The paper proves a gravity conjecture in higher dimensions.
problem The validity of Wheeler's thin sandwich conjecture in arbitrary dimensions.
method Extending results from 3D to nD, the paper shows geometric conditions for the conjecture's validity.
result The thin sandwich problem is well-posed on any compact n-dimensional manifold, n ≥ 3.
Paper analyzes RTSE on AE manifolds and closed manifolds, showing well-posedness and parametrization of solutions.
problem Analyzing well-posedness of RTSE on AE manifolds and closed manifolds.
method Analyzes RTSE on AE manifolds and closed manifolds, considering specific conditions for well-posedness.
result On AE manifolds, RTSE solutions parametrize an open subset in the space of ECE solutions.
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
Proves Pólya's conjecture for thin products and Riemannian manifolds.
problem Proving Pólya's conjecture for specific geometric domains.
method Analyzes thin products and Riemannian manifolds, proving inequalities for eigenvalues.
result Proves Pólya's conjecture for thin products and related Riemannian manifolds.
Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite knot K, we use the companion torus T to provide a lower bound for w(K), proving th…
The paper proves that certain thin links have only Z2 torsion in Khovanov homology.
problem Understanding torsion in Khovanov homology for thin links.
method Local analysis of Khovanov homology for links supported on two adjacent diagonals.
result Khovanov homology of thin links has only Z2 torsion over a specified range of homological gradings.
The Ham Sandwich Theorem helps find bounds on Laplacian eigenvalues.
problem Finding bounds on Laplacian eigenvalues for convex domains.
method Applied Gromov's ham sandwich method to get monotonicity and inequalities.
result Universal inequalities for Neumann eigenvalues of Laplacian on convex domains.
Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.
problem Properties of twisted Mazur pattern satellite knots.
method Use bordered Floer theory to analyze knots and calculate genus.
result Prove non-thinness of Qn(K) and calculate 3-genus in terms of n and K. New method solves Einstein constraint equations with non-constant mean curvature.
problem Solving Einstein constraint equations with non-constant mean curvature.
method Drift method, which compensates for greater analytic complexity.
result Method can handle metrics with conformal Killing but not true Killing vector fields.
New financial model with sandwiched volatility for option pricing.
problem Developing a new financial model for option pricing.
method Introducing a new model with stochastic volatility driven by a Gaussian Volterra process, ensuring the solution is sandwiched between two arbitrary Hölder continuous functions.
result Developed an algorithm for pricing options with discontinuous payoffs using Malliavin calculus.
Study symplectic fillings of sandwiched singularities.
problem Contrast deformation theory and symplectic topology of Milnor fibers.
method Develop an analog of de Jong--van Straten's theory in the symplectic setting using spinal open books and nearly Lefschetz fibrations.
result Minimal symplectic fillings of links are generated by certain immersed disk arrangements.
Two knots with unique surgery properties.
problem Characterizing strongly invertible L-space knots.
method Examined surgeries and knot properties.
result Found knots whose surgeries are never Khovanov thin.
In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over \Q an…
The paper proves a geodesic sandwich theorem and applies it to manifold inequalities.
problem Proving inequalities in Riemannian manifolds with bounded curvature.
method Geodesic convex functions and sectional curvature bounds.
result Gradient of convex functions is orthogonal to geodesics.
The paper analyzes defects on structured surfaces and calculates stress and shape.
problem Analyzing defects on structured surfaces and their effects on stress and shape.
method Classified and quantified defects, derived strain incompatibility relations, and applied to shells.
result Determined internal stress field and deformed shape for shells with defects.
Geometric proof of curve characterization using loop-bundles.
problem Characterization of simple curves in terms of the Goldman-Turaev bracket.
method Combining combinatorial approach with geometric proof.
result Geometric proof of curve characterization conjecture.
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagra…
We prove torsion constraints in Khovanov homology for certain links.
problem Understanding torsion in Khovanov homology for specific knot types.
method Establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over Z2. result Prove no 2k-torsion for k>1 in Khovanov homology of Z2-thin links. In math.GT/0106017 it was shown that thin position on Heegaard spines can be a useful tool for analyzing the topology of knots in 3-space. The proof there (specifically, of the Goda-Teragaito conjecture) requires masses of technical detail; it is easy to lose track of the underlying ideas. The present paper gives an ov…
Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.
problem Finding invariant measures for contact Hamiltonian systems.
method Splitting the system into Reeb and Liouville dynamics; using invariant measures and symplectic sandwiches.
result Invariant measure found for Reeb dynamics; characterization of Liouville dynamics invariant measure.
This article, written to appear as a chapter in "The Springer Handbook of Spacetime", is a review of the initial value problem for Einstein's gravitational field theory in general relativity. Designed to be accessible to graduate students who have taken a first course in general relativity, the article first discusses …
This paper develops a method to estimate the rate-distortion function for general data sources.
problem Estimating the rate-distortion function for general data sources.
method Develops an algorithm for sandwiching the R-D function of a general (not necessarily discrete) source using i.i.d. data samples.
result Estimates R-D sandwich bounds for various data sources, including natural images, indicating potential for improving compression methods.
