Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
3-manifolds have covers with infinitely many ideal triangulations.
problem Proving the existence of infinitely many geometric ideal triangulations in certain 3-manifolds.
method Using separability of peripheral subgroups and conjugacy separability theorems.
result Every cusped hyperbolic 3-manifold has a cover with infinitely many geometric ideal triangulations.
Infinite type surfaces can be perfectly divided into triangles.
problem Triangulating surfaces of infinite type.
method Showed arcs can be completed into triangulations if they intersect curves a finite number of times.
result Any surface of infinite type admits an ideal triangulation.
Study ideals in the Goldman Lie algebra S by mapping to a simpler algebra.
problem Identify and classify ideals in the Goldman Lie algebra S. method Construct an algebra homomorphism to a simpler structure and classify ideals.
result Found infinite classes of ideals in both Z-module and Q-module cases. This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…
New infinite series of hyperbolic polytopes with special growth rates found.
problem Finding new infinite series of non-compact hyperbolic polytopes.
method Constructing infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes.
result Growth rates of the constructed polytopes are Perron numbers.
New method to parametrize infinite Riemann surfaces with bounded triangulations.
problem Parametrizing infinite Riemann surfaces with bounded triangulations.
method Introducing bounded ideal triangulations and proving real-analyticity of the parametrization.
result Real-analytic parametrization of Teichmüller spaces for infinite surfaces with bounded triangulations.
Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Study ideal circle patterns (ICPs) and develop a uniform Ring Lemma via pointed Gromov-Hausdorff convergence.
result Establish existence and rigidity of embedded ICPs and infinite ideal polyhedra (IIP).
The article studies random infinite ideal hyperbolic polyhedra and their dual graphs, establishing new boundary theories.
problem Uniformization and boundary theory of random infinite ideal hyperbolic polyhedra and their dual graphs.
method Combinatorics, geometry, analysis, and random walks perspectives.
result Characterization of the ICP type of IAG and convergence of simple random walk to the boundary.
Paper finds infinite family of minimal triangulations for complex 3D shapes.
problem Finding minimal ideal triangulations for complex 3D shapes.
method Examined Dehn fillings on specific links to find minimal triangulations.
result Found an infinite family of minimal ideal triangulations for a specific type of 3D shape.
Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.
problem Establishing convergence and long-time existence of the combinatorial p-th Calabi flow.
method Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns.
result Sharp criterion for convergence in finite case and long-time existence in infinite case for p≥2. The study classifies complex parallelisable nilmanifolds with unobstructed deformations.
problem Characterizing complex parallelisable nilmanifolds with unobstructed deformations.
method Analyzing Lie algebras associated with nilmanifolds and their verbal ideals.
result There are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension 19, and infinitely many in dimension 20.
New jellyfish found in various flows.
problem Existence of geometrically distinct shapes in flows.
method Analyzing elastic, curve diffusion, and ideal flows.
result Infinitely many distinct shapes discovered.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
Essential tori in certain 3-manifolds are missed by ideal points in character varieties.
problem Essential tori in 3-manifolds are not detected by ideal points in character varieties.
method Infinite families of 3-manifolds are constructed to show the existence of essential tori not detected by ideal points in character varieties over any algebraically closed field.
result Essential tori in 3-manifolds are missed by ideal points in character varieties over any algebraically closed field.
Paper provides new Alexander ideal-based obstruction to 0-concordance of knotted surfaces.
problem Tackles the 0-concordance problem for knotted surfaces in S4. method Uses Alexander ideals to induce a homomorphism and prove non-sliceness.
result Alexander ideal determines 0-concordance classes and non-sliceness.
Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.
Defines ideal simplicial volume for manifolds with boundary, showing it's bounded by classical volume.
problem Measuring the complexity of manifolds with boundary.
method Introduces ideal simplicial volume, compares it to classical volume, and computes specific examples.
result Ideal simplicial volume is bounded by classical volume and can be strictly smaller for certain manifolds.
Review of diffusion models for SBI in non-ideal data scenarios.
problem Inference of parameters from complex simulation outputs with intractable likelihoods.
method Diffusion models for likelihood-free inference, addressing model misspecification, unstructured observations, and missing data.
result Improved robustness and efficiency in SBI methods for non-ideal data scenarios.
Study axisymmetric ideal fluids on 3-manifolds, proving Fredholm properties.
problem Riemannian geometry of axisymmetric ideal fluids.
method Proving Fredholm properties of L2 exponential map for axisymmetric flows. result Axisymmetric diffeomorphisms form a totally geodesic submanifold.
We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\m…
Detecting closed essential surfaces in 3-manifolds is challenging.
problem Detecting closed essential surfaces in 3-manifolds using ideal points of character varieties or algebraic non-integral representations.
method Using Chesebro's module-theoretic interpretation of Culler-Shalen theory, constructing examples of 3-manifolds with no algebraic non-integral representations.
result Construction of an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representations into PSL(2, C).
