The paper generalizes Cartan Geometry using Polacek and Siegel's approach.
problem Formulating sigma model dynamics in a covariant way.
method Using Polacek and Siegel's generalised curvature and torsion approach within the generalised metric formalism.
result Almost all higher generalised tensors correspond to covariant derivatives of the generalised Riemann tensor.
Constructs a unique Levi-Civita connection for generalised metrics.
problem Non-uniqueness of generalised Levi-Civita connections.
method Geometrically constructs a canonical generalised Levi-Civita connection.
result Decomposes the generalised Riemann curvature tensor in terms of classical geometric data.
The paper applies generalised geometry to semi-Riemannian immersions and hypersurfaces.
problem Analyzing semi-Riemannian immersions and hypersurfaces using generalised geometry.
method Develops the pullback of generalised metrics and divergence operators, introduces generalised exterior curvature, and derives Gauß-Codazzi equations.
result Establishes the constraint equations for the initial value formulation of the generalised Einstein equations.
New approach to T-duality using Courant algebroids.
problem Developing a new framework for T-duality.
method Relational description of Courant algebroids and weakened isometries.
result Existence and uniqueness of T-dual backgrounds.
Defines T-duality and generalised Ricci flow relations using Courant algebroid relations.
problem Establishing compatibility between T-duality and generalised Ricci flow.
method Introducing Courant algebroid relations, invariant divergence operators, and generalised isometries.
result T-duality is compatible with generalised Ricci flow, and T-dual solutions are also solutions of generalised Ricci flow.
We define (p,q) hermitian geometry as the target space geometry of the two dimensional (p,q) supersymmetric sigma model. This includes generalised Kähler geometry for (2,2), generalised hyperkähler geometry for (4,2), strong Kähler with torsion geometry for (2,1) and strong hyperkähler with torsion geometry f…
Classifies invariant generalised Killing spinors on Lie groups.
problem Classifying invariant generalised Killing spinors on Lie groups.
method Complete classification using invariant properties and computational methods.
result Existence of non-trivial invariant generalised Killing spinors implies all invariant spinors are generalised Killing with the same endomorphism.
Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the …
The study explores properties of a specific type of spacetime.
problem Discussing geometric and physical properties of hyper-generalised quasi-Einstein spacetime.
method Analyzing various types of pseudosymmetry and Ricci symmetry over the spacetime.
result Proved the existence of a non-trivial hyper-generalised quasi-Einstein spacetime.
This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields …
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.
The paper explores linear generalised complex structures over vector bundles.
problem Understanding holomorphic vector bundles in a generalized geometry context.
method Adapted linear splitting and equivalence to C-multiplication and C-Lie algebroid structure. result Generalised complex Lie algebroids are expressed as complex conjugated Lie bialgebroids.
Study generalised Einstein metrics on Lie groups, classifying various types.
problem Classify left-invariant generalised Einstein metrics on Lie groups.
method New algebraic reformulation, classification based on Lie bracket and scalar product.
result Full classification of solvable and almost Abelian generalised Einstein Lie groups.
The study finds conditions for Sasakian manifolds and generalised Ricci solitons.
problem Conditions for Sasakian manifolds and generalised Ricci solitons.
method Finding necessary conditions for both Sasakian structures and generalised Ricci solitons on odd-dimensional manifolds.
result Necessary conditions for both Sasakian structures and generalised Ricci solitons on odd-dimensional manifolds.
Hydrodynamic structures linked to F-manifolds.
problem Hydrodynamic equations and their Hamiltonian structures.
method Introducing generalised (bi-)Hamiltonian structures and associating them with (bi-)flat F-manifolds.
result Generalised (bi-)Hamiltonian structures of hydrodynamic type can be associated with (bi-)flat F-manifolds.
We present a general formalism for incorporating the string corrections in generalised geometry, which necessitates the extension of the generalised tangent bundle. Not only are such extensions obstructed, string symmetries and the existence of a well-defined effective action require a precise choice of the (generalise…
In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia, and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised s…
Polynomial invariant derived from birack labelling of knots.
problem Developing a polynomial invariant for a broader class of knot theories.
method Generalizing biquandle colouring to birack labelling, reducing to biquandle invariant.
result Polynomial invariant for a class of knot theories.
We reformulate ten-dimensional type II supergravity as a generalised geometrical analogue of Einstein gravity, defined by an O(9,1)×O(1,9)⊂O(10,10)×R+ structure on the generalised tangent space. Using the notion of generalised connection and torsion, we introduce the analogue of the Levi-C…
New bounds link flat minima to good generalisation in overparameterized models.
problem Understanding the relationship between flat minima and generalisation in overparameterized machine learning models.
method Combining PAC-Bayes, Poincaré, and Log-Sobolev inequalities to derive generalisation bounds involving gradient terms.
result Flat minima positively influence generalisation performance, highlighting the benefits of the optimisation phase.
We expand upon a claim made in a recent paper [arXiv:1411.5721] that generic minimally supersymmetric AdS backgrounds of warped flux compactifications of Type II and M theory can be understood as satisfying a straightforward weak integrability condition in the language of Ed(d)×R+ generalised geomet…
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.
