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The Arnold conjecture for singular symplectic manifolds
In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of
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A polyfold proof of the Arnold conjecture
We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a...
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An Arnold-type principle for non-smooth objects
In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example...
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An Arnold-type prinicple for non-smooth objects
In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous.
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Another look at the Hofer-Zehnder conjecture
We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks’...
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Another look at the Hofer–Zehnder conjecture
We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks’...
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Infinitesimal Center Problem on Zero Cycles and the Composition Conjecture
AbstractWe study the analog of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in...
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Variational approach to Arnold diffusion
Arnold diffusion was conjectured by Arnol’d (1964) as a typical phenomena of topological instability in classical mechanics. In this paper, we give a...
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Half a Century with the Problem of the Gradient of an Analytic Function
This is a review concerning the study of the local dynamics of a gradient vector field of a real analytic function. Starting from the famous Thom’s...
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Computers As a Novel Mathematical Reality: IV. The Goldbach Problem
AbstractIn this part I pursue the discussion of the role of computers in additive number theroy. I sketch the definitive solution of the ternary =...
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Symmetric Bases for Finite Element Exterior Calculus Spaces
In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange,...
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On purely-maximal ideals and semi-Noetherian power series rings
Tarizadeh and Aghajani conjectured that each purely-prime ideal is purely-maximal (Tarizadeh and Aghajani in Commun Algebra 49(2):824–835, 2021,...
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Sarnak’s Conjecture for Irregular Flows on Infinite-dimensional Torus
Sarnak’s Disjointness Conjecture states that the Möbius function is disjoint with any zero-entropy flow. This note establishes this conjecture, with...
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Completeness of derived interleaving distances and sheaf quantization of non-smooth objects
We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves...
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Second-Order Bounds on Correlations Between Increasing Families
Harris’s correlation inequality states that any two monotone functions on the Boolean hy-percube are positively correlated. Talagrand [11] started a...
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Conjectures on Stably Newton Degenerate Singularities
We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the...