A new model optimizes Bloom filters using machine learning.
problem Improving the efficiency of Bloom filters for data sets.
method Modeling learned Bloom filters with machine learning, optimizing with sandwiching method.
result Optimized learned Bloom filters provide improved performance.
Study predicts evolution patterns for pretzel knots, revealing abrupt transitions and hidden non-linearity.
problem Predicting evolution of Khovanov polynomials for pretzel knots.
method Conjectured explicit evolution formulas, revealed abrupt transitions, and identified additional Lyapunov exponents.
result Abrupt transitions and hidden non-linearity in evolution of Khovanov polynomials for thick knots.
We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of ra…
We prove some rigidity theorems for configurations of closed disks. First, fix two collections C and C~ of closed disks in the Riemann sphere C^, sharing a contact graph which (mostly-)triangulates C^, so that for all corresponding pairs of intersecting dis…
We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental…
New families of non-tiling domains satisfy Pólya's conjecture.
problem Finding non-tiling domains that satisfy Pólya's conjecture.
method Analyzing partitioning and eigenvalue orders of domains.
result Existence of families of non-tiling domains satisfying Pólya's conjecture.
Maximal extractable value in CFMMs can degrade or improve routing quality, with reordering MEV showing logarithmic impact.
problem Maximal extractable value in constant function market makers (CFMMs) and its impact on routing quality.
method Game theoretic analysis of MEV in CFMMs, constructing price of anarchy and analyzing reordering MEV.
result Conditions under which reordering MEV shows logarithmic impact, and implications for MEV searchers and CFMM designers.
New thin position for graphs in 3-manifolds combines ideas from knot theory and 3-manifold topology.
problem Defining a new thin position for graphs in 3-manifolds.
method Combining thin position concepts from knot theory and 3-manifold topology.
result Defines new invariants of knots, links, and graphs in 3-manifolds.
The paper examines rigidity of thin domains under specific boundary conditions.
problem Linear geometric rigidity of shallow thin domains with zero Dirichlet boundary conditions.
method Analyzes two scaling regimes for ε in (h, √h] and (√h, 1), proving rigidity formulas.
result Rigidity does not depend on curvature in the small parameter regime ε ∈ (h, √h].
Paper proves SVV model reproduces power-law skew in implied volatilities.
problem Reproducing power-law behavior in implied volatility skew.
method Analytical proof using Malliavin calculus and Volterra kernel selection.
result SVV model reproduces power-law skew under correct kernel choice.
If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.
We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.
A theorem divides hyperplanes evenly with a line through the origin.
problem Dividing hyperplanes evenly with a line.
method Direct proof using measures on hyperplanes.
result A line through the origin divides hyperplanes evenly.
The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.
problem Understanding thin subgroups in special linear groups.
method Constructing and embedding non-arithmetic hyperbolic manifolds into SL(n+1)(R).
result Non-arithmetic lattices in SO(n,1) can be embedded into SL(n+1)(R) as thin subgroups.
Study on wormholes in modified gravity using generalized geometry.
problem Existence and conditions for thin shell wormholes in F(R)-gravity. method Used Colombeau algebra to define generalized geometry and analyze wormholes.
result Suitable quadratic F can satisfy the null energy condition (NEC). We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The exi…
In this paper, we propose new efficient algorithms to verify the null space condition in compressed sensing (CS). Given an (n−m)×n (m>0) CS matrix A and a positive k, we are interested in computing αk={z:Az=0,z=0}max{K:∣K∣≤k}max ∥zK∥1∥z∥1, where …
We describe a natural decomposition of a normal complex surface singularity (X,0) into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…
Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and…
Abby Thompson proved that if a link K is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…
Polyak-Ruppert CLT for SA-Adam with momentum and non-convergent adaptive preconditioning
problem Adaptive optimizers combining momentum and non-convergent preconditioning
method Proving positive drift stability and a non-autonomous Polyak-Ruppert CLT for SA-Adam
result The iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich
New method characterizes thin links via Conway spheres and tangle decompositions.
problem Characterize thin links without relying on specific knot invariants.
method Developed a relative version of thinness for tangles and used it to characterize thinness via tangle decompositions along Conway spheres.
result Characterized thin links via Conway spheres and tangle decompositions.
The paper explores knots' height, trunk, and representativity, finding gaps and bounds.
problem Investigating the properties of knots and their invariants.
method Analyzing conjectures, defining new invariants, and comparing knot positions.
result Found gaps between height and minimal height, and bounds on representativity.
Regularized Stein thinning improves MCMC output approximations.
problem Pathologies in Stein thinning leading to poor approximations.
method Theoretical analysis and regularization to improve KSD.
result Regularized Stein thinning alleviates pathologies and improves efficiency.
The paper studies eigenvalues in negatively curved spaces, showing their relationship to the thick-thin decomposition.
problem Investigating the relationship between eigenvalues and the geometric structure of negatively curved spaces.
method Analyzing the eigenvalues of the Laplacian on the thick part of a manifold's thick-thin decomposition.
result Eigenvalues in the thick part of a negatively curved manifold are significantly smaller than those in the entire manifold.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.