Study shows flows with critical forcing term satisfy area change formula.
problem Analyzing flows with critical forcing terms and their area change.
method Identifying minimal conditions for Brakke flows to satisfy area change formula.
result Generalized BV flows with critical forcing terms have a lower bound for extinction time.
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.
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.
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.
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.
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. 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…
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. New affine BV-capacity differs from classic in higher dimensions.
problem Classic BV-capacity limitations in higher dimensions.
method Geometric-measure-theoretic study of affine BV-capacity.
result Affine BV-capacity is distinct from classic in higher dimensions.
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…
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.
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.
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. 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.
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…
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.
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.
Analyzes soap films using BV functions and covering spaces.
problem Solving Plateau's problem with soap films.
method Covering space method with constrained BV functions.
result Examples of soap films not modelable with Reifenberg method.
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.
Perturbs gauge theories on manifolds with boundary.
problem Quantum gauge theories on manifolds with boundary.
method Cohomological symplectic (BV-BFV) formalism for a general perturbative quantization scheme.
result Explicit examples of abelian BF theory and its perturbations.
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 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.
Explains a new method for quantizing field theories with boundaries.
problem Quantizing field theories on manifolds with boundaries.
method BV-BFV formalism for perturbative quantization.
result Demonstrates Chern-Simons theory as a new example.
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…
The paper proves a regularity theorem for Brakke flows near triple junctions.
problem Understanding the structure of triple junctions in Brakke flows.
method Establishes the ε-regularity theorem for k-dimensional Brakke flows near static, multiplicity-one triple junctions.
result The regular structure of triple junctions persists under weak mean curvature flow.
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.
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.
Survey discusses recent advances on Brakke flows and their properties.
problem Existence and regularity of Brakke flows.
method Analyzes recent progress including new theorems and proofs.
result Proof that branching singularities trigger dynamical instability.
Quantizes symplectic manifolds using BV quantization and Fedosov's method.
problem Quantize symplectic manifolds rigorously.
method Import physical interpretation, use Fedosov's deformation quantization, and BV quantization of sigma models.
result Deduce algebraic index theorem via semi-classical analysis and Feynman diagram computations.
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 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…
Study shows diffeomorphism groups cannot embed certain groups, especially free products.
problem Embedding free products into diffeomorphism groups.
method Analyzing groups and diffeomorphisms on compact manifolds.
result Free products cannot embed into Diff1+bv(M) for certain groups. 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.
Abstract study shows Milnor metric from BF theory partition function.
problem Milnor metric on manifold determinant line.
method Batalin--Vilkovisky formalism, BV pushforward to cohomology.
result Recovery of Milnor metric from BF theory partition function. 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.
The abstract discusses connecting quantum mechanics and algebraic index theories.
problem Exploring the connection between quantum mechanics and algebraic index theories.
method Explains how the classical algebraic index theorem can be proved in terms of BV quantization of topological quantum mechanics and 2d chiral CFT.
result Shows how the generating function of all genus Gromov-Witten invariants on elliptic curves is mirror equivalent to an elliptic chiral index.