Generalizes L-infinity spaces to sheaves over nilpotent dg manifolds.
problem Formal moduli problems and L-infinity spaces.
method Extending L-infinity spaces to sheaves of L-infinity algebras over nilpotent dg manifolds.
result Characteristic classes of the new L-infinity space recover primary invariants of Lie algebroids.
We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop…
The procedure "Lie group --> Lie algebra" has a generalization "simplicial manifold --> L_infinity algebra", or yet better, "presheaf on the category of surjective submersions --> L_infinity algebra". We describe this generalization, together with its higher-order extensions.
We define the notion of action of an L-infinity algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particula…
Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group.…
Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a…
Multisymplectic geometry admits an operation that has no counterpart in symplectic geometry, namely, taking the product of two multisymplectic manifolds endowed with the wedge product of the multisymplectic forms. We show that there is an L-infinity-embedding of the L-infinity-algebra of observables of the individual f…
New approach to Lagrangian field theories using pro-finite structures and L-infinity algebras.
problem Formulating Lagrangian field theories with locality constraints.
method Using pro-finite structures and L-infinity algebras to define local observables and a pre-multisymplectic form.
result Definition of L-infinity algebra of local observables based on Lagrangian cohomology.
Establishes higher T-duality for super M-branes.
problem Generalizing T-duality for super p-branes.
method Super L-infinity-algebraic T-duality for super WZW-terms.
result Spherical T-duality of super M5-branes.
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infin…
New method for flux quantization on phase space stacks.
problem Defining and constructing flux-quantized phase space stacks.
method Observation of flux densities and characterization of Cauchy data.
result Flux-quantized phase space stacks have classifying spaces with rational Whitehead L-infinity algebra.
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (Ω^*[1](\fol), d_\fol…
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an L∞-algebra associated with the problem. result Smooth parametrization of the space of structures around a given one.
New algebra structure for Legendrian knots preserves contact homology invariants.
problem Constructing an L∞ algebra for Legendrian knots. method Combining rational Symplectic Field Theory and combinatorial methods.
result Invariant Poisson algebra of Legendrian links under isotopy.
Unified framework for various adversarial attacks on deep networks.
problem Vulnerability of deep neural networks to adversarial attacks.
method ADMM (Alternating Direction Method of Multipliers) for generating adversarial examples.
result ADMM-based attacks achieve highest success rates and minimal distortion.
We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field Q admits a structure of L-infinity algebra with the Lie derivative LQ as unary …
Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.
problem Relating Lie bialgebroids to homotopy Poisson structures on supermanifolds.
method Introduces L-infinity bialgebroids and higher Koszul brackets to connect these structures.
result Shows that (TM,T∗M) has an L-infinity bialgebroid structure for homotopy Poisson structures. Constructs L∞ structure on symplectic cohomology.
problem None explicitly stated; focuses on construction.
method Constructs L∞ structure on symplectic cohomology. result Symplectic cohomology gains an L∞ structure. A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise i…
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
Explains how pre-symplectic structures can be changed.
problem Understanding how pre-symplectic structures can be deformed.
method Uses Dirac geometry to explain the geometric origin of L∞-algebra controlling deformations. result Discovers the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures. Study on deformations of pre-symplectic structures using an L-infinity algebra.
problem Deformation theory of pre-symplectic structures.
method Parametrization of deformations using Koszul L-infinity algebra.
result A quotient of the Koszul L-infinity algebra is isomorphic to the L-infinity algebra controlling foliations.
The study of geometric structures around transversals using deformation spaces.
problem Understanding local behavior of geometric structures around singular foliations.
method Using deformation spaces to study local behavior of geometric structures.
result Obtained normal form theorems around transversals for various geometric structures.
Improved training boosts certified robustness of L-infinity distance nets.
problem Certified robustness of L-infinity distance nets is not as strong as conventional networks.
method Improved training process combining scaled cross-entropy and clipped hinge loss with a decaying mixing coefficient.
result Certified accuracy of L-infinity distance nets improved from 33.30% to 40.06% on CIFAR-10.
The paper proves a generalized inverse function theorem for curved L∞ spaces.
problem Proving a generalized inverse function theorem for curved L∞ spaces. method Obstruction theory for L∞ homomorphisms and homotopy transfer theorem for curved L∞ algebras. result A morphism of curved L∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. Motivated by the definition of homotopy L∞ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a 2-category with invertible 2-morphisms, and that certain fiber product property holds in this 2-category. In a subsequent pa…
An L∞-algebra is built on symplectic manifold homology.
problem No specific problem stated; focuses on construction of algebra.
method Construction of an L∞-algebra on symplectic manifold homology. result The constructed L∞-algebra naturally projects to a Lie algebra extension. Goto proved deformation smoothness for special geometric structures.
problem Deformation smoothness of geometric structures.
method Used L∞-algebra and homotopy abelian properties. result Unified and provided new proofs of deformation smoothness.
We define a manifold M where objects c∈M are curves, which we parameterize as c:S1→Rn (n≥2, S1 is the circle). Given a curve c, we define the tangent space TcM of M at c including in it all deformations h:S1→Rn of c. In this paper we study geometries on the manifold of curves, pr…
Lie algebroids linked to L∞ spaces in derived geometry.
problem Relating Lie algebroids to L∞ spaces in derived geometry. method Constructing a faithful functor from Lie algebroids to L∞ spaces and showing the relationship between representations and vector bundles. result Lie algebroids provide an essentially unique L∞ space, and a shifted-symplectic structure on a dg Lie algebroid produces a shifted-symplectic structure on the associated L∞ space. We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the ca…
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and L∞-algebras. result Extension of conserved currents to spacetime vector fields.
New L∞ liftings derived from Chern-Simons classes for coherent sheaves.
problem Liftings of semiregularity maps for coherent sheaves on complex manifolds.
method Introducing Chern-Simons classes for curved DG-pairs and proving canonical liftings.
result Canonical L∞ liftings of Buchweitz-Flenner semiregularity maps. Split Courant algebroids linked to special algebra structures.
problem Understanding the structure of split Courant algebroids.
method Established a correspondence with multiplicative curved L∞-algebras. result Split Courant algebroids correspond to multiplicative curved L∞-algebras. Paper constructs observables using multisymplectic geometry and algebraic methods.
problem Building observables in multisymplectic geometry.
method Uses L∞-algebras, Gerstenhaber algebras, BV-modules, and constraint triples. result Reconstructs and explains recent geometric results.
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of L∞-morphisms twisted by gauge equivalent elements. result Globalized formalities with different covariant derivatives are homotopic.
We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the high…
Study shows equivalence in foliations and pre-symplectic forms aligns with gauge equivalence.
problem Deformation theory of foliations and pre-symplectic forms.
method Proved geometric equivalence agrees with algebraic gauge equivalence using L∞-algebras. result Gauge equivalences for foliations and pre-symplectic structures are consistent.
We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …
Reviewing Q-manifolds, modular classes, and applications.
problem Obtaining invariant volumes on Q-manifolds.
method Exploring Q-manifolds, modular classes, and applying to specific examples.
result Applications to L∞-algebroids and higher Poisson manifolds. I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…
Study of manifolds with special holonomy using Frölicher-Nijenhuis bracket.
problem Understanding manifolds with special holonomy.
method Use of Frölicher-Nijenhuis bracket to define cohomologies and L∞-algebras. result Definition and computation of Frölicher-Nijenhuis cohomology.
Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.
problem Improving generative models' performance.
method Integrates L∞-algebra into Generative Flow Models, leveraging higher geometry and symmetries. result Substantial performance improvements on Gaussian Mixture Model datasets.