arXiv research

We complete the reduction scheme in the whole LP category, introduced in [7] to perform Lagrangian reduction by stages. We answer affirmatively the open question of whether reduction can be done in the whole category and analyze the Noether theorem on LP-bundles, the relationship with Hamiltonian reduction by stages an…

Proposes a new category of bundles for Lagrangian reduction in field theory.

In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introdu…

Computing PL geometric category in 2D is NP-hard.

Survey and generalization of implosion and contraction in symplectic and hyperkähler geometry.

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian.…

A closed 3-form $H \in Ω^3_0(M)$ defines an extension of $Γ(TM)$ by $Ω^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian redu…

We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample t…

During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the pr…

The paper presents counterexamples to LS-category conjectures and constructs maps between manifolds.

Reduces Poisson manifolds with Hamiltonian Lie algebroids.

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Wald…

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also …

Adapts category theory to manifold learning for better understanding and stability.

Modern deep learning-based recommendation systems exploit hundreds to thousands of different categorical features, each with millions of different categories ranging from clicks to posts. To respect the natural diversity within the categorical data, embeddings map each category to a unique dense representation within a…

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

This work extends reduction processes for nonholonomic discrete mechanical systems.

Study minimal networks on spheres and balls near standard metrics.

POTD estimates SDR subspace using optimal transport for binary response.

For statistical learning, categorical variables in a table are usually considered as discrete entities and encoded separately to feature vectors, e.g., with one-hot encoding. "Dirty" non-curated data gives rise to categorical variables with a very high cardinality but redundancy: several categories reflect the same ent…

This paper proposes a method to reduce complexity in GLMs with categorical predictors.

The paper introduces SRFs and I-Poisson manifolds, linking foliations to Riemannian geometry.

Reduces symplectic manifolds with singularities for quantum reduction.

Introduces a new method for symplectic reduction along submanifolds.

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary sup…

We compare several approaches to learn an Optimal Map, represented as a neural network, between probability distributions. The approaches fall into two categories: ``Heuristics'' and approaches with a more sound mathematical justification, motivated by the dual of the Kantorovitch problem. Among the algorithms we consi…

Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by the classes (or categories). To this end, we set up a model in which each class i…

Develops algorithm to reduce real-world inequality.

Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representatio…

Develops quantum character theory for complex reductive groups.

TQFT invariants are either easy or hard to compute, depending on the TQFT type.

Two novel methods estimate multiple FDR directions for binary categorical responses.

An algorithm for efficient computation of equivariant neural network layers.

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

Extends Higgs fields theory to complex fiber bundles.

The paper classifies Lie algebroids and their connections, modulating principal objects.

This paper explains spectral clustering and its equivalence to PCA, breaking it into fully connected and multi-connected cases.

Natural Language Processing models help encode categorical process inputs.

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit red…

We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is val…

The paper proves a Serre-Swan Theorem for coisotropic algebras.

Studies amenable category's monotonicity and its relation to topological complexity.

To each oriented surface S, we associate a differential graded category Ko(S). The homotopy category Ho(Ko(S)) is a triangulated category which satisfies properties akin to those of the contact categories studied by K. Honda. These categories are also related to the algebraic contact categories of Y. Tian and to the bo…

We study twisted N=2 superconformal gauge theory on a product of two Riemann surfaces Sigma and C. The twisted theory is topological along C and holomorphic along Sigma and does not depend on the gauge coupling or theta-angle. Upon Kaluza-Klein reduction along Sigma, it becomes equivalent to a topological B-model on C …

Study logarithmic flat connections on principal bundles using Lie groupoids.

Study identifies methods for handling sparse data in web settings with contextual bandits.

Formulates a new connection between topological and geometric categories.

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free c…