This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept …
Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.
problem Understanding and characterizing generalised spin structures and their properties.
method Definitions, basic notions, connections, covariant Lie derivative, covariant Cartan calculus, symmetry algebra.
result Characterization of homogeneous generalised spin structures.
We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises …
Unified description of string and brane worldvolumes using auto-parallel vector fields.
problem Describing worldvolumes of strings and branes in arbitrary backgrounds.
method Introducing auto-parallel generalised vector fields and their properties.
result Unified worldvolume equations for strings and branes.
We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new in…
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and L-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic …
We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form RD−1,1×M. Using the language of Ed(d)×R+ generalised…
This work uncovers algorithm-dependent regularisation in diffusion models.
problem Understanding and improving generalisation in high-dimensional diffusion models.
method Algorithmic stability and score stability analysis.
result Identifies multiple sources of implicit regularisation unique to diffusion models.
Neuro-symbolic agent learns systematic generalisation from formal instructions.
problem Achieving zero-shot generalisation of formally specified tasks.
method Combines deep reinforcement learning with temporal logic.
result Systematic learning emerges with convolutional layers and abstract operators.
We reinterpret the generalised Lie derivative of M-theory E6 generalised geometry as hamiltonian flow on a graded symplectic supermanifold. The hamiltonian acts as the nilpotent derivative of the tensor hierarchy of exceptional field theory. This construction is an M-theory analogue of the Courant algebroid and reve…
Unified framework for exceptional and generalised geometry, and Poisson-Lie duality.
problem Unified framework for exceptional and generalised geometry.
method Introducing G-algebroid, generalising Lie and Courant algebroids.
result Classification of 'exact' algebroids and compatibility with supergravity.
Paper simplifies link classification in 3-sphere using braids and templates.
problem Classify links in the 3-sphere.
method Simplified braid description, generalised T-links, bunch algorithm.
result Established upper volume bound for 3-manifolds.
Study shows bounds on volumes of weakly generalised alternating knots.
problem Volume bounds for weakly generalised alternating knots.
method Analysis of weakly generalised alternating knots in 3-manifolds.
result Upper volume bound does not hold for weakly generalised alternating knots.
The study examines sequences of 3D generalised monopoles with uniform bounds.
problem Analyzing sequences of 3D generalised monopoles with uniform L2-norm bounds.
method Focus on Swann bundles and uniform lower bounds on hyperKahler potential.
result Convergent subsequences of generalised monopoles over compact subsets of Y' are found.
Deep Reinforcement Learning has shown great success in a variety of control tasks. However, it is unclear how close we are to the vision of putting Deep RL into practice to solve real world problems. In particular, common practice in the field is to train policies on largely deterministic simulators and to evaluate alg…
We construct a toric generalised Kähler structure on CP2 and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.
In this paper we calculate the integral Pontrjagin homology ring of the based loop space on some generalised symmetric spaces with a toral stationary subgroup. In the Appendix we show that the method can be applied to other type generalised symmetric spaces as well.
Defines relations between Dirac structures and spinors using Courant algebroid relations.
problem Defines relations between Dirac structures and spinors using Courant algebroid relations.
method Uses Courant algebroid relations to define relations between Dirac structures and spinors.
result Proves existence results for T-dual structures and demonstrates compatibility with Type II supergravity equations.
Y-algebroids generalize symmetry structures for string/M-theory compactifications.
problem Capturing local symmetry structures for string/M-theory compactifications to various dimensions.
method Define Y-algebroids based on a tensor Y, classify M-exact E7-algebroids, and relate to E7(7)×R^+ generalised geometry.
result Y-algebroids match the generalised tangent space of E7(7)×R^+ generalised geometry, including possible fluxes.
Strong Kähler with Torsion is the target space geometry of (2,1) and (2,0) supersymmetric nonlinear sigma models. We discuss how it can be represented in terms of Generalised Complex Geometry in analogy to the Gualtieri map from the geometry of (2,2) supersymmetric nonlinear sigma models to Generalised Kähler Geo…
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
New PAC-Bayes bounds use Wasserstein distances to improve generalization.
problem Lack of geometric properties in existing PAC-Bayes bounds.
method Developed new PAC-Bayes bounds with Wasserstein distances.
result Optimization guarantees translate to good generalization abilities.
New method optimises learning via surrogate PAC-Bayes bounds.
problem Computational intractability of optimising generalisation bounds.
method Iteratively optimising surrogate training objectives derived from PAC-Bayes bounds.
result Iteratively optimising surrogates implies optimising original generalisation bounds.
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.
Deep neural networks achieve stellar generalisation on a variety of problems, despite often being large enough to easily fit all their training data. Here we study the generalisation dynamics of two-layer neural networks in a teacher-student setup, where one network, the student, is trained using stochastic gradient de…
Constructs brane current algebras from QP-manifolds, generalizing string currents.
problem Constructing brane current algebras from QP-manifolds.
method Using Poisson algebra and QP-manifolds (symplectic L∞-algebroids), the paper derives a universal geometric form for Poisson brackets of brane currents. result Derives a universal expression for 't Hooft anomaly in the presence of fluxes.