New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.
problem Characterizing holomorphic tensors on Vaisman manifolds.
method Using the parallelism of the Lee form and properties of the Lee field.
result The Kodaira dimension of Vaisman manifolds is invariant under certain quotients.
We generalize results of Lee, Gornik and Wu on the structure of deformed colored sl(N) link homologies to the case of non-generic deformations. To this end, we use foam technology to give a completely combinatorial construction of Wu's deformed colored sl(N) link homologies. By studying the underlying deformed higher r…
Study Einstein-Yang-Mills fields on specific manifolds, proving field deformations.
problem Deforming Einstein-Yang-Mills fields over conformally compact manifolds.
method Deformation theory using 0-calculus of Mazzeo and Melrose. result Any small perturbation of boundary data can be realized as an Einstein-Yang-Mills field.
Proves a conjecture about knotted spheres using plane Floer homology.
problem Proving a conjecture about knotted spheres.
method Using Daemi's plane Floer homology as a key tool.
result Proves a conjecture regarding the odd Khovanov cobordism maps associated to knotted spheres.
We generalize the works of Lee [arXiv:math/0210213v3] and Gornik [arXiv:math/0402266v2] to construct a basis for generic deformations of the colored sl(N)-homology defined in [arXiv:1002.2662v1]. As applications, we construct non-degenerate pairings and co-pairings which lead to dualities of generic deformations of the…
New homology invariant for links in surfaces, a deformation of APS.
problem Defining a new homology invariant for links in surfaces.
method A 1-parameter family of homology invariants, motivated by instanton Floer homology.
result The new invariant recovers APS homology and has a stronger detection property.
Paper proves optimal systolic inequality for manifolds with positive triRic curvature.
problem Optimal systolic inequality for manifolds with positive triRic curvature.
method Stable weighted k-slicing, volume comparison theorem, and metric deformation. result Proves an optimal systolic inequality and characterizes the equality case.
New q-deformed integers help compute Jones polynomials efficiently.
problem Computing Jones polynomials of rational links efficiently.
method Defining q-deformed integers from pairs of coprime integers and using them to compute Jones polynomials.
result Efficient algorithm for computing Jones polynomials of rational links.
The paper bounds higher Steklov eigenvalues of graphs on surfaces.
problem Bounding higher Steklov eigenvalues of graphs on surfaces.
method Using metrical deformation via probability flows, the upper bound is derived.
result The upper bound of higher Steklov eigenvalues is established.
Study on contracting maps and their rigidity under curvature constraints.
problem Rigidity of contracting maps between manifolds with positive curvature.
method Analysis of curvature pinching and contracting conditions involving singular values.
result Established the relation between curvature pinching and contracting conditions.
We use the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the Khovanov complex to give an alternative way to compute Rasmussen s-invariants of knots. By disregarding generators away from homological degree 0 we can considerably improve …
We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a s…
This paper studies CR manifolds and embeddability in complex spaces.
problem Characterize embeddable deformations of 3D CR manifolds.
method Analyzes complex functions on CR manifolds and uses spherical harmonics.
result Space of embeddable deformations is a Frechet submanifold near the origin.
Study on deformations of LC Spin(7) instantons simplifies the problem.
problem Deformation theory of instantons on locally conformal Spin(7) manifolds.
method Reformulated linearized deformation equations using a t-parameter family of Dirac operators, demonstrating cancellation of torsion terms.
result The deformation space H^1 is governed by Levi-Civita geometry, reducing the problem to a torsion-free setting.
Calabi-Yau theorem extended to Vaisman manifolds.
problem Uniqueness of Vaisman metrics and their characterization.
method Analyzing the Lee form and Lee class properties.
result Vaisman metrics uniquely determined by volume and Lee class.
We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.
We introduce a class of hermitian metrics with {\em Lee potential}, that generalize the notion of l.c.K. metrics with potential introduced in \cite{ov} and show that in the classical examples of Calabi and Eckmann of complex structures on $S^{2p+1}\x S^{2q+1}$, the corresponding hermitian metrics are of this type. Thes…
The study examines Lee metrics on groups and their properties.
problem Characterizing groups that admit Lee metrics.
method Analyzing conditions for groups to have or not have Lee metrics, studying specific families of groups, and providing tables for groups of order ≤ 31.
result Conditions for groups to have Lee metrics, including specific families and non-cyclic groups.
We give a complete description of all locally conformally Kähler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dr…
New theorem on Lee classes for LCK manifolds with potential.
problem Determining Lee classes on LCK manifolds with potential.
method Analyzing cohomology classes of Lee forms and proving the result for Vaisman manifolds.
result The set of Lee classes on LCK manifolds with potential forms an open half-space in H1(M,R). Defines a Rasmussen invariant for links in RP^3.
problem Defines a new invariant for links in the real projective space.
method Constructs a Lee-type deformation of Khovanov homology for links in RP^3.
result Shows the s-invariant gives constraints on link cobordisms in I × RP^3.
Synthetic proof of Gannon-Lee theorem for spacetimes.
problem Proving incompleteness in globally hyperbolic spacetimes.
method Synthetic null energy condition and synthetically asymptotically regular trappedness condition.
result Generalized classical incompleteness theorem to weighted spacetimes.
