Starting from the candidate Bloch-Beilinson filtration on Chow groups of 0-cycles constructed by J. Lewis, we develop and describe geometrically a series of Hodge-theoretic invariants defined on the graded pieces. Explicit formulas (in terms of currents and membrane integrals) are given for certain quotients of the inv…
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6 results for “0-cycles”
Study stabilizes arithmetic statistics of rational maps over finite fields.
problem Stability of arithmetic statistics of rational maps over finite fields.
method Representation stability and arithmetic statistics of spaces of 0-cycles.
result Arithmetic quantities associated to rational maps over finite fields stabilize as degree increases.
The paper predicts coincidences in homological densities of spaces of 0-cycles on manifolds.
problem Predicting coincidences in homological densities of spaces of 0-cycles on manifolds.
method Using Weil's analogy and Björner--Wachs theory of lexicographic shellability.
result The topological predictions are true and provide new homological stability theorems.
Weyl law for 1-cyclesmath.DG
Proves the Weyl law for 1-cycles in manifolds.
problem Proving the Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds.
method Using parametric versions of the coarea inequality and isoperimetric inequality, along with a localized approximation method.
result Proves the Weyl law for 1-cycles in manifolds.
New method uses topology to analyze data bandwidth.
problem Analyzing the evolution of topological features in changing data.
method Persistence Flamelets, a multiscale version of Persistence Landscape.
result Persistence Flamelets can provide insights into KDE bandwidth parameters.
The study examines the behavior of monopoles as their mass increases and finds that they abelianize near singular points.
problem The behavior of monopoles as their mass increases and the formation of singular points.
method Analysis of mass-renormalized energy measures and convergence of fields to a reducible monopole.
result The mass-renormalized energy measures concentrate at singular points, and the fields abelianize near these points.