smooth approximations of the metric, Sobolev version of Friedrichs' Lemma, comparison theory of RCD-spaces, rigidity theorem for compact manifolds.
result Asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass.
Study on smoothness of 4D Willmore-type hypersurfaces.
problem Investigating smoothness of critical points of a 4D Willmore-type energy.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the energy are smooth.
We classify generalised supersymmetric fluxbranes in type II string theory obtained as Kaluza-Klein reductions of the Minkowski space vacuum of eleven-dimensional supergravity. We obtain two families of smooth solutions which contains all the known solutions, new solutions called nullbranes, and solutions interpolating…
Extends Gelfand duality to various geometric and analytical categories.
problem Generalizing Gelfand duality to different types of manifolds and bundles.
method Unified cohomological argument for manifolds and suitable classes of functions for bundles.
result Gelfand duality extended to real analytic and Stein manifolds, and to vector, affine, and jet bundles.
This work uses differential topology to address challenges in DNNs.
problem Challenges in Deep Neural Networks: expressibility, optimisability, and generalisability.
method Modeling the dataset as a smooth manifold and applying differential topology to loss landscape, expressibility, and generalisability.
result A differential topological view offers new insights into DNNs' challenges.
Motivated by the definition of the smooth manifold structure on a suitable mapping space, we consider the general problem of how to transfer local properties from a smooth space to an associated mapping space. This leads to the notion of smoothly local properties. In realising the definition of a local property at a pa…
Billiard trajectories (broken generalised geodesics) are considered in the exterior of an obstacle K with smooth boundary on an arbitrary Riemannian manifold. We prove a generalisation of the well-known Santalo's formula. As a consequence, it is established that if the set of trapped points has positive measure, then…
New GLPs split Lévy bridges into non-overlapping subprocesses.
problem Creating multivariate stochastic processes with specific properties.
method Defining GLPs by splitting Lévy bridges and using time changes.
result GLPs have terminal values and increments with generalised multivariate Liouville distributions.
We study supersymmetric AdSD backgrounds of eleven-dimensional or type II supergravity preserving N supersymmetries using generalised geometry. We show that a large class correspond precisely to spaces admitting a generalised GD,N structure with a weak integrability condition, which we cal…
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
This work improves generalisation bounds using chaining and information theory.
problem Improving generalisation bounds for supervised learning algorithms.
method Developed a theoretical framework linking generalisation bounds to their chained counterparts, derived new bounds using Wasserstein distance.
result Chained generalisation bounds can be tighter than standard bounds, especially for concentrated hypothesis distributions.
CRL improves recommendation systems by reducing distribution shift.
problem Offline metrics fail to predict online performance due to distribution shift in recommender systems.
method Proposes an information-theoretic disentanglement criterion and a variational lower bound for better generalisation under distribution shift.
result CRL variants deliver substantial online gains in listener engagement compared to baseline models.
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally …
The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one …
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the μ--H…
Improved model for non-smooth signals with complex spectra.
problem Current models struggle with non-smooth signals and complex spectral structures.
method CGPCM and RGPCM models with causality and Bayesian nonparametric interpretations, improved variational inference.
result Proposed models show better performance on synthetic and real-world data.
Introduces a new geometric framework for non-perturbative BV-theory.
problem Non-perturbative generalization of BV-theory in infinite-dimensional spaces.
method Derived differential geometry and homotopical algebraic geometry.
result Concrete model of derived smooth stacks for encoding non-perturbative BV-theory.
Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for …
New approach to compute generalization performance using known risk distribution.
problem Computing generalization performance in machine learning.
method Assumes known risk distribution ρ(r), computes expected error using empirical risk minimization, and considers power-law behavior of ρ(r). result Corrected typical behavior of generalization performance due to chance correlations in training set.
Recently Gay and Kirby described a new decomposition of smooth closed 4-manifolds called a trisection. This paper generalises Heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear n-mani…
Paper proves non-zero generalization boost for equivariant models.
problem Improving generalization in machine learning models.
method Analyzes simplest case of linear models, focusing on invariant/equivariant properties.
result First provably non-zero improvement in generalization for invariant/equivariant models.
We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists. W…
We consider a generalised complex Monge-Ampère equation on a compact Kähler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related real PDE in a ball), as long as the Hessian is bounded below by a pre-determined co…
Generalizes Kauffman's clock theorem to surfaces.
problem Proving a lattice structure on graph states in various surfaces.
method Using matchings and graph orientations, extending Propp's results.
result Two generalizations of Kauffman's theorem for more surfaces.
Graph-dependent implicit regularisation improves Distributed SGD for convex problems.
problem Improving convergence rates in distributed stochastic subgradient descent.
method Graph-dependent implicit regularisation strategies for Distributed SGD.
result Established statistical learning rates retaining centralised guarantees.
Generalizes Connes's tangent groupoid for sub-Riemannian geometry.
problem Calculating tangent cones in sub-Riemannian geometry.
method Constructs a completion of MimesMimesR+imes using sub-Riemannian metric. result Calculates all tangent cones in Gromov-Hausdorff distance.
We present local estimates for solutions to the Ricci flow, without the assumption that the solution has bounded curvature. These estimates lead to a generalisation of one of the pseudolocality results of G.Perelman in dimension two.
Local generalization of frame bundles using a weakened Maurer-Cartan equation.
problem Characterizing the local structure of frame bundles with a weakened Maurer-Cartan equation.
method Introducing generalised frame bundles and studying their properties.
result Generalised frame bundles can have singular underlying manifolds.
We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of certain rational homology balls bounded by lens spaces.
We study the geometric properties of holomorphic distributions of totally null m-planes on a (2m+ε)-dimensional complex Riemannian manifold (M,g), where ε∈0,1 and m≥2. In particular, given such a distribution N, say, we obtain algebraic conditions on the Weyl tensor and t…
Smooth FFT from B-fields and D-branes.
problem Constructing a smooth functorial field theory from B-fields and D-branes.
method Definition of a smooth bordism category, transgression, functoriality, thin homotopy invariance, positive reflection structure.
result Generalizes open-closed TQFTs to include target spaces and open strings.
The paper generalizes a result on smooth mapping class groups and proves a property of Dehn twists in 4-manifolds.
problem Properties of Dehn twists in smooth 4-manifolds.
method Generalization of a previous result and alternative proof of a consequence.
result Dehn twists along the boundary of simply-connected 4-manifolds are trivial after connected sums.
Develops PAC-Bayesian framework for physics-informed machine learning.
problem Lack of statistical generalisation understanding for PIML models.
method PAC-Bayesian framework with multi-task perspective, incorporating physical structure.
result High-probability generalisation guarantees with unbounded losses.