Study relaxed curvature for surfaces, focusing on energy and BV properties.
problem Defining curvature for non-parametric surfaces with BV and measure properties.
method Examined inscribed polyhedral surfaces to approximate relaxed energy, analyzed BV properties and total curvature.
result Properties of functions with finite relaxed energy, analyzed Schwarz-Peano counterexample.
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.
Characterizes a specific type of Courant algebroid with a Calabi-Yau structure.
problem Understanding specific types of Courant algebroids with Calabi-Yau structures.
method Explains how a homotopy BV algebra with certain properties characterizes these algebroids.
result A Courant algebroid with a Calabi-Yau structure is a homotopy BV algebra with specific properties.
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…
We define a jet-space analog of the BV-Laplacian, avoiding delta-functions and infinite constants; instead we show that the main properties of the BV-Laplacian and its relation to the Schouten bracket originate from the underlying jet-space geometry.
New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.
This is a paper about geometry of (iterated) variations. We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "δ(0)=0" and "logδ(0)=0" within BV-approach to quantisation of gauge systems. Remarkably, the ge…
Explains BV Laplacian on half-densities in simple terms.
problem None explicitly stated; focuses on explanation.
method Didactical review of BV Laplacian on half-densities.
result Explains BV Laplacian concept in plain language.
Extended equivariant BV formalism to manifolds with boundaries.
problem Handling manifolds with boundaries in equivariant BV formalism.
method Extension of AKSZ theories to manifolds with boundaries.
result Successfully applied to manifolds with boundaries.
A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending o…
This paper is devoted to a geometric-measure-theoretic study of the brand new affine BV-capacity which is essentially different from the classic BV-capacity in dimension greater than one.
Study BV operators on holomorphic polyvector fields on toric varieties.
problem Existence of BV operators in Gerstenhaber algebras.
method Analyzing BV operators on holomorphic polyvector fields on smooth compact toric varieties.
result Necessary and sufficient condition for BV operators existence.
Constructs Lie-Rinehart algebra for Einstein's equations.
problem Initial value problem constraints for Einstein's equations.
method BV-BFV approach to boundary value problems, constructing L∞-algebroid. result Lie-Rinehart algebra comes from slight generalization of Lie algebroid.
Let A=A0+A1+A2+... be a Gerstenhaber algebra generated by A0 and A1. Given a degree -1 operator D on A0+A1, we find the condition on D that makes A a BV-algebra. Subsequently, we apply it to the Gerstenhaber or BV algebra associated to a Lie algebroid and obtain a global proof of the corresponden…
Deep BV automates brain ventricle segmentation from 3D ultrasound images of mouse embryos.
problem Manual segmentation of brain ventricles from high-frequency ultrasound images is tedious, time-consuming, and requires specialized expertise.
method A fully automated system using two modules: localization and segmentation. Localization identifies a 3D bounding box containing the entire brain ventricle. Segmentation then segments the detected bounding box into brain ventricle and background.
result Achieves a Dice Similarity Coefficient (DSC) of 0.8956 for BV segmentation on an unseen test set, surpassing the previous state-of-the-art method by 25%
Proves sufficiency of countable test plans for BV functions on metric spaces.
problem Recovering BV functions and their measures on arbitrary metric spaces.
method Proves sufficiency of countable test plans on arbitrary metric measure spaces and geodesics on CD(K,N) spaces. result Countable test plans are sufficient for BV functions and their measures on metric spaces.
Integral currents with boundary of finite mass are integral.
problem Integral currents with boundary of finite mass are integral.
method De Giorgi's structure theorem for integer-valued BV functions and a cylindrical projection argument. result Integral currents with boundary of finite mass are integral.
Study of nonlinear PDEs using derived geometry and BV formalism.
problem Understanding non-linear PDEs via derived geometric methods.
method Derived enhancement of de Rham complex, algebro-geometric techniques, BV formalism.
result Natural derived enhancement of de Rham complex for nonlinear PDEs.
Study of quasimorphisms and bounded cohomology in braided Thompson groups.
problem Investigate quasimorphisms and bounded cohomology in braided versions of Thompson groups.
method Analyze quasimorphisms and bounded cohomology of various braided Thompson groups.
result Found infinite-dimensional spaces of quasimorphisms in some braided Thompson groups and trivial second bounded cohomology in others.
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.
Deep learning models can adaptively estimate functions with varying smoothness using regularization.
problem Estimating functions with heterogeneous smoothness in Besov or BV classes.
method Introduced a Parallel NN variant of deep ReLU networks with ℓ2 regularization equivalent to promoting ℓp-sparsity. result Achieves minimax rates for Besov and BV classes with exponentially closer performance as depth increases.
Study quantization schemes on Kähler manifolds linking star products and BV quantizations.
problem Quantization of structures on Kähler manifolds.
method Construct Fedosov's star products and Batalin-Vilkovisky (BV) quantizations.
result One-loop exactness of BV quantizations, leading to a cochain level formula.
