Geometric group theory explores groups through their geometric properties.
problem Understanding groups via geometric properties.
method Cayley and Schreier graphs, ping-pong lemma, quasi-isometries, growth of groups, hyperbolicity.
result Gromov's theorem on groups of polynomial growth and amenability.
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
Study geometric and analytical properties of ρ-Einstein solitons.
problem Characterize geometric and analytical features of ρ-Einstein solitons. method Analyze the spectrum of the drifted Laplacian operator and prove volume growth estimates.
result Establish new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. Word metrics from large balls in injective spaces are exponentially generic and growth tight.
problem Understanding genericity and growth in word metrics from injective spaces.
method Geometric arguments and large ball considerations.
result Exponential genericity and growth tightness for word metrics.
In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show t…
The study counts ends on shrinkers using geometric covering methods.
problem Counting the number of ends on shrinkers.
method Geometric covering method to study the number of ends.
result Proves that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree.
The Cheeger-Gromoll theorem is adapted for groups, revealing geometric properties of virtually abelian groups.
problem Understanding the curvature of groups and its relation to geometric properties.
method Developed a new curvature notion for groups and proved a splitting theorem.
result Geometric characterization of virtually abelian groups.
The paper proves properties of geometric flows on noncompact manifolds.
problem Existence criteria for geometric flows on noncompact affine Riemannian manifolds.
method Obtained existence criteria through a geometric flow on noncompact affine Riemannian manifolds.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature and bounded geometry are diffeomorphic to \(\mathbb{R}^n\) if their tangent bundle has maximal volume growth.
Study pentagon growth with laser-cut models.
problem Explore topological and geometric properties of pentagon cell growth.
method Cell growth process in Euclidean plane, physical representations created with laser cutter.
result Aesthetic and geometric insights from pentagon growth models.
The paper proposes a method to determine the geometric priors of relational data.
problem Identifying geometric structure in heterogeneous, high-dimensional data.
method Combinatorial approach analyzing nearest-neighbor structures and local neighborhood growth rates.
result The method can identify the geometric priors of suitable embedding spaces for relational data.
Geodesics grow infinitely in certain Finsler manifolds.
problem Understanding the growth of geodesic chords in Finsler manifolds.
method Analyzing properties of forward complete Finsler manifolds with infinite fundamental group.
result Geodesics grow infinitely in specific Finsler manifolds.
Study on geometric flows and rigidity of solitons.
problem Understanding rigidity and properties of gradient solitons.
method Identifying Hamilton's identity for geometric flows and proving its utility.
result Recovery of results about rigidity and properties for arbitrary geometric flows.
Einstein metrics are blocked by manifold features and group growth.
problem Existence of Einstein metrics on specific 4-manifolds.
method Analysis of collapsing and group growth effects.
result Several 4-manifolds cannot support Einstein metrics due to specific features.
For any given C∞ immersion r:S1→R2 such that the set NSr=R2−∪s∈S1(r(s)+drs(Ts(S1))) is not empty, a simple geometric model of crystal growth is constructed. It is shown that our geometric model of crystal growth never form…
Survey on handlebody groups and their properties.
problem Understanding handlebody groups and their properties.
method Survey and recent trends in geometric group theory.
result Application of the cheap α-rebuilding property to handlebody group homology growth. We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterizati…
We study local control of the mechanism with the growth vector (4,7). We study controllability and extremal trajectories on the nilpotent approximation as an example of the control theory on Lie group. We give solutions of the system an show examples of local extremal trajectories.
We investigate the relation between economic growth and equality in a modified version of the agent-based asset exchange model (AEM). The modified model is a driven system that for a range of parameter space is effectively ergodic in the limit of an infinite system. We find that the belief that "a rising tide lifts all…
We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reform…
We investigate fourth order Paneitz equations of critical growth in the case of n-dimensional closed conformally flat manifolds, n≥5. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the Q-cu…
We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature. For instance, we obtain power bounds of degree two plus epsilon in length for the number of billiard orbits between almost all pairs of po…
This paper analyzes the robust growth rate of leveraged ETFs under uncertain parameters.
problem Analyzing the robust long-term growth rate of leveraged ETFs with uncertain parameters.
method Derive worst-case parameters using comparison principle and martingale extraction method.
result Explicitly obtain robust long-term growth rates under various models.
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
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.
