Study groups with polynomial growth, finding structure and applications.
problem Understanding groups with polynomial growth structure.
method Structure theorem for locally compact groups of polynomial growth.
result Applications on various growth functions and relations to FC-G series.
Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the f-Laplacian, proving estimates under curvature assumptions. result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.
problem Classifying polynomial growth solutions to drift-harmonic equations on specific types of manifolds.
method Inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.
result All drift-harmonic functions with polynomial growth asymptotically separate variables and dimensions of spaces are computed.
Study polynomial growth harmonic functions on infinite penny graphs.
problem Finite-dimensional property of polynomial growth harmonic functions on infinite penny graphs.
method Asymptotically sharp dimensional estimate for ancient solutions of the heat equation.
result Proved the asymptotically sharp dimensional estimate.
Study on Monge-Ampère equations with polynomial growth rates.
problem Analyzing solutions to Monge-Ampère equations with polynomial right-hand sides.
method Utilizing polynomial growth analysis to study regularity and growth rates of solutions.
result Translators for sub-affine-critical curvature flows are smooth and convex with specific growth rates.
We study ancient solutions of polynomial growth to heat equations on graphs, and extend Colding and Minicozzi's theorem [CM19] on manifolds to graphs: For a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dime…
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…
Compact groups with polynomial growth have specific embeddings.
problem Understanding groups with polynomial growth.
method Embedding into semidirect products of Lie and compact groups.
result Groups can be embedded as co-compact subgroups.
The paper defines higher invariants for groups of polynomial growth and proves their convergence.
problem Defining and proving convergence of higher invariants for groups of polynomial growth.
method Using delocalized cyclic cocycles and a determinant map construction.
result A well-defined pairing between delocalized cyclic cocyles and K-theory classes of C*-algebraic secondary higher invariants.
Polynomial growth elements found in all subgroups of Out(F_n).
problem Understanding polynomial growth in subgroups of Out(F_n).
method Analyzing conjugacy classes and elements of Out(F_n).
result Polynomial growth elements exist in all subgroups of Out(F_n).
Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.
problem Characterizing ancient solutions on an infinite strip with polynomial and exponential growth.
method Analyzing parabolic equations on an infinite strip, proving properties of ancient solutions.
result Ancient solutions on the strip are constant if they grow polynomially, and have a finite-dimensional space for slower exponential growth.
In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We a…
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.
problem Analyzing ancient caloric functions on manifolds beyond volume doubling.
method Time polynomial structure result on ancient caloric functions with polynomial growth.
result Finiteness result for ancient caloric functions is essentially sharp, except for multi-end cases.
Study shows torsion homology growth vanishes for certain free-by-cyclic groups.
problem Understanding torsion homology growth in free-by-cyclic groups.
method Analyzing polynomially growing monodromy and showing vanishing homology torsion.
result Integral torsion equals ℓ2-torsion for these groups, verifying a conjecture. Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non…
Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
problem Finding upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
method Discretizing a bounded domain and using comparison theorems.
result The $k^{\mbox{th}}$ eigenvalue tends to 0 proportionally to 1/∣B∣d−11. We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians:…
We prove a Liouville property for any f-harmonic function with polynomial growth on a complete noncompact smooth metric measure space (M,g,e−fdv) when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.
Study harmonic functions on submanifolds and their cones.
problem Understanding harmonic functions on minimal submanifolds and their cylindrical cones.
method Using the method from [6, 7], analyzing polynomial growth.
result Dimensions of harmonic functions match if cone growth degree differs.
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.
problem Understanding polynomial growth of holomorphic functions on Kähler-Ricci shrinkers.
method Analyzing scalar curvature conditions to prove finite generation of the ring of holomorphic functions.
result The ring of holomorphic functions with polynomial growth on Kähler-Ricci shrinkers is finitely generated.
Study on the growth of colored Jones polynomial for figure-eight knot cables.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot cables.
method Analyzing the asymptotic growth of the N-dimensional colored Jones polynomial of a cable of the figure-eight knot. result The growth rate of the colored Jones polynomial is exponential and related to the Chern-Simons invariant.
The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.
problem Estimating dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
method Proved upper bounds for dimensions and existence of sections using polynomial growth.
result Upper bounds for dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such mani…
We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.
Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.
problem Asymptotic behavior of colored Jones polynomials of figure-eight knot.
method Analyzes growth rate of polynomial evaluated at specific points.
result Growth rate determined by Chern-Simons invariant of an affine representation.
Efficiently samples arbitrary compact bodies with polynomial complexity.
problem Uniform sampling from arbitrary compact bodies efficiently.
method Warm start algorithm under isoperimetry and volume growth condition.
result Substantial generalization of known results for convex and star-shaped bodies.
In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…
We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
Study of minimal surfaces in a specific symmetric space with polynomial growth.
problem Asymptotic geometry of minimal surfaces in a symmetric space.
method Homeomorphism between Hitchin components and maximal surfaces, identification of convex embeddings, local limits of equivariant surfaces.
result Identification of planar maximal surfaces as local limits of equivariant surfaces.
Polynomial growth bounds for eigenfunctions on non-compact spaces.
problem Growth of eigenfunctions on non-compact spaces with Gaussian weight.
method Analyzing non-compact manifolds with Gaussian weight and drift Laplacian.
result Showed same polynomial growth bounds as Euclidean space for general tensors.
In this paper, we show that there exists a nonconstant CR holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of Yau uniformization conject…
We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than n2logn. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functio…
Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.
problem Analyzing properties of Kähler manifolds with nonnegative bisectional curvature.
method Established precise relations among minimal degree, volume growth, and scalar curvature decay.
result Unified understanding of Kähler-Ricci flow through polynomial growth holomorphic functions.
Study examines boundedness of oscillating singular integrals on specific Lie groups.
problem Investigating boundedness of oscillating singular integrals on Lie groups of polynomial growth.
method Presented kernel criteria in terms of sub-Riemannian structure and Fourier analysis.
result Extended classical oscillating conditions for boundedness of oscillating convolution operators.
Let F be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold M. We show that if the fundamental group of each leaf of F has polynomial growth of degree k for some non-negative integer k, then the foliation F is without holonomy.
In this paper we provide a lower bound for the long time on-diagonal heat kernel of minimal submanifolds in a Cartan-hadamard ambient manifold assuming that the submanifold is of polynomial volume growth. In particular cases, that lower bound is related with the number of ends of the submanifold.
Study on colored Jones polynomial of figure-eight knot for complex parameters.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot.
method Analyzing the asymptotic growth rate of the polynomial for complex parameters with small imaginary part.
result Growth rate of polynomial is related to the Chern-Simons invariant for large real part of the parameter and to the reciprocal of Alexander polynomial for small real part.
The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.
problem Investigating Q-curvature and volume entropy on conformally flat manifolds.
method Introducing a new volume entropy, establishing identities, and proving rigidity results.
result Each polynomial growth polyharmonic function on such manifolds is of finite dimension, and the Cohn-Vossen inequality achieves equality under specific conditions.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial …
Efficient algorithm for sampling from arbitrary compact bodies.
problem Sampling from arbitrary compact bodies efficiently.
method Warm start algorithm with polynomial complexity.
result Substantial generalization of known results for convex and star-shaped bodies.
New polynomial helps compute flow growth rates in 3D manifolds.
problem Computing growth rates of pseudo-Anosov flows in 3-manifolds.
method Modified veering polynomial and combinatorial flow graph.
result Computes growth rates of pseudo-Anosov flows after cutting.