We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.
Jintai DingDepartment of Mathematical Science, University of Cincinnati, USAZheng ZhangDepartment of Mathematical Science, University of Cincinnati, USAJoshua DeatonDepartment of Mathematical Science, University of Cincinnati, USA
Advances in Mathematics of Communications, 15, (1), 65-72, 2021.2
We present a cryptanalysis of a signature scheme HIMQ-3 due to Kyung-Ah Shim et al , which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems in 2017. We will show that inherent to the signing process is a leakage of information of the private key. Using this information one can forge a signature.
Bobo HuaSchool of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, ChinaYong LinDepartment of Mathematics, Information School, Renmin University of China, Beijing 100872, ChinaYanhui SuCollege of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China
In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.
Yong LinDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaShuang LiuYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, ChinaHongye SongDepartment of Mathematics, Renmin University of China, Beijing 100872, China; Beijing International Studies University, Beijing 100024, China
We prove the equivalence between some functional inequalities and the ultracontractivity property of the heat semigroup on infinite graphs. These functional inequalities include Sobolev inequalities, Nash inequalities, Faber–Krahn inequalities, and log-Sobolev inequalities. We also show that, under the assumptions of volume growth and CDE(n, 0), which is regarded as the natural notion of curvature on graphs, these four functional inequalities and the ultracontractivity property of the heat semigroup are all true on graphs.
Yong LinDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaHongye SongSchool of General Education, Beijing International Studies University, Beijing 100024, China; Department of Mathematics, Renmin University of China, Beijing 100872, China
Analysis of PDEsDifferential Geometrymathscidoc:2207.03005
We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.
The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,1) and CD(K,n).
Yong LinDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of ChinaShuang LiuDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of ChinaYunyan YangDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China
The Journal of Geometric Analysis, 27, 1667–1679, 2016.9
We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our estimate follows from the graph structure of the gradient form and the Laplacian operator. Though our assumption on graphs is slightly stronger than that of Bauer et al. (J Differ Geom 99:359–405, 2015), our estimate can be easily applied to nonlinear differential equations, as well as differential inequalities. As applications, we estimate the greatest lower bound of Cheng’s eigenvalue and an upper bound of the minimal heat kernel, which is recently studied by Bauer et al. (Preprint, 2015) by the Li-Yau estimate. Moreover, generalizing an earlier result of Lin and Yau (Math Res Lett 17:343–356, 2010), we derive a lower bound of nonzero eigenvalues by our gradient estimate.
Yong LinDepartment of Mathematics, Renmin University of China, Beijing, 100872, P. R. ChinaShuang LiuDepartment of Mathematics, Renmin University of China, Beijing, 100872, P. R. ChinaYun Yan YangDepartment of Mathematics, Renmin University of China, Beijing, 100872, P. R. China
Acta Mathematica Sinica, English Series, 32, 1350–1356, 2016.10
Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359–409 (2015)]. In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li–Yau inequality by the global gradient estimate, we can get similar results.
We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.
Bing-Long ChenDepartment of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaXiaokui YangMorningside Center of Mathematics, Institute of Mathematics, Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve f:C→X.