Study proves a new flow method for mean curvature with volume change analysis.
problem Existence of BV flow via mean curvature flow.
method Elliptic regularization to prove existence of generalized BV flow.
result Proves existence of generalized BV flow with volume change expression.
Proves existence of multi-phase flows from arbitrary initial data.
problem Non-uniqueness issue in Brakke flows.
method Global existence proof for multi-phase mean curvature flow.
result Validates explicit identity for evolving grain volumes.
Study cohomology of odd symplectic manifolds, linking to Lagrangian submanifolds and BV Laplacians.
problem Understanding cohomology classes on odd symplectic manifolds and their relation to Lagrangian submanifolds.
method Investigates complexes of differential, integral, and pseudo forms, introduces new operators, and proves cohomology isomorphisms.
result Proves isomorphism between de Rham cohomology and BV Laplacian cohomology on odd symplectic manifolds.
Study quantum aspects of 1-form symmetries using BV-BRST cohomology.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
Study quantum aspects of 1-form symmetries using BV-BRST cohomology and gerbes.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
Survey on quantization methods on Kähler manifolds.
problem None explicitly stated; focuses on methods.
method Deformation quantization, geometric quantization, Berezin-Toeplitz quantization, BV quantization.
result New relationships among quantization methods on Kähler manifolds.
New bin-wise scaling methods improve prediction uncertainty calibration for machine learning.
problem Improving prediction uncertainty calibration for machine learning regression.
method Adaptations of Binwise Variance Scaling (BVS) with alternative loss functions and feature-based binning.
result Improved adaptivity and consistency in prediction uncertainty calibration.
New family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups BVn,r(H) are finitely generated if H is finitely generated. This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms Gp:TpM→[0,∞] are given. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doub…
Introduces a new operator generating higher Koszul brackets on differential forms.
problem Developing a new operator for higher Koszul brackets on differential forms.
method Introducing a formal ℏ-differential operator Δ generating higher Koszul brackets on differential forms. result Established properties of the introduced BV type operator and its inclusion in a one-parameter family.
We investigate the perturbative aspects of Rozansky-Witten's 3d σ-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of reg…
The goal of this note is to give a brief overview of the BV-BFV formalism developed by the first two authors and Reshetikhin in [arXiv:1201.0290], [arXiv:1507.01221] in order to perform perturbative quantisation of Lagrangian field theories on manifolds with boundary, and present a special case of Chern-Simons theory a…
Paper approximates BV functions using neural networks with ReLU activation.
problem Approximating BV functions with neural networks.
method Studied convergence of stochastic gradient flow and proved Poincaré inequality for penalized cost function.
result Localization theorem: Error of constrained problem is of order R−1/9 with respect to unconstrained problem. The Poisson--Weil sigma model, worked out by us recently, stems from gauging a Hamiltonian Lie group symmetry of the target space of the Poisson sigma model. Upon gauge fixing of the BV master action, it yields interesting topological field theories such as the 2--dimensional Donaldson-Witten topological gauge theory a…
Extends BV functions and divergence-measure fields to metric spaces.
problem Defining BV functions and divergence-measure fields in metric spaces.
method Employing differential structure developed by N. Gigli, extending BV functions and divergence-measure fields to metric spaces.
result Gauss-Green formulas established for BV functions and divergence-measure fields in metric spaces.
The perturbative Chern-Simons theory is studied in a finite-dimensional version or assuming that the propagator satisfies certain properties (as is the case, e.g., with the propagator defined by Axelrod and Singer). It turns out that the effective BV action is a function on cohomology (with shifted degrees) that solves…
Let M be a compact one--manifold, and let Diff1+bv(M) denote the group of C1 orientation preserving diffeomorphisms of M whose first derivatives have bounded variation. We prove that if G is a group which is not virtually metabelian, then (G×Z)∗Z is not realized …
We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the BV fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regul…
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.
The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylins…
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For H2-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the L1-lower s…
In the popular approach of "Bayesian variable selection" (BVS), one uses prior and posterior distributions to select a subset of candidate variables to enter the model. A completely new direction will be considered here to study BVS with a Gibbs posterior originating in statistical mechanics. The Gibbs posterior is con…
Estimates BV functions from noisy data using Voronoi diagrams.
problem Estimating multivariate BV functions from scattered noisy data.
method Form Voronoi diagram, solve optimization problem with discrete TV regularization.
result Voronoigram is minimax rate optimal for BV functions.
Develops BV function and finite perimeter set theory on Riemannian manifolds.
problem Theory of BV functions and finite perimeter sets on arbitrary Riemannian manifolds.
method Localization framework combining Euclidean and metric measure space techniques.
result Recovery of key Euclidean results in Riemannian setting.