We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks …
New triangulations for twist knots, proving volume conjecture.
problem Computing the volume of twist knot complements.
method Ideal triangulations and H-triangulations, using Thurston's method and volume functional.
result Proved the Teichmüller TQFT volume conjecture for all twist knots.
We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a b…
We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform ir…
Study metric learning from limited preference comparisons, showing how low-dimensional structure can still reveal metric information.
problem Learning metric from limited pairwise preference comparisons.
method Ideal point model, divide-and-conquer approach for low-dimensional structure.
result Metric can be jointly identified even with limited comparisons when items exhibit low-dimensional structure.
This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2, in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume t…
Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.
problem Understanding the structure of convolution algebras on Lie groupoids.
method Analyzing smooth functions and invariant subsets to prove H-unitality.
result H-unitality of groupoid algebras and their quotients, leading to excision properties.
Essential triangulations of certain manifolds are connected via specific moves.
problem Connecting essential triangulations of certain manifolds.
method Essential triangulations are connected via 2-3 and 3-2 moves alone, ignoring those for which no 2-3 move preserves essentiality.
result Essential triangulations of certain manifolds are connected via 2-3 and 3-2 moves alone.
The paper studies ideal flows of closed curves, classifying critical points and proving flow behavior.
problem Analyzing the generalised ideal flow of closed planar curves.
method Completely classifies critical points and proves properties of the m-ideal flow. result For m>1, the m-ideal flow of closed curves converges to a round multiply-covered circle. The paper explores infinite-dimensional nonholonomic and vakonomic systems.
problem Understanding dynamics of infinite-dimensional systems with constraints.
method Visualizing and revisiting classical and new examples of nonholonomic and vakonomic systems.
result Infinite-dimensional systems exhibit both nonholonomic and vakonomic dynamics.
Given a reduced alternating diagram for a link, we obtain conditions that guarantee that the link complement has a complete hyperbolic structure, crossing arcs are the edges of an ideal geodesic triangulation, and every crossing arc is isotopic to a simple geodesic. The latter was conjectured by Sakuma and Weeks in 199…
This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-Bäcklund equivalence, introduced in Fliess et al. (1999…
We study discrete, cocompact, isometric actions of groups on Hadamard spaces, and the induced actions on ideal boundaries. For a class of groups generalizing fundamental groups of three-dimensional graph manifolds, we find a set of invariants for the action which determine the boundary action up to equivariant homeomor…
Essential triangulations connect via specific moves in 3-manifolds.
problem Finding essential triangulations in 3-manifolds with infinite boundary components.
method Using 2-3, 3-2, 0-2, and 2-0 moves to connect essential triangulations.
result Essential triangulations are connected via specific moves.
Geometric framework for Newton's equations on diffeomorphism groups.
problem Modeling fluid dynamics and related systems on geometric spaces.
method Geodesic approach and infinite-dimensional information geometry.
result Unified framework for various fluid dynamics equations.
Highly twisted knots can be geometrically triangulated.
problem Proving geometric triangulations for hyperbolic 3-manifolds.
method Using highly twisted knots and extending Gueritaud and Schleimer's work.
result Sufficiently highly twisted knots admit a geometric triangulation.
Toolbox for stochastic Euler equations using Ebin-Marsden theory.
problem Applying geometric framework to stochastic PDEs.
method Combining infinite-dimensional geometry and stochastic analysis.
result Local well-posedness of maximal solutions for incompressible Euler equation with noise.
The study connects curves and cohomology on manifolds.
problem Understanding quasiregular curves and their cohomology.
method Analyzing de Rham cohomology and Künneth ideals.
result Non-trivial algebra homomorphisms between manifold cohomology and exterior algebra.
The paper shows that arithmetic hyperbolic 3-orbifolds have many non-commensurable pairs with similar geodesic spectra.
problem Understanding the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic 3-orbifolds.
method Using a bounded gaps result for prime ideals in number fields, the paper constructs infinitely many non-commensurable pairs of arithmetic hyperbolic 3-orbifolds with similar geodesic spectra.
result Arithmetic hyperbolic 3-orbifolds can have many non-commensurable pairs with similar geodesic spectra.
In this paper an approach is proposed to represent a class of dissipative mechanical systems by corresponding infinite-dimensional Hamiltonian systems. This approach is based upon the following structure: for any non-conservative classical mechanical system and arbitrary initial conditions, there exists a conservative …
Researchers introduce a family of hyperbolic Brunnian links and calculate their volumes.
problem Calculating volumes of hyperbolic Brunnian links.
method Dehn fillings on cusped manifolds with volumes related to ideal right-angled hyperbolic antiprisms.
result Upper bounds for volumes of 3-manifolds S3∖Br(k1,…,kn) are obtained. Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.