Roger and Yang's algebra has a finite set of generators.
problem Identifying the finite set of generators for the Kauffman bracket arc algebra.
method Provided an explicit, finite set of generators.
result The Kauffman bracket arc algebra is finitely generated.
The paper constructs finite generating sets for complex algebraic structures.
problem Finite generation of specific algebraic structures.
method Explicit construction of finite generating sets for γ2IAn and γ2Inb. result Explicit finite generating sets for γ2IAn and almost explicit for γ2Inb. Finite generating sets exist and are generic for certain homeomorphism groups.
problem Existence and genericity of finite topological generating sets for homeomorphism groups.
method Analyzing specific groups like Diff+1(I) and Diff+1(S1), and using Baire category theory. result Genericity of finite topological generating sets in certain homeomorphism groups.
Category theory generalizes finite type invariants using diagrams systems.
problem Generalizing finite type invariants using category theory.
method Relating generating sets for generalized finite type theories with diagrams systems.
result Demonstrates the correspondence between finite type theories and diagrams systems.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
Develops algorithm for finite generating set of liftable mapping class groups of regular abelian covers.
problem Finding finite generating sets for liftable mapping class groups of regular abelian covers.
method Algorithm based on a result providing generating sets for groups acting on graphs with finite quotients.
result Provides finite generating sets for LModp(Sg) for various regular abelian covers. We consider the fundamental group π of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of π has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that…
Study rigidity of geodesic planes in geometrically finite manifolds with cusps.
problem Rigidity of geodesic planes in geometrically finite manifolds with rank 1 cusps.
method Analysis of horocycles recurrence and density criteria for immersions.
result Closed immersions give rise to surfaces with finitely generated fundamental groups.
We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus g≥3. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.
We found a finite set of maps to generate a subgroup of mapping class groups for non-orientable surfaces.
problem Generating a finite set of maps for the level 2 twist subgroup of mapping class groups of non-orientable surfaces.
method Used crosscap pushing maps and Dehn twists along non-separating loops and curves.
result Found a finite generating set for the level 2 twist subgroup of mapping class groups of non-orientable surfaces.
Study rectifiability of finite perimeter sets in RCD(K,N) spaces.
problem Understanding sets of finite perimeter in RCD(K,N) spaces.
method Developed a Gauss-Green integration by parts formula and proved rectifiability of the reduced boundary.
result Rectifiability of the reduced boundary for sets of finite perimeter over RCD(K,N) spaces.
Suppose a group G is quasi-isometric to a free product of a finite set S of finitely generated abelian groups; let S′ denote the set of ranks of the free abelian parts of the groups in S. Then G is commensurable with the free product of Z with a Zn for each n occurring in S′.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
Set Flow models sets of data, learns dependencies, and achieves state-of-the-art likelihoods.
problem Modeling and sampling from finite, potentially high-dimensional, non-i.i.d. sets of data.
method Extends RealNVPs to handle finite sets, maintaining invertibility and exact log-likelihood evaluation.
result Achieves state-of-the-art likelihoods on 3D point clouds.
Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper…
Regular languages describe contracting geodesics in groups.
problem Characterizing groups with infinite contracting geodesics.
method Analyzing geodesics in Cayley graphs with a contracting property.
result Groups with infinite contracting geodesics are either virtually Z or acylindrically hyperbolic.
In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general…
The paper extends lossy coding to nonlinear latent representations.
problem Learning finite-dimensional coding schemes with nonlinear reconstruction maps.
method Generalizes Maurer--Pontil framework to nonlinear maps, connects to generative modeling, and provides generalization bounds.
result Established a connection to approximate generative modeling and presented generalization bounds.
We obtain an index of the complexity of a random sequence by allowing the role of the measure in classical probability theory to be played by a function we call the generating mechanism. Typically, this generating mechanism will be a finite automata. We generate a set of biased sequences by applying a finite state auto…
Study on self-similar sets on Riemannian manifolds with new separation conditions.
problem Analyzing self-similar sets on Riemannian manifolds with new separation conditions.
method Formulated weak separation and finite type conditions for conformal iterated function systems on Riemannian manifolds.
result Obtained formulas for Hausdorff dimensions of self-similar and graph self-similar sets.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
Study Kauffman bracket skein modules of Seifert fibered spaces.
problem Understanding the structure of Kauffman bracket skein modules.
method Investigate spanning sets and module structure.
result Kauffman bracket skein modules are finitely generated.
Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Extends inf-convolution to countable risk measures for risk sharing.
problem Limited inf-convolution theory to finite sets of risk measures.
method Extends inf-convolution to countable sets, investigates properties and results.
result Generalizes known properties and results to countable case.
This work analyzes batch MARL with networked agents, providing finite-sample bounds.
problem Understanding the theoretical foundation of decentralized batch MARL with networked agents.
method Developed batch MARL algorithms for two settings: collaborative and competitive networks, without a central controller.
result Quantified finite-sample errors of estimated action-value functions for both settings.
This paper derives finite generating sets for liftable mapping class groups of certain branched covers of tori.
problem Tackles the structure of liftable mapping class groups of specific branched covers of tori.
method Uses Reidemeister-Schreier rewriting process and Birman-Hilden theory to derive finite generating sets.
result Derives finite generating sets for LModpk(S1,2) for all k≥2. We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of exp…
In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups G1 and G2 of an infinite co-volume Kleinian group $G \subset \Isom(\mathbf{H}^3)$ having Λ(G1)=Λ(G2) are commensurable. In particular, it is proved tha…
Uniformly finite Cannon--Thurston fibers in most hyperbolic settings.
problem Existence and finiteness of Cannon--Thurston maps.
method Analysis of proper maps between hyperbolic metric spaces.
result Uniform finiteness of Cannon--Thurston fibers in most known settings.
The paper develops algorithms to detect stability and Morse properties in various groups.
problem Detecting stability and Morse properties in finitely generated groups.
method Various detection and decidability algorithms for stability and Morse properties in specific types of groups.
result The algorithms provide a way to determine if a subgroup is stable or Morse in specific group types.
The paper studies properties of mapping class groups for surfaces with infinite fundamental groups.
problem Characterizing the algebraic and topological properties of mapping class groups for surfaces with infinite fundamental groups.
method Analyzes the isomorphism and automorphism properties of pure mapping class groups for surfaces with finite genus, and shows their algebraic and topological structure.
result The pure mapping class group detects the homeomorphism type of the surface and is residually finite if and only if the surface has finite genus.
Extended mapping class groups can be generated by two elements for low genus cases.
problem Generating finite order elements to represent extended mapping class groups.
method Proving finite order generating sets for specific genus cases.
result Extended mapping class groups can be generated by two elements for genus 3 and 4, but not for genus 1.
Cube category simplifies set modeling.
problem Modeling set operations efficiently.
method Introducing interval-preserving monotone functions between finite Boolean lattices.
result Cube category facilitates model structures equivalent to simplicial sets.
It is known that the level 2 principal congruence subgroup of GL(n;Z) has a finite generating set. In this paper, we give a finite presentation of the level 2 principal congruence subgroup of GL(n;Z).
Study irrational pencils on complex manifolds, finding non-finitely generated homology.
problem Understanding the homology of the kernel induced by irrational pencils on complex manifolds.
method Analyzing critical points and homology of fundamental groups of complex manifolds.
result Homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated.
The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable surface. Then, either G is virtually abelian or it has infinite girth; moreover, if we …
Study infinite group presentations and their Dehn functions.
problem Behavior of Dehn functions for groups with infinite sets of relators.
method Analyzing Dehn functions for specific groups and presentations.
result Discovered continuum many distinct growth types of Dehn functions for the group Z^2.
A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite …
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
New theory uses probability sets for data variability, improving machine learning.
problem Variability in data distribution causes learning issues.
method Uses convex sets of probabilities (credal sets) to model data variability.
result Derives bounds for risk of models learned from multiple training sets.
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
Characterizes when geodesics in groups are generic.
problem Understanding genericity of geodesics in groups.
method Characterizes Gromov hyperbolicity via contracting elements.
result Genericity of contracting geodesics in groups.
Exponential proportion of pseudo-Anosovs in mapping class groups.
problem Proportion of non-pseudo-Anosov mapping classes in a ball of radius R.
method Using word metric and finite generating sets, we show that the proportion decreases exponentially.
result Proportion of non-pseudo-Anosov mapping classes decreases exponentially.
Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is…
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
Finite rigid sets in arc complexes help classify surfaces.
problem Classifying surfaces based on their arc complexes.
method Constructing finite rigid sets in arc complexes of surfaces.
result Isomorphic arc complexes imply homeomorphic surfaces.
New examples show finite covers of surfaces have limited homology.
problem Finite covers of surfaces and their homology limits.
method Constructing specific finite covers and subgroups.
result Homology of finite covers is limited by simple closed curves.