arXiv subjects

Bingyu Zhang

Publications and source records attributed to Bingyu Zhang.

13 recordsLinked to original sources

arXiv2026

Density of fibers for the filtered Fukaya category of $T^*N$

We answer a question of Biran and Cornea about the density of iterated cones of fibers in the Fukaya category of a cotangent bundle. We prove that indeed if we take a dense set of basepoints, the iterated cones of the cotangent fibres are dense in the Filtered Fukaya category. In an appendix we prove that the space of exact Lagrangians in a symplectic manifold is never totally bounded for the spectral distance (unless it is empty). This was implicit in \cite{MCA-VH-CV} for $n=1$ and proved for cotangent bundles of negatively curved manifolds in \cite{A-B-C}.

math.SG↗

arXiv2025

A remark on Continuous K-theory and Fourier-Sato transform

In this note, we prove a generalization of Efimov's computation for the universal localizing invariant of categories of sheaves with certain microsupport constraints. The proof is based on certain categorical equivalences given by the Fourier-Sato transform, which is different from the original proof. As an application, we compute the universal localizing invariant of the category of almost quasi-coherent sheaves on the Novikov toric scheme introduced by Vaintrob.

math.AT↗

arXiv2025

Contact non-squeezing at large scale via generating functions

Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $πR_2^2 \leq K \leq πR_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq πR_2^2 \leq πR_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points.

math.SG↗

arXiv2025

Almost mathematics, Persistence module, and Tamarkin category

We give a precise unification of three theories that are widely used by symplectic geometers: (Almost) modules over the Novikov ring, Persistence modules, and the Tamarkin category. Our method provides new input in this direction, especially in relation to Vaintrob's Novikov/log-perfectoid mirror symmetry for Novikov toric schemes. The results of this paper can also be treated as a study of persistent homology from a higher algebra point of view. As applications, we establish a version of homological mirror symmetry over the Novikov ring for toric varieties and propose a conjecture for homological mirror symmetry over the Novikov ring for log Calabi-Yau varieties.

math.SG↗

arXiv2025

On the Hochschild cohomology of Tamarkin categories

To any open subset of a cotangent bundle, Tamarkin has associated a certain quotient of a category of sheaves. Here we show that the Hochschild cohomology of this category agrees with filtered symplectic cohomology.

math.SG↗

arXiv2025

Non-linear microlocal cut-off functors

To any conic closed set of a cotangent bundle, one can associate four functors on the category of sheaves, which are called non-linear microlocal cut-off functors. Here we explain their relation with the microlocal cut-off functor defined by Kashiwara and Schapira, and prove a microlocal cut-off lemma for non-linear microlocal cut-off functors, adapting inputs from symplectic geometry. We also prove two Künneth formulas and a functor classification result for categories of sheaves with microsupport conditions.

math.SG↗

arXiv2023

Idempotence of microlocal kernels and $S^1$-equivariant Chiu-Tamarkin invariant

In this article, we present some results and constructions about the Chiu-Tamarkin invariant motivated by the idempotence of microlocal kernels, including: (1) a natural explanation for the definition of the $\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant; (2) a graded commutative product on the non-equivariant Chiu-Tamarkin invariant; and (3) a construction of the $S^1$-equivariant Chiu-Tamarkin invariant. As applications, we: (1) construct a sequence of symplectic capacities $(\overline{c}_k)_{k\in \mathbb{N}}$ and prove that it coincides with the symplectic capacities $({c}_k)_{k\in \mathbb{N}}$ we defined using the $\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant under certain conditions; and (2) prove a Viterbo isomorphism. In the Appendix, we provide a proof of admissibility for all open sets in a cotangent bundle under the setup of triangulated categories.

math.SG↗

arXiv2023

Capacities from the Chiu-Tamarkin complex

In this paper, we construct a sequence $(c_k)_{k\in\mathbb{N}}$ of symplectic capacities based on the Chiu-Tamarkin complex $C_{T,\ell}$, a $\mathbb{Z}/\ell$-equivariant invariant coming from the microlocal theory of sheaves. We compute $(c_k)_{k\in\mathbb{N}}$ for convex toric domains, which are the same as the Gutt-Hutchings capacities. Our method also works for the prequantized contact manifold $T^*X\times S^1$. We define a sequence of "contact capacities" $([c]_k)_{k\in\mathbb{N}}$ on the prequantized contact manifold $T^*X\times S^1$, and we compute them for prequantized convex toric domains.

