The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.
In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongatio…
Paper applies Newman-Penrose formalism to ACM manifolds.
problem Classifying compact ACM manifolds with η-Einstein metrics. method Newman-Penrose formalism applied to ACM manifolds.
result Classification of compact ACM manifolds with η-Einstein metrics. In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.
Study on cohomology of G2 manifolds, proving almost formality.
problem Cohomology of compact torsion-free G2 manifolds.
method Analysis of the interplay between exterior derivative and derivation, using Hodge theory and G2 structure properties.
result Proves compact torsion-free G2 manifolds are 'almost formal'.
Holomorphic bundles on complex manifolds with boundary are studied, extending results from Donaldson's work.
problem Extending holomorphic structures to complex manifolds with boundary.
method Analyzing formally integrable almost complex structures and their extensions to holomorphic structures.
result Holomorphic structures can be extended to a neighborhood of a strictly pseudoconvex boundary.
Under appropriate assumptions, we generalize the concept of linear almost Poisson struc- tures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolution…
New algebraic structures for Hermitian geometry cohomologies.
problem Understanding cohomologies of Hermitian manifolds.
method Introducing BV-algebras and homotopy BV-algebras.
result Cohomologies of Hermitian manifolds are endowed with homotopy hypercommutative algebra structures.
This paper formalizes Q-learning and linear TD convergence using Lean 4.
problem Formalizing convergence properties of Q-learning and linear TD learning. method Formal verification using Lean 4 theorem prover and Mathlib library.
result Formalized almost sure convergence of Q-learning and linear TD learning. Models close to formal for Vaisman and quasi-Sasakian manifolds, including nilmanifolds.
problem Formality of quasi-Sasakian and Vaisman manifolds.
method Providing models that are as close as possible to being formal for compact manifolds with transversely Kaehler structures.
result Classification of corresponding nilmanifolds.
Characterizes structures on generalized tangent bundles and CRF-structures.
problem Understanding structures on generalized tangent bundles and CRF-structures.
method Equivalent characterizations and spinor formalism for CRF-structures.
result Characterization of generalized complex manifolds as products and infinitesimal deformations of CRF-structures.
The paper explores the formality of low-dimensional manifolds using algebraic structures.
problem Investigating the formality of low-dimensional manifolds.
method Introducing Poincaré DGCAs of Hodge type and small algebra/quotient algebras to study equivalence classes of manifolds.
result A (r−1) connected Poincaré DGCA of Hodge type is A∞-quasi-isomorphic to an A3-algebra, with the formality determined by a specific Harrison cohomology class. We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equa…
We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these …
New equivariant formality concepts solve the toral rank conjecture.
problem Toral rank conjecture and equivariant formality of actions.
method Rational homotopy theory, Hirsch-Brown models, A-infinity algebras.
result Actions with new properties satisfy the toral rank conjecture.
New curvature equations obstruct integrability of complex structures.
problem Understanding curvature obstructions to integrability of complex structures.
method Direct approach using Nijenhuis tensor derivatives and curvature scalars.
result Certain complex structures cannot coexist with non-flat constant curvature metrics.
The study finds conditions for the existence of extremal toric almost Kähler metrics.
problem Conditions for the existence of extremal toric almost Kähler metrics.
method Observation and application of recent results on K-stability and Abreu equation.
result Existence of extremal toric almost Kähler structures is equivalent to uniform K-stability.
One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation δ such that dδ-lemma holds. We first show that such derivation δ generates a (1,1)-tensor field (we denote it by R). Then, we show that the supercommutation of d and δ…
Extends Double Field Theory with new kinematical structure.
problem Formalizing Double Field Theory on para-Hermitian manifolds.
method Constructing a canonical connection and generalised Lie derivative for a Leibniz algebroid.
result Integrability conditions for symmetry algebra closure under non-flat and non-constant η and ω. Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topologi…
The abstract discusses constructing manifolds with specific algebraic models using Lie group actions.
problem Constructing manifolds with prescribed algebraic models through Lie group actions.
method Using principal K-bundles and algebraic models, the abstract describes methods to construct manifolds with specific properties. result The existence of algebraic models for manifolds with prescribed properties has implications on the structure of cohomology jump loci and representation varieties.
Paper proves a quantitative estimate for transforming almost complex structures into standard ones.
problem Transforming almost complex structures into standard ones on bounded domains.
method Proves existence of global diffeomorphisms under Hölder-Zygmund conditions.
result Existence of a global diffeomorphism in a specified Hölder-Zygmund class.
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space RqN, the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over RqN based on the R-matrix formalism, we…
MALT improves adversarial attacks by targeting classes more efficiently.
problem Naive targeting of adversarial attacks based on classifier confidence.
method MALT - Mesoscopic Almost Linearity Targeting, based on medium-scale almost linearity assumptions.
result MALT wins over AutoAttack on CIFAR-100 and ImageNet datasets, five times faster.
