New formula splits instanton Floer homology invariants.
problem Expressing Seiberg-Witten invariants in terms of other invariants.
method Observation of a splitting formula in instanton Floer homology.
result A new formula for instanton Floer homology invariants.
New formulas for knot invariants under specific surgeries.
problem Defining and proving splitting formulas for knot invariants.
method Defined a functorial extension of the Kricker invariant and proved splitting formulas for null Lagrangian-preserving surgery.
result Splitting formulas for the Kricker invariant under null Lagrangian-preserving surgery.
We describe a relation between Atiyah-Patodi-Singer boundary condition and a global elliptic boundary condition which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hoermander index as…
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
The paper proves integral formulas for manifolds with multiple orthogonal distributions.
problem Understanding geometric properties of manifolds with multiple orthogonal distributions.
method Develops integral formulas for Riemannian manifolds with k>2 orthogonal complementary distributions. result Generalizes known formulas for k=2 and applies to manifold splitting and immersions. Integral formulas for metric-affine spaces with specific distributions.
problem Finding geometrical obstructions for distributions and foliations.
method Integral formulas involving Ricci and scalar curvatures, second fundamental forms, and integrability tensors.
result Splitting of manifolds and geometrical obstructions for distributions and foliations.
The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.
problem Quantifying rigidity in manifolds with nonnegative Ricci curvature.
method Investigates pinching of Colding's monotone functionals and constructs k-splitting functions. result Quantitative control of splitting functions by pinching at independent points controls the distance to the nearest cone.
Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.
problem Vertical isomorphisms of Fedosov dg manifolds associated with Lie pairs.
method Construction of Fedosov dg manifolds via splitting and connection, proving unique isomorphisms using iteration formula.
result Existence and uniqueness of vertical isomorphisms between Fedosov dg manifolds.
Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.
problem Calculating the Atiyah-Patodi-Singer index without invertibility of boundary operator.
method Using an asymptotic gluing formula for eta invariants and a splitting principle.
result Formula expressing index in terms of eta invariants of domain-wall massive Dirac operators.
First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting formula (Theorem 4.12) for the spectral flow of a curve of selfadjoint elliptic opera…
We derive a decomposition formula for the spectral flow of a 1-parameter family of self-adjoint Dirac operators on an odd-dimensional manifold M split along a hypersurface Σ (M=X∪ΣY). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula tak…
New formula proves skein modules are finite for 3-manifolds.
problem Proving skein modules are finite for closed 3-manifolds.
method Using Heegaard splittings and algebraic computation.
result Skein modules are finite-dimensional, resolving a conjecture.
For rational homology 3-spheres, there exist two universal finite-type invariants: the Le-Murakami-Ohtsuki invariant and the Kontsevich-Kuperberg-Thurston invariant. These invariants take values in the same space of "Jacobi diagrams", but it is not known whether they are equal. In 2004, Lescop proved that the KKT invar…
We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.
This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
Two tropical gluing formulas help calculate Gromov-Witten invariants.
problem Calculating Gromov-Witten invariants of symplectic manifolds.
method Tropical geometry applied to exploded manifolds.
result Generalizes existing formulas for Gromov-Witten invariants.
We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Pato…
We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimen…
We first present three graphic surgery formulae for the degree n part Zn of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form ∑I⊂N(−1)♯IZn(MI) where N is the set of com…
The paper studies a splitting theorem for a specific invariant of 4-manifolds.
problem Obstructing metrics of positive scalar curvature and finding integral homology 3-spheres.
method Proves a splitting formula for the Seiberg-Witten invariant in terms of the Frøyshov invariant and Lefschetz number.
result New classes of integral homology 3-spheres with infinite order in the homology cobordism group.
It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvatur…
Abstract: Geometrically describes Poisson cohomology groups around symplectic leaves.
problem Understanding the first Poisson cohomology groups around symplectic leaves.
method Splitting theorems for infinitesimal automorphisms of coupling Poisson structures.
result Derives criteria for vanishing of first Poisson cohomology groups.
New formulas for p-capacitary potentials in convex domains.
problem Analyzing geometric properties of p-capacitary potentials.
method Monotonicity formulas derived from p-Laplace equation solutions.
result New characterizations of rotationally symmetric solutions and domains.
Clustering evaluation measures are frequently used to evaluate the performance of algorithms. However, most measures are not properly normalized and ignore some information in the inherent structure of clusterings. We model the relation between two clusterings as a bipartite graph and propose a general component-based …
We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked poin…
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
Study shows splitting schemes can approximate WFR flows faster than the exact flow.
problem Improving sampling efficiency in Wasserstein-Fisher-Rao gradient flows.
method Investigates operator splitting techniques to numerically approximate WFR flows.
result A judicious choice of step size and operator ordering can lead to faster convergence of split schemes to the target distribution.
