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 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.
Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.
problem Convergence of Allen-Cahn solutions to multiphase mean curvature flow.
method Conditional convergence result of Allen-Cahn solutions to De Giorgi type BV-solutions of multiphase mean curvature flow.
result De Giorgi type BV-solutions are unique in a weak-strong sense.
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.
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.
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…
Volumetric analysis of brain ventricle (BV) structure is a key tool in the study of central nervous system development in embryonic mice. High-frequency ultrasound (HFU) is the only non-invasive, real-time modality available for rapid volumetric imaging of embryos in utero. However, manual segmentation of the BV from H…
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 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.
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.
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.
The paper proves smoothness of transition layers in the Allen-Cahn equation.
problem Proving uniform C2,α regularity for transition layers. method Utilizes Allen-Cahn monotonicity formula, Lipschitz approximation, and blowups.
result Shows uniform C2,α regularity for transition layers converging to smooth mean curvature flows. 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.
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.
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…
Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the BV quantization of a one-dimensional sigma model with target X…
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.
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.
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.
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 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…
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…
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…
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…
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 …
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…
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…
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.
In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a …
We introduce BV-algebra structures on the homology of the space of framed long knots in Rn in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The …
We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by C1+bv diffeomorphisms on the circle, which generalizes a result of Farb-Franks, and which parallels a result of Ghys and Burger-Monod concerning differentiable actions of higher ran…
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.
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.
Quantization of (-1)-shifted derived Poisson manifolds via BV-infinity operators.
problem Quantizing (−1)-shifted derived Poisson manifolds. method Using BV-infinity operators on the space of Berezinian half-densities, proving quantization via lifting of Maurer-Cartan elements.
result Quantization of (−1)-shifted derived Poisson manifolds is equivalent to the vanishing of the second Poisson cohomology group.