The paper resolves kinks on curves on surfaces with punctures.
problem Resolving kinks of curves on surfaces with punctures.
method Application of the Diamond Lemma.
result The resolution of kinks of a curve on a surface with punctures is uniquely determined up to homotopy.
New knot invariants created using multiple skein relations.
problem Creating more powerful knot invariants.
method Introducing new ways to smooth crossings and using a system of skein equations.
result A simplified version of the modified invariant is easier to compute and generalizes the two-variable Kauffman polynomial.
Diamonds help compute volatility models efficiently.
problem Computing volatility models in forward variance form.
method Application of diamond trees and forests.
result Efficient computation of volatility models.
Study of circle homeomorphisms with square summable diamond shears.
problem Characterizing circle homeomorphisms with specific summability properties.
method Analysis of homeomorphisms in modular coordinates and comparison to Weil-Petersson class.
result Sharp results comparing new class to Weil-Petersson class and Hölder classes.
Develops a diamond price index for online auction platforms.
problem Tracking market trends of wholesale diamond prices.
method Modelling diamond prices to create a hedonic index.
result Provides a basis for constructing derivatives for collectables.
Study defines new products for Lorentzian spaces and analyzes causal diamonds.
problem Understanding causal diamonds in Lorentzian spaces.
method Introduced taxicab and uniform products for Lorentzian pre-length spaces. Defined D(RimesTX) space and analyzed its properties. result The space D(RimesTX) is geodesic and globally hyperbolic for complete X. Soft diamond regularizers improve deep learning performance and sparsity.
problem Improving deep learning performance and sparsity of trained weights.
method New soft diamond synaptic weight priors based on thick-tailed symmetric alpha stable probability curves.
result Soft diamond regularizers outperform state-of-the-art methods in deep learning tasks.
We give a geometrically intrinsic construction of a global time function for relatively compact diamond-shaped regions in arbitrary spacetimes. In the case of Minkowski spacetime, the flow of diffeomorphisms associated to a suitably normalized gradient of this time function becomes the conformal isotropy subgroup of th…
Defines doubling conditions for Lorentzian spaces linked to curvature bounds.
problem Relating doubling conditions to curvature bounds in Lorentzian geometry.
method Defines doubling conditions using chronological diamonds and proves implications with timelike curvature bounds.
result Relates doubling to curvature bounds in Lorentzian geometry.
New tool helps classify invariant subvarieties in degenerations.
problem Classifying invariant subvarieties in degenerations.
method Introducing diamonds of GL(2,R)-invariant subvarieties.
result Classified a rich collection of diamonds.
The geometry of causal diamonds or Alexandrov open sets whose initial and final events p and q respectively have a proper-time separation τ small compared with the curvature scale is a universal. The corrections from flat space are given as a power series in τ whose coefficients involve the curvature at the cen…
Diamond method controls FDR for trustworthy feature interaction discovery in ML models.
problem Limited interpretability of ML models due to black box nature.
method Diamond method integrates model-X knockoffs framework to control FDR for non-additive interactions.
result Diamond method ensures accurate discovery of feature interactions with FDR control.
The paper establishes inequalities for causal diamonds in curved and Minkowski spacetimes.
problem Formulating inequalities for geometric quantities of causal diamonds in curved and Minkowski spacetimes.
method Formulating inequalities involving the red-shift factor and analyzing its monotonicity and maximal value.
result Established inequalities for causal diamonds in curved and Minkowski spacetimes, including a universal inequality for all spacetime dimensions.
Paper introduces a new convergence for Lorentzian spaces using causal diamonds.
problem No specific problem stated; focuses on a new geometric convergence.
method Uses causal diamonds to define a new convergence for Lorentzian spaces.
result Proves a pre-compactness theorem for Lorentzian spaces.
Generalizes global hyperbolicity to higher signatures and proves compactness.
problem Global hyperbolicity in higher pseudo-Riemannian signatures.
method Generalization and proof of compactness of causal diamonds.
result Existence of solutions to a Plateau problem and characterization of holonomies.
In this note we prove that the volume of a causal diamond associated with an inertial observer in asymptotically de Sitter 4-dimensional space-time is monotonically increasing function of cosmological time. The asymptotic value of the volume is that of in maximally symmetric de Sitter space-time. The monotonic property…
3-manifolds foliar if surfaces minimize Thurston norm, leading to taut foliations.
problem Understanding foliar 3-manifolds via minimizing surfaces.
method Defining double-diamond taut sutured manifolds and analyzing their implications.
result Many slopes are strongly realized in foliations of 3-manifolds, including knot complements.
Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.
problem No specific problem stated; focuses on defining a new geometric dimension.
method Introduces a one-parameter family of volume measures and a doubling condition for causal diamonds.
result Defines a geometric dimension for synthetic spacetimes, distinguishing between spacelike and null subspaces.
In a previous paper we obtained formulae for the volume of a causal diamond or Alexandrov open set I+(p)∩I−(q) whose duration τ(p,q) is short compared with the curvature scale. In the present paper we obtain asymptotic formulae valid when the point q recedes to the future boundary I+ of an asymp…
Two surfaces minimize variance of Gaussian curvature.
problem Minimizing the variance of Gaussian curvature on triply periodic minimal surfaces.
method Interpreting branch values of Gauss map, expressing variance as integrals of exponentials of Green's functions, analyzing Hessian.
result The P and D surfaces are local minimizers of the variance of Gaussian curvature.
Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
Whitehead's Lemma is a simple result of Stallings folds.
problem None explicitly stated; Whitehead's Lemma is observed as a consequence.
method Observation of Whitehead's Lemma as a consequence of Stallings folds.
result Whitehead's Lemma is a simple result of Stallings folds.
Study on Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
problem Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
method Elementary variational approach, Lorentzian isoperimetric problem, uniform estimate of causal diamonds.
result Heisenberg group has Lorentzian Hausdorff dimension 4 and satisfies neither timelike curvature-dimension nor measure contraction properties.
New curvature measure for causal sets derived from optimal transport.
problem Capturing Ricci curvature in causal sets.
method Using Lorentzian optimal transport, novel curvature defined along maximal chains.
result Recovery of timelike Ricci curvature from order-theoretic data.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
New CR-structures lemma simplifies CR-manifold deformation proof.
problem Deformation unobstructedness of CR-manifolds.
method New Tian-Todorov lemma applied to CR-manifolds.
result Reproved deformation unobstructedness of CR-manifolds.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Study several weak forms of a lemma on compact complex manifolds.
problem Understanding weak forms of a lemma on compact complex manifolds.
method Complete unified study of weak forms of the $\ddb-$Lemma.
result Unified understanding of various weak forms of the $\ddb-$Lemma.
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Selberg's Lemma fails for certain curved manifolds.
problem Applying Selberg's Lemma to negatively curved Hadamard manifolds.
method Proving the failure of Selberg's Lemma for specific groups of manifolds.
result Selberg's Lemma does not hold for discrete isometry groups of negatively curved Hadamard manifolds.
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
We show that K3 surfaces with non-symplectic automorphisms of prime order can be used to construct new compact irreducible G2-manifolds. This technique was carried out in detail by Kovalev and Lee for non-symplectic involutions. We use Chen-Ruan orbifold cohomology to determine the Hodge diamonds of certain complex thr…
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.
problem Improving the conditions under which wave front germs imply map germs.
method Generalization of Zakalyukin's lemma for frontals and applications to surface singularities.
result The paper provides a more general version of Zakalyukin's lemma for map germs.
Paper proves almost Schur Lemma on smooth metric measure spaces.
problem Proving a specific lemma on metric measure spaces.
method Proves almost Schur Lemma using closed smooth metric measure spaces.
result Implications of the lemma for X. Cheng's and De Lellis-Topping's results.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
Proves a quantitative closing lemma for negatively curved manifolds.
problem Closing lemma for negatively curved manifolds.
method Quantitative closing lemma proof.
result Study of partner and pseudo-partner orbits for self-crossing closed geodesics.
Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.
problem Construct minimal surfaces over Pitot quadrilaterals.
method Develops a fully explicit framework using harmonic diffeomorphisms and Weierstrass data.
result Constructs a unique minimal surface \(Σ^\diamond\) that maximizes Gaussian curvature.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.
Study joins and intersections of subgroups in free groups, disproving and repairing a lemma.
problem Disproving and repairing a lemma on joins and intersections of subgroups in free groups.
method Analyzing graphs of joins and intersections of finitely generated subgroups of a free group.
result Showed how to disprove and repair a lemma of Imrich and Müller on these graphs.
Proves Poincaré lemma on specific sub-Riemannian structures.
problem Solving an open problem on Poincaré lemma for corank 1 sub-Riemannian structures.
method Combines Poincaré lemma on Riemannian manifolds with a path integral formula.
result Necessary and sufficient 'curl-vanishing' conditions for Poincaré lemma.
Corrects a lemma in a 2009 paper about contact homology and Floer homology.
problem An error in a lemma about contact homology and Floer homology.
method None, as it is a correction of an existing lemma.
result Corrects an error in a previously published lemma.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.