Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
Let S be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of S lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when S becomes Euclidean, i.e. very small.
This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesi…
We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finite…
We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in…
A new method shrinks a complex structure without much change.
problem Understanding the geometry of Bing's wild involution.
method Producing a counterintuitive construction to shrink the Bing decomposition without much change.
result A method to shrink a complex structure (Bing's decomposition) without much change.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
Flow on curves in inversive geometry converges to loxodromics.
problem Gradient flow for curve length in inversive geometry.
method Invariant gradient flow for invariant length functional.
result Solutions exist for all time and converge to loxodromic curves.
Models such as Sequence-to-Sequence and Image-to-Sequence are widely used in real world applications. While the ability of these neural architectures to produce variable-length outputs makes them extremely effective for problems like Machine Translation and Image Captioning, it also leaves them vulnerable to failures o…
I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can "progr…
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
Prediction markets can be manipulated by traders who can move contract settlements, harming price discovery.
problem Manipulation of settlement times in prediction markets leads to unfair wealth transfer and harms price discovery.
method Developed a model showing how settlement manipulation transfers wealth and harms price discovery, and observed real-world effects on Polymarket's Bitcoin contract.
result Manipulators capture significant profits from retail traders, especially when settlement times are short.
Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.
problem Understanding how edge subdivision impacts the Perron eigenvalue of tree Ricci matrices.
method Compressing branches into scalar feedback functions via Schur complement, reducing the spectral problem to a one-dimensional Chebyshev equation.
result Edge subdivision can decrease, preserve, or increase the Perron eigenvalue of tree Ricci matrices.
Uniformly branching trees are equivalent to certain metric spaces.
problem Characterizing metric spaces equivalent to uniformly branching trees.
method Proving equivalence between trivalent quasiconformal trees and uniformly branching trees.
result Any two uniformly branching trees are quasisymmetrically equivalent.
Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical ℓ∞-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in …
Model analyzes debt recycling strategies under various fiscal regimes and jurisdictions.
problem Understanding debt recycling dynamics and their impact on repayment times and equity growth.
method Developed a calibrated model incorporating mortgage interest rates, borrowing costs, and tax shields.
result Introducing positive interest rates without tax shields contracts success regions and lengthens repayment times, but tax shields partially reverse these effects.
We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
Uniformly perfect Morse boundaries characterize geometric properties of groups.
problem Characterizing geometric properties of groups using Morse boundaries.
method Introducing and geometrically characterizing uniformly perfect Morse boundaries for proper geodesic metric spaces.
result The Morse boundary of any finitely generated, non-elementary group is uniformly perfect if it is nonempty.
We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in Rn or non-positively curved n-dimensional simply connected manifold then X×Rn is integrally hyperspherical. If a un…
Projective varieties remain stable under close polarizations, extending to Kähler cones.
problem Maintaining stability of projective varieties under close polarizations.
method Uniformly valuative stability definition and extension to Kähler cones.
result Openness of uniformly valuative stability on the Kähler cone of projective manifolds.
CAI automates extraction and validation of corporate GHG emission metrics.
problem Manual extraction of corporate GHG emission metrics is labor-intensive and error-prone.
method CAI uses LLMs to automate extraction and validation of metrics from corporate disclosures.
result CAI improves data collection efficiency and accuracy by automating the process.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We defi…
Classifies 3-manifolds with uniformly positive scalar curvature.
problem Classifying 3-manifolds with uniformly positive scalar curvature.
method Analyzes properties of 3-manifolds with mean convex boundaries and uniformly positive scalar curvature.
result 3-manifolds with uniformly positive scalar curvature are homeomorphic to sums of spherical 3-manifolds and S1imesS2. In this paper, we prove the compactness theorem for gradient Ricci solitons. Let (Mα,gα) be a sequence of compact gradient Ricci solitons of dimension n≥4, whose curvatures have uniformly bounded L2n norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded vol…
Uniformly positive scalar curvature implies a lower bound on injectivity radius.
problem Bounding curvature and scalar curvature in three-manifolds.
method Analyzing bounded sectional curvature and uniformly positive scalar curvature properties.
result Uniform lower bound on injectivity radius.