math.SG↗

arXiv2023

2D Magnetic Heterostructures: Spintronics and Quantum Future

The discovery of two-dimensional (2D) magnetism within atomically thin structures derived from layered crystals has opened up a new realm for exploring magnetic heterostructures. This emerging field provides a foundational platform for investigating unique physical properties and exquisite phenomena at the nanometer and molecular/atomic scales. By engineering 2D interfaces using physical methods and selecting interlayer interactions, we unlock the potential for extraordinary exchange dynamics. This potential extends to high-performance and high-density magnetic memory applications, as well as future advancements in neuromorphic and quantum computing. This review delves into recent advances in 2D magnets, elucidates the mechanisms behind 2D interfaces, and highlights the development of 2D devices for spintronics and quantum information. Particular focus is placed on 2D magnetic heterostructures with topological properties, promising for a resilient and low-error information system. Finally, we discuss the trends of 2D heterostructures for future electronics, considering the challenges and opportunities from physics, material synthesis, and technological prospective.

cond-mat.mes-hall↗

arXiv2019

Local Exact Controllability to the Trajectories of the Korteweg-de Vries-Burgers Equation on a Bounded Domain with Mixed Boundary Conditions

This paper studies the internal control of the Korteweg-de Vries-Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear.

math.OC↗

arXiv2019

Lower regularity solutions of non-homogeneous boundary value problems of the sixth order Boussinesq equation in a quarter plane

In this article, we study an initial-boundary-value problem of the sixth order Boussinesq equation on a half line with nonhomogeneous boundary conditions: \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x>0\mbox{, }t>0,\] \[u(x,0)=φ(x), u_t(x,0)=ψ''(x),\] \[ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t),\] where $β=\pm1$. It is shown that the problem is locally well-posed in $H^s(\mathbb{R}^+)$ for $-\frac12<s\leq 0$ with initial condition $(φ,ψ)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$ and boundary condition $(h_1,h_2,h_3) $ in the product space $H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s-1}{3}}(\mathbb{R}^+)\times H^{\frac{s-3}{3}}(\mathbb{R}^+)$.

math.AP↗

arXiv2016

On Sharpness of the Local Kato Smoothing Property of Dispersive Wave Equations

Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations $$ w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb{R}^n, \ t\in \mathbb{R} , $$ enjoy the local smoothing property $$ q\in H^s (\R ^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac{m-1}{2}}_{loc} \left (\R^n\right )\Big ) , $$ where $m$ is the order of the pseudo-differential operator $P(D)$. This property, now called local Kato smoothing, was first discovered by Kato for the KdV equation and implicitly shown later by Sjölin for the linear Schrödinger equation. In this paper, we show that the local Kato smoothing property possessed by solutions general dispersive equations in the 1D case is sharp, meaning that there exist initial data $q\in H^s \left (\R \right )$ such that the corresponding solution $w$ does not belong to the space $ L^2 \Big (-T,T; H^{s+\frac{m-1}{2} +ε}_{loc} \left (\R\right )\Big )$ for any $ε>0$.

math.AP↗

arXiv2015

Neumann boundary controllability of the Korteweg-de Vries equation on a bounded domain

In this paper we study boundary controllability of the Korteweg-de Vries (KdV) equation posed on a finite domain $(0,L)$ with the Neumann boundary conditions: u_t+u_x+uu_x+u_{xxx}=0 in (0,L)x(0,T), u_{xx}(0,t)=0, u_x(L,t)=h(t), u_{xx}(L,t)=0 in (0,T), u(x,0)=u_0(x) in (0,L). We show that the associated linearized system u_t+(1+β)u_x+u_{xxx}=0 in (0,L)x(0,T), u_{xx}(0,t)=0, u_x(L,t)=h(t), u_{xx}(L,t)=0 in (0,T), u(x,0)=u_0(x) in (0,L) is exactly controllable if and only if the length $L$ of the spatial domain $(0,L)$ does not equal to $-1$ or does not belong to set R_β:={\frac{2π}{\sqrt{3(1+β)}}\sqrt{k^{2}+kl+l^{2}}:k,l\in\mathbb{N}^{\ast}}\cup{\frac{kπ}{\sqrt{1+β}}:k\in\mathbb{N}^{\ast}} and the nonlinear system is locally exactly controllable around a constant steady state $β$ if the associated linear system is exactly controllable.

math.AP↗