Extends unique continuation theorem to manifolds with boundary, proving zero set codimension.
problem Proving unique continuation for exterior differential forms on manifolds with boundary.
method Extends Aronszajn-Krzywicki-Szarski theorem to manifolds with boundary, assuming suitable boundary conditions.
result Hausdorff dimension of zero sets of harmonic forms and eigenfields has codimension at least 2.
Paper links set derivatives to its orthogonal projections.
problem Understanding the relationship between set derivatives and projections.
method Derives equations from topological link between Minkowski functional partial derivatives and set boundary.
result System of equations for orthogonal projections derived.
Improved efficient learning of neighbor representations for large datasets.
problem Efficiently learn neighbor representations for large datasets.
method Differentiable Boundary Sets algorithm that overcomes computational issues and improves accuracy.
result Significant reduction in training time and improved classification accuracy.
This paper classifies components of meromorphic differential strata.
problem Understanding the boundary of multi-scale compactification of meromorphic differentials.
method Classifying connected components of residueless meromorphic differentials.
result Classification of connected components of strata of residueless meromorphic differentials.
We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.
Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.
problem Exact Lagrangian submanifolds with Legendrian boundary in unit ball.
method Uses Liouville form and boundary unique continuation for differential forms.
result Equatorial n-disk rigidity for compact exact Lagrangian self-similar submanifolds. The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
Classifies connected components of meromorphic differentials with residue conditions.
problem Understanding the structure of meromorphic differentials with residue constraints.
method Analyzes the multi-scale compactification and residue conditions.
result Classified connected components of generalized strata of meromorphic differentials.
Proves well-posedness for elliptic problems on domains with singular points.
problem Elliptic boundary value problems on domains with singular points.
method Conformal changes of metric, differential geometry of manifolds with boundary and bounded geometry.
result Proves well-posedness and regularity results for elliptic boundary value problems.
Research on refined algebraic domains respecting differential geometry.
problem Understanding shapes and regions of real algebraic curves.
method Investigates points in two curves, singular points, inflection points, and points of double tangent lines, considering differential geometry.
result Proves fundamental properties and investigates examples of refined algebraic domains.
We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct L2 harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity prop…
Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
Bayesian approach for solving systems of linear PDEs with boundary conditions.
problem Modeling data efficiently with prior knowledge from systems of linear PDEs.
method Construct multi-output Gaussian process priors using Gröbner bases and pullback parametrizations.
result Gaussian process priors can represent solutions to systems of linear PDEs adhering to boundary conditions.
New method uses Gaussian processes for solving linear PDEs with boundary conditions.
problem Solving linear PDEs with boundary conditions.
method Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs).
result Significant accuracy and resource improvements over existing methods.
We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invar…
Connected boundaries of strata of differentials are always connected in various compactifications.
problem Understanding the connectedness of boundaries of differentials' strata in various compactifications.
method Explicit degeneration techniques, algebraic compactifications, and properties of Teichmüller curves.
result The boundaries of differentials' strata are always connected in any complete algebraic compactification.
Smooth solutions found for hydrodynamic equations.
problem Geodesic equations on diffeomorphism groups.
method Proving smoothness of solutions and constructing exponential maps.
result Smooth solutions match boundary conditions.
Derives numerical formulas for elliptic differential operators on specific groupoids.
problem Index problem of elliptic differential operators on boundary groupoids.
method Similar to Moroianu and Nistor's renormalized trace approach, focusing on eta and Atiyah-Singer terms.
result For q≥3, K-theoretic and Fredholm indices are given by the Atiyah-Singer term. Holomorphic bundles on complex manifolds with boundary are studied, extending results from Donaldson's work.
problem Extending holomorphic structures to complex manifolds with boundary.
method Analyzing formally integrable almost complex structures and their extensions to holomorphic structures.
result Holomorphic structures can be extended to a neighborhood of a strictly pseudoconvex boundary.
The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted dif…
New proof of Gaffney's inequality for differential forms on manifolds with boundary.
problem Proving Gaffney's inequality for differential forms on manifolds with boundary.
method Variational approach combined with Bochner's technique.
result New proof of Gaffney's inequality for differential forms.
The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of ze…
Eigenvalue bounds for forms on warped manifolds studied.
problem Eigenvalue bounds for differential forms on warped product manifolds.
method Geometric eigenvalue bounds in warped product manifolds with non-negative Ricci curvature and strictly convex boundary.
result Escobar type lower bounds and sharp bounds for specific cases.
New Poincaré inequality for differential forms on manifolds.
problem Developing inequalities for differential forms on manifolds.
method Proving a new Poincaré-type inequality and deriving new inequalities involving mean and scalar curvatures.
result Characterized the limiting case of a new inequality involving mean and scalar curvatures of the boundary.