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
Constructs uniformly positive scalar curvature metrics on open manifolds
problem Finding uniformly positive scalar curvature metrics on open manifolds
method Using Morse functions and exhaustion
result Proving the existence of uniformly positive scalar curvature metrics
Improved rigidity of Delaunay triangulated plane.
problem Rigidity of Delaunay triangulated plane under discrete conformality.
method Modifying Wu's proof to weaken the uniformly acute condition to the uniformly Delaunay condition.
result Improved rigidity result for Delaunay triangulated plane.
New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. For a domain Ω⊂Rn, we introduce the concept of a uniformly Cm defining function. We characterize uniformly Cm defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly Cm defining functions. Some of ou…
We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.
Study shows convergence speed for Fekete points on specific sets.
problem Understanding convergence speed for Fekete points on certain sets.
method Demonstrates (Cα,Cα′)-regularity for uniformly polynomially cuspidal sets. result Established convergence speed for Fekete points on these sets.
Characterizes a specific homology group for certain graphs.
problem Understanding the first uniformly finite homology group with Z coefficients. method Analyzes uniformly locally finite graphs, characterizes the group for trees and Z2 coefficients, and identifies three phenomena for general graphs. result Necessary conditions for non-vanishing of the group in transitive graphs.
We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.
Study asymptotic behavior of Weingarten surfaces at infinity.
problem Understanding the behavior of Weingarten surfaces at infinity.
method Derive asymptotic expansion and solve Dirichlet problem.
result Established maximum principle and solved Dirichlet problem.
Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.
problem Determining asymptotic Chow stability of uniformly K-stable toric varieties.
method Detailed study of triangulations of moment polytope neighborhoods and analysis of Futaki-Ono invariant.
result Every uniformly K-stable polarized smooth toric variety with vanishing Futaki-Ono invariant is asymptotically Chow polystable.
The paper proves scalar curvature decay for uniformly contractible manifolds with finite asymptotic dimension.
problem Proving decay of scalar curvature for uniformly contractible manifolds with finite asymptotic dimension.
method Using index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control, and Lipschitz control for topological K-theory of finite dimensional simplicial complexes.
result The scalar curvature decays to zero at a rate depending only on the contractibility radius and the diameter control of the asymptotic dimension.
Consider the unnormalized Ricci flow (gij)t=−2Rij for t∈[0,T), where T<∞. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t∈[0,T) then the solution can be extended beyond T. We prove that if the Ricci curvature is uniformly bounded…
Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.
problem Characterizing uniformly elliptic Weingarten spheres in S2xR.
method Proving bounded second fundamental form and applying Hopf's result.
result Rotational uniformly elliptic Weingarten surfaces in S2xR are congruent to the canonical example.
We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times t \in[0, T), then the curvature tensor has to be uniformly bounded as well.
Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.
problem Compactness of manifolds with specific curvature and boundary conditions.
method Monotone quantities constructed from positive proper harmonic functions with Neumann condition.
result Proves compactness of manifolds with nonnegative Ricci curvature and uniformly convex boundary.
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
problem Understanding Legendrian isotopy in cable links of uniformly thick knots.
method Introduced new technique of Legendrian surgeries to classify Legendrian knots in negative cables of twist knots.
result Found new phenomena of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.
No 5D aspherical manifolds can have uniformly positive scalar curvature.
problem Proving the non-existence of metrics with positive scalar curvature on certain 5D manifolds.
method Uniform acyclicity and toric symmetrization of stable μ-bubbles.
result Compact aspherical 5-manifolds cannot have metrics with uniformly positive scalar curvature.
New algorithm estimates causal effects for non-Gaussian data.
problem Estimating causal effects in non-Gaussian distributions.
method Generalized k-Triangle Faithfulness Assumption and Edge Estimation Algorithm.
result Uniformly consistent estimates of causal effects.
We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.
A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that conv…