Almost all local minima in neural networks are strongly convex.
problem The prevalence of strongly convex neighborhoods around local minima in neural network optimization landscapes.
method Rigorous analysis of shallow neural networks with analytic activation functions, dividing parameter space into efficient and redundant domains.
result For shallow neural networks on the efficient domain, almost all local minima are strongly convex.
We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
problem The topology of Kähler manifolds is largely determined by the geometry due to its rigidity.
method We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
result We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
This paper studies geometric structures on manifolds with specific symplectic properties.
problem Understanding geometric structures on manifolds with quaternionic skew-Hermitian properties.
method Equivalent definitions, intrinsic torsion, classification of geometries, explicit connections.
result Classification of symmetric spaces with invariant torsion-free structures.
The paper defines Morse-Bott invariants for critical sets of circles.
problem Homological invariants from Morse-Bott data on unions of circles.
method Axiomatic approach to moduli spaces and evaluation maps, defining homological invariants.
result Construction of a homotopy invariant cascade homology functor.
The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.
problem Understanding the fixed points and cohomology of almost complex torus manifolds.
method Using directed labeled multigraphs and Hirzebruch genera to encode and analyze the manifolds.
result Almost complex torus manifolds have positive Todd genus and at least n+1 fixed points.
We propose a new approach for Collaborative Filtering which is based on Boolean Matrix Factorisation (BMF) and Formal Concept Analysis. In a series of experiments on real data (Movielens dataset) we compare the approach with the SVD- and NMF-based algorithms in terms of Mean Average Error (MAE). One of the experimental…
The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems simil…
Proposes a new gauge theory for fuzzy geometries using finite-dimensional algebras.
problem Modeling fuzzy geometries in noncommutative geometry.
method Introduces a Yang-Mills-Higgs matrix model based on gauge matrix spectral triples.
result States Yang-Mills-Higgs theory as an explicit random multimatrix model.
The paper proves neural networks are almost always surjective, impacting model safety.
problem Ensuring neural networks can generate any output, including harmful content.
method Analyzing fundamental neural architectures and generative models.
result Many neural architectures are almost always surjective, allowing for arbitrary outputs.
Large neural networks learn low-dimensional representations that balance complexity and regularity.
problem Understanding the tradeoff between low-dimensional representations and complexity in deep neural networks.
method Computed finite depth corrections to reveal a measure of regularity that bounds the pseudo-determinant of the Jacobian.
result Proved the conjectured bottleneck structure in learned features as network depth increases, showing almost all hidden representations are approximately low-dimensional and weight matrices have singular values close to 1.
Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.
problem Global invertibility in nonlinear elasticity with a vanishing nonlocal self-repulsion term.
method Proves global invertibility in the Γ-limit of elastic energy with a vanishing nonlocal self-repulsion term. result Global invertibility can be obtained in the Γ-limit of the elastic energy with a vanishing nonlocal self-repulsion term. The paper connects higher-dimensional mechanics to Lie n-algebroids.
problem Understanding interactions in higher-dimensional gauge systems.
method Comparing BV/BRST formalism with Lie n-algebroids and defining polytorsion.
result Relates topological n-branes to differential geometry on Lie n-algebroids.
Model-based Bayesian Reinforcement Learning (BRL) allows a found formalization of the problem of acting optimally while facing an unknown environment, i.e., avoiding the exploration-exploitation dilemma. However, algorithms explicitly addressing BRL suffer from such a combinatorial explosion that a large body of work r…
We argue that the Einstein gravity theory can be reformulated in almost Kahler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation (NGT) on manifolds enabled with nonholonomic distribut…
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diff…
New method verifies formulas for causal interventional distributions.
problem Deciding if a given formula correctly identifies an interventional distribution.
method Proposed a falsifier to check if a formula is identifying.
result Falsifier can induce an almost-surely correct verifier for certain models.
SepNE separates network embedding for efficiency and scalability.
problem Inseparable network embedding methods waste resources on irrelevant nodes.
method Formalized separated matrix factorization, SepNE learns node embeddings independently.
result SepNE outperforms state-of-the-art methods on large networks.
Geometrically proves WKB solutions of Schrödinger equations are resurgent.
problem Understanding resurgent behavior of WKB solutions on Riemann surfaces.
method Purely geometric approach using holomorphic Lie groupoids and spectral curves.
result Formal WKB solutions are Borel summable in almost all directions.
We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\times U(1)\times U(1)). For all n>1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles …
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.
The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.
problem Quantization of compact symplectic manifolds with higher Landau levels.
method Develops Berezin-Toeplitz quantization using a Bochner Laplacian with specific spectral properties.
result The quantization provides a formal star-product for the lowest Landau level.
3D BF theory on certain 3-manifolds evaluated via residues and large k limits.
problem Singular and ill-defined partition function of 3D BF theory.
method Direct evaluation of path integral for specific 3-manifolds, using residues and large k limits of Chern-Simons matrix integrals.
result 3 definitions of the integral offer insights into the sum/integral over all flat connections.