We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…
Extends machine learning models for analytic boundary conditions in differential equations.
problem Inclusion of data in differential equations using symbolic algorithms.
method Combines computer algebra with Gaussian processes and extends to analytic boundary conditions using Gröbner and Janet bases of Weyl algebras.
result Describes divergence-free flow in domains bounded by analytic functions.
Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.
problem Biharmonic Steklov problems with Neumann boundary conditions.
method Introduced a biharmonic Steklov problem and proved its well-posedness. Established eigenvalue estimates using Kuttler-Sigillito inequalities.
result Eigenvalue estimates for the biharmonic Steklov problem with Neumann boundary conditions.
We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair (X,W), where X is any non-compact Calabi-Yau manifold and W is any holomorphic complex-valued function defined on X whose critical set is compact. The models are constructed at cochain level …
Derives integral formula for differential forms on compact spaces with applications.
problem Integral formula for differential forms on compact spaces with boundary.
method Derives a weighted Reilly type integral formula.
result Lower bounds for spectrum and eigenvalues of differential forms.
Study boundary value problems for elliptic operators on manifolds.
problem Characterize and analyze boundary conditions for first-order elliptic differential operators.
method Develops a new framework for elliptic boundary conditions, proving equivalence and regularity of solutions.
result Elliptic boundary conditions yield a Fredholm operator on compact manifolds.
SageMath package diffstrata calculates intersection theory on abelian differentials.
problem Computing intersection theory on the boundary of strata of abelian differentials.
method Explicit combinatorial description of the boundary, implemented algorithms in SageMath.
result Computes the Euler characteristic of strata using intersection theory.
The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary co…
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
Recent research connects Hörmander's old work to modern boundary Laplacian analysis.
problem How close is the Dirichlet-to-Neumann map to the boundary Laplacian?
method Investigates techniques from Hörmander's 1950s manuscript to solve modern boundary Laplacian problems.
result Obtained results for DtN maps on non-smooth boundaries, Helmholtz equation, and differential forms.
New BdryMatérn GP model for reliable boundary integration on irregular domains.
problem Incorporating boundary information in Gaussian process models for complex phenomena.
method Proposes a novel BdryMatérn GP framework with a new covariance kernel derived via path integral and stochastic PDE.
result Sample paths from the BdryMatérn GP satisfy desired boundaries with smoothness control on derivatives.
Proves path connectedness of asymptotically flat metrics with boundary.
problem Proving path connectedness of asymptotically flat metrics with boundary.
method Generalization of Marques' result to compact manifolds with boundary, differential topology, and a new proof.
result Space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on R^3\B^3 is path connected.
On a time-oriented Lorentzian manifold (M,g) with non-empty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets S⊂M of sources is uniquely determined by measurements of the intersection of future light cones from points in S …
Generalizes differentiation under integral sign to submanifolds with corners.
problem Closing a gap in mathematical literature for evolving submanifolds with corners.
method Proves generalizations of the Reynolds Transport Theorem for submanifolds with corners.
result Provides a unified treatment of integral theorems for unbounded cases.
We consider a notion of conservation for the heat semigroup associated to a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitabl…
Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
Clarifies when solutions to stochastic PDEs stay near given subsets.
problem Understanding the proximity of solutions to stochastic PDEs to given subsets.
method Analyzes distance between closed sets and solutions to stochastic PDEs.
result Clarifies conditions for solutions to stay near given subsets.
We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Study solves inverse problems for real principal type operators using unique data sets and ray transforms.
problem Determining coefficients in real principal type equations from boundary data.
method Unique data sets, bicharacteristic ray transforms, and propagation of singularities.
result Global uniqueness results for determining coefficients in nonlinear real principal type equations.
Study shows swirling flows destabilize boundary regions.
problem Stability of swirling flows near flat boundaries.
method Differential geometric approach to prove instability.
result Proves flows have a destabilizing effect near boundaries.
Study on biharmonic Steklov problem on differential forms.
problem Characterize and estimate eigenvalues of biharmonic Steklov problem.
method Introduce boundary conditions, prove properties, derive inequalities.
result Characterize smallest eigenvalue and prove spectrum properties.
Global calculus for manifolds with boundary, solving evolution problems.
problem Global solvability of evolution problems on manifolds with boundary.
method Established global functional calculus and Gårding inequality for pseudo-differential operators without local coordinates.
result Global solvability for a class of evolution problems.
Constructs a support-preserving homotopy for differential forms with boundary decay estimates.
problem Non-uniqueness of chain homotopies in de Rham complexes with boundary decay properties.
method Constructs a specific chain homotopy with desirable support propagation and boundary decay estimates.
result Obtains a support-preserving right inverse of the divergence operator with optimal decay estimates.