Paper uses neural networks to calibrate Lee-Carter models for multiple populations.
problem Calibrating Lee-Carter models for multiple populations with neural networks.
method Developed neural network architectures to fit Lee-Carter and Poisson Lee-Carter models simultaneously.
result Smooth and less sensitive parameter estimates, improved forecasting performance.
Proves Gannon-Lee theorem for C1 spacetimes.
problem Classical singularity theorems for C1 spacetimes. method Proves theorem for C1 spacetimes, shows geodesic properties. result Gannon-Lee theorem holds for C1 spacetimes. Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
Classifies surfaces with T-singularities and ample canonical class.
problem Classifying surfaces with T-singularities and specific properties.
method Using KSBA moduli space and techniques for surfaces with T-singularities.
result Identifies surfaces with only T-singularities in the KSBA space and proves non-smoothability conditions.
Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alter…
Study on Lee classes of complex surfaces, proving connectedness and bounds.
problem Understanding Lee classes of complex surfaces with LCS structures.
method Analyzing deRham classes of Lee 1-forms and using properties of PSH functions.
result Connectedness of Lee deRham classes and explicit negative upper bound on hyperbolic Kato surfaces.
This note is devoted to partial study of recurrent equation dω=β∧ω, based on linear algebra of exterior forms. Such equation was considered by Lee, for non-degenerate 2-form. In this note we approach general case, when ω is arbitrary. Particularly, we extend results obtained by Lee, on odd-forms.
Study examines Weyl structures on Riemannian manifolds with vanishing Lee form.
problem Characterizing Weyl structures on Riemannian manifolds with specific properties.
method Analyzes Weyl structures reducible in the direction of the Lee form, proving conditions for flatness or exactness.
result Proves every homogeneous Kenmotsu manifold is isometric to real hyperbolic space.
We discuss a remarkable formula discovered by Jerison and Lee to classify constant scalar curvature pseudohermitian structures on the sphere. We show that the formula is valid in the wider context of Einstein pseudohermitian manifolds. As an application we prove a uniqueness result that generalizes the theorem of Jeris…
Alternative definition of dDT connections for Spin(7) manifolds.
problem Defining deformed Donaldson-Thomas connections for Spin(7) manifolds.
method Real Fourier-Mukai transform applied to compute new connections.
result Alternative definition of dDT connections is compatible with existing connections.
In this paper, we extend the classical Ho-Lee binomial term structure model to the case of time-dependent parameters and, as a result, resolve a drawback associated with the model. This is achieved with the introduction of a more flexible no-arbitrage condition in contrast to the one assumed in the Ho-Lee model.
The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
We show that the page at which the Lee spectral sequence collapses gives a bound on the unknotting number, u(K). In particular, for knots with u(K)<3, we show that the Lee spectral sequence must collapse at the E_2 page. An immediate corollary is that the Knight Move Conjecture is true when u(K)<3.
In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of Δn2 where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentatio…
Calculates knot X-torsion order using spectral sequences.
problem Calculating the X-torsion order of knots. method Using the reduced Bar-Natan--Lee--Turner spectral sequence.
result Example of X-torsion order 4. We give a simple proof of Lee's result from [Adv. Math. 179 (2005) 554-586; arXiv:math.GT/0210213], that the dimension of the Lee variant of the Khovanov homology of a c-component link is 2^c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type theorem for tang…
New link homology theories for RP3 yield distinct invariants.
problem Extending link homologies for links in RP3. method Introducing new Lee and Bar-Natan spectral sequences for links in RP3. result New Lee and Bar-Natan invariants are distinct from existing ones.
Developed a new homology theory for graph chromatic polynomials.
problem Categorification of chromatic polynomials.
method Introduced a new homology theory HLee(G) and developed a spectral sequence. result Spectral sequence converges to HLee(G), supporting H∗(G). In this paper a multi-factor generalization of Ho-Lee model is proposed. In sharp contrast to the classical Ho-Lee, this generalization allows for those movements other than parallel shifts, while it still is described by a recombining tree, and is stationary to be compatible with principal component analysis. Based on…
Study shows torsion order bounds band-unlinking number for knot cobordisms.
problem Understanding bounds on knot cobordisms using torsion orders.
method Established an inequality involving local maxima, genus, and torsion orders of Khovanov homology.
result Torsion order gives a lower bound for the band-unlinking number.
The exterior derivative dθ of the Lee form θ of almost Hermitian manifolds is studied. If ω is the Kähler two-form, it is proved that the Rω-component of dθ is always zero. expressions for the other components, in [λ01,1] and in [[λ2,0]], of dθ are also obtained. They are given in ter…
Study mSpin(7)-dDT connections on manifolds with mSpin(7)-structures.
problem Understanding moduli spaces of mSpin(7)-dDT connections. method Introduced and studied mSpin(7)-dDT connections using fully nonlinear PDEs. result Moduli space MmSpin(7)′ has finite expected dimension and smoothness under certain conditions. New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
Characterizes braid types and estimates twist coefficients.
problem Understanding braid types and their properties.
method Birman-Ko-Lee left canonical form of braids.
result Characterization of almost strongly quasipositive braids and estimates of fractional Dehn twist coefficient.
Paper proves uniqueness of Einstein metrics on balls.
problem Uniqueness of conformally compact Einstein metrics on balls.
method Compactness result for conformally compact Einstein metrics.
result Global uniqueness of Graham-Lee metrics on balls.