arXiv research

Develops differential KO-theory with constructions and applications.

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kähler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.

In this paper, by introducing a notion of local quasi holomorphic frame, we obtain a curvature formula for almost Hermitian manifolds which is similar to that of Hermitian manifolds. Moreover, as applications of the curvature formula, we extend a result of H.S. Wu and a result of F. Zheng to almost Hermitian manifolds.

Hom and Wu introduced a knot concordance invariant called $ν^+$, which dominates many concordance invariants derived from Heegaard Floer homology. In this paper, we give a full-twist inequality for $ν^+$. By using the inequality, we extend Wu's cabling formula for $ν^+$ (which is proved only for particular positive cab…

Study distance one surgeries between specific lens spaces.

Survey on recent breakthrough linking curvature and Kobayashi hyperbolicity.

We consider a non-Gaussian option pricing model, into which the underlying log-price is assumed to be driven by an $α$-stable distribution. We remove the a priori divergence of the model by introducing a Mellin regularization for the Lévy propagator. Using distributional and $\mathbb{C}^n$ tools, we derive an analytic …

Directly proves Wu's theorem on negative curvature metrics.

Private learning can be used to efficiently solve online learning problems.

The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.

In this paper, we study the uniqueness in the de Rham-Wu decomposition for pseudo-Riemannian manifolds.

Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for $\bar{\partial}$-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type $(1,1)$. Cappell and Mil…

We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization of the fact that two plane curves are regularly homotopic if and only if they hav…

Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equation $z_{yy} + (1/z)_{xx} + 2 = 0$ possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformatio…

3-manifolds with convex boundary are rigid in certain curvature conditions.

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fi…

Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most stochastic processes used in traditional option pricing models can be cast as special cases of time-changed Lévy processes. In particular these are models which can be tailored to exhibit correlated jumps in both the log price of assets and t…

We introduce invariants of spatial graphs related to the Wu invariant and the Simon invariant, and apply them to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of embedded graphs.

We study an analogue of the analytic torsion for elliptic complexes that are graded by $\mathbb{Z}_2$, orignally constructed by Mathai and Wu. Motivated by topological T-duality, Bouwknegt an Mathai study the complex of forms on an odd-dimensional manifold equipped with with the twisted differential $d_H = d+H$, where …

Classifies invariant hypersurfaces with singularities.

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

Counting HCMU sphere components using weighted trees.

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…

Improved rigidity of Delaunay triangulated plane.

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kähler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds. When the metric is Kähler, this is just the holomorph…

Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.

Derives formulas for differential forms on weighted manifolds.

It is explained how to find the de~Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Ri…

In this paper we give explicit formulas of differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…

Gelfand and MacPherson provided a new formula for calculating Pontrjagin classes.

We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent…

Geodesic completeness proven for certain Lorentzian spaces.

New formulas for Bertrand curves lead to harmonicity conditions.

WU-UCT parallelizes MCTS with linear speedup and limited performance loss.

In this paper, we shall use the Kähler geometry formulation to study the global behavior of the Ricci flow on $R^2$. The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which corresponds an eternal Ricci flow) obtained by L.F.Wu. We can use the classificatio…

Paper derives localization formula for $S^1$-action and equivariant η-invariants.

Formula for complex SVD backpropagation developed.

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

Sharp Steklov eigenvalue estimates for differential forms on manifolds.

In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $β_3$-invariants) which are transverse braid invariants. This family includes Wu's $ψ_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $β_3$-inv…

The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization…

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 on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

Formula extended for elliptic operators, yielding eigenvalue estimates.

The paper extends Montel's theorem to complex Finsler manifolds.

In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic diff…

Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.

The paper simplifies FLRW photon propagators using geometric embeddings.