Derives a new formula for optimal stopping problems with exploding derivatives.
problem Optimal stopping problems with complex boundary conditions.
method Develops a change of variable formula for functions with exploding derivatives near a surface.
result Derives a formula similar to Itô's but with less restrictive conditions.
M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behavi…
The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.
problem Integral and variation formulas for mixed scalar curvature in multi-product manifolds.
method Generalizes results from pseudo-Riemannian almost product manifolds to multi-product structures.
result Generalizes formulas for mixed scalar curvature in multi-product manifolds.
The paper derives an integral formula for mixed scalar curvature of singular distributions.
problem Differential geometry of singular distributions on Riemannian manifolds.
method Proves divergence theorem and Codazzi equation for singular distributions.
result Derives an integral formula for mixed scalar curvature of singular distributions.
Two new rational formulae for normal implied volatility are presented.
problem Calculating normal implied volatility using iterative methods.
method Two explicit rational formulae that avoid iteration and logarithms.
result Accurate and fast formulae for normal implied volatility.
Study parallel tractors and cotractors on almost Grassmannian structures.
problem Characterize parallel tractors and cotractors on almost Grassmannian structures.
method Provide explicit formulae for splitting operators, first BGG operators, and prolongation connections. Characterize solutions of the BGG operators geometrically.
result Describe the geometry of the zero locus of solutions of the first BGG operators.
Matrix formulas for knot invariants derived from Tait graphs.
problem Computing knot invariants for alternating links.
method Squarefree matrix extraction from Tait graph vertices.
result Explicit formulas for CWRk for k≥4. Paper analyzes holdout cross-validation for large non-Gaussian covariance estimation.
problem Estimating large covariance matrices for non-Gaussian data.
method Use of Weingarten calculus and Ledoit-Péché formula for theoretical error derivation.
result Optimal train-test split ratio is proportional to square root of matrix dimension.
We study manifolds with split-complex structure and apply some general results to the study of Lorentz surfaces. In particular, we apply our results to timelike minimal immersions. The conformal realization of these surfaces is obtained using a representation based on loop groups. The classical Weierstrass representati…
We study the (standard) cohomology Hst∙(E) of a Courant algebroid E. We prove that if E is transitive, the standard cohomology coincides with the naive cohomology Hnaive∙(E) as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its stan…
This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.
problem Analyzing the Bochner formula for Riemannian flows and deriving eigenvalue estimates.
method The approach involves studying the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M, splitting it into two parts, and establishing eigenvalue estimates.
result Established an eigenvalue estimate of the basic Laplacian on basic forms, and discussed the limiting case of the estimate.
Spectral sequence connects link surgeries to Khovanov homology.
problem Understanding Khovanov homology of link surgeries.
method Developed a spectral sequence relating rational Khovanov homology to surgeries.
result Explicit splitting formula for Jones polynomial derived.
We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansion…
Formula for spectrum linking braid and bridge indices.
problem Understanding the relationship between braid and bridge indices.
method Foliation theory applied to booklink embeddings.
result Formula for link spectra of braid and bridge indices.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
Quantitative analysis of order-splitting behavior in Japanese stock market.
problem Understanding and quantifying the order-splitting behavior of traders in the Japanese stock market.
method Analysis of a large dataset of trading accounts over nine years, clustering traders into order-splitting and random traders, and applying statistical methods to analyze metaorder length and sign correlation.
result The metaorder length distribution follows power laws with exponent α, and the sign correlation exponent γ is approximately α-1, supporting the LMF model.
A new numerical scheme approximates nonlinear filtering densities for noisy and partial measurements.
problem Approximating nonlinear filtering densities for noisy and partial measurements.
method Deep splitting scheme applied to the Fokker--Planck equation followed by Bayes' formula.
result Convergence rate established for the numerical scheme under parabolic Hörmander condition.
The paper finds formulas for flat models of certain Lie algebras.
problem Finding formulas for flat models of Lie algebras.
method Solving linear algebraic equations based on Lie algebra representations.
result Formulas for flat models of Lie algebras f4 and e6. We introduce a new approach to the study of timelike minimal surfaces in the Lorentz-Minkowski space through a split-complex representation formula for this kind of surface. As applications, we solve the Björling problem for timelike surfaces and obtain interesting examples and related results. Using the Björling repre…
This paper proves a symplectic formula for SU(n) generalized Casson invariants.
problem Establishing a relationship between gauge theory and generalized Casson invariants.
method Using intersection numbers of Lagrangian submanifolds.
result Analogous identification result for SU(n) generalized Casson invariants.
New map constructed from equivariant spectra for manifold study.
problem Understanding equivariant parametrized h-cobordism in non-manifold settings.
method Constructed a map from suspension G-spectrum to equivariant A-theory spectrum, compatible with tom Dieck splitting formulas.
result Fiber of constructed map is wedge of stable h-cobordism spectra.