Let X be a Hadamard manifold and Γ a discrete group of isometries of X which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the geometric limit set of Γ in order to derive a new asymptotic estimate for the growth rate of closed geodesics in not necessar…
We prove that a complete noncompact Kähler manifold Mnof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}n and we show that the manifold is topologically {\bf R}2n. In particular, when Mn is a Kähler surface of positive bisecti…
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
Study growth of LP wealth in G3Ms affected by trading fees and arbitrage.
problem Analyzing profitability of LPs in G3Ms under trading fees and arbitrage.
method Stochastic reflected diffusion processes to model G3M dynamics.
result Long-term expected logarithmic growth of LP wealth calculated.
The paper studies 3D manifolds with positive scalar curvature and volume growth.
problem Understanding the geometry of 3D manifolds with positive scalar curvature.
method Analyzes volume and geometric properties of 3D complete manifolds with positive scalar curvature, considering different curvature conditions.
result Volume growth estimates for 3D manifolds with positive scalar curvature, answering Gromov's question affirmatively.
Uniform models for random 3-manifolds with controlled metrics.
problem Understanding the geometric properties of random 3-manifolds.
method Two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings.
result The diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk.
We develop a simple theoretical framework for the evolution of weighted networks that is consistent with a number of stylized features of real-world data. In our framework, the Barabasi-Albert model of network evolution is extended by assuming that link weights evolve according to a geometric Brownian motion. Our model…
Study on harmonic functions in spaces with collapsing behaviors.
problem Harmonic functions on spaces with inhomogeneous collapsing behaviors at infinity.
method Analysis of complete and incomplete spaces with nonnegative Ricci curvature.
result Any nonconstant harmonic function yields a definite exponential growth rate.
The study generalizes a specific geometric correspondence to higher dimensions.
problem Understanding nondegenerate lines on holomorphic contact manifolds.
method Analyzing nondegenerate lines and corresponding distributions on higher-dimensional manifolds.
result A generalization of the (2,3,5)-distributions to higher dimensions. Ancient solutions of Ricci flow with Type I growth are classified.
problem Understanding ancient solutions of Ricci flow with specific curvature growth.
method Analyzing ancient solutions with Type I curvature growth in arbitrary dimensions.
result Ancient solutions with Type I growth are classified into specific types.
Grimaldi-Pansu metrics are constructed for manifolds with multiple ends.
problem Volume growth on manifolds with more than one end.
method Constructing Riemannian metrics with bounded geometry and uniform bounds for volume growth.
result Uniform bounds for volume growth of Grimaldi-Pansu metrics in certain manifolds.
A new growth model for dynamic networks using Markovian latent points.
problem Modeling temporal dynamic networks with latent points and distances.
method Markovian latent space dynamic with Euclidean Sphere sampling and connection probabilities based on geodesic distances.
result Theoretical guarantees for non-parametric estimation of the latitude and envelope functions.
The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.
problem Characterizing complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature.
method Using a geometric flow on noncompact affine Riemannian manifolds, constructing Hessian metrics, and proving diffeomorphism.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature are diffeomorphic to R^n if their tangent bundle has maximal volume growth.
This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth manifolds are deduced. When the sequences of polyhedra are obtained by the action…
The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.
problem Understanding the visible range from a point on harmonic manifolds.
method Analyzing Poisson Boolean models on harmonic manifolds, focusing on the geometric mechanism of tube volumes around geodesic segments.
result The visible range from a point on harmonic manifolds follows an exponential distribution.
The paper studies the growth of closed geodesics on hyperbolic surface amalgams.
problem Understanding the growth of closed geodesics on hyperbolic surface amalgams.
method Analyzing topological and volume entropies, and their dependence on geometric data.
result Entropy can increase exponentially with pasting length in the absence of a lower bound on the systole.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
Evolution of planar curves under a nonlocal geometric equation is investigated. It models the simultaneous contraction and growth of carbonate particles called ooids in geosciences. Using classical ODE results and a bijective mapping we demonstrate that the steady parameters associated with the physical environment det…
This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an orbifold with another triangulation. This method is used to show that eventually p…
We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.
We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the critical exponent of a cusp-uniform action of G on a hyperbolic metric space. As a c…
Study on braids' invariant growth and counting functions.
problem Counting elements of braid group modulo center with positive extremal length.
method Use extremal length with totally real horizontal boundary values to define an invariant.
result Number of elements with positive extremal length grows exponentially.
We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbound…
Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A com…