The paper addresses statistical consistency in functional flow matching with rigorous mathematical proofs.
problem Statistical consistency in functional flow matching under scattered or adaptive refinement.
method Strong L2 convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions. result Proves strong L2 convergence with quantitative bounds for orthogonal projections and point-sensor extensions. Sharp bounds derived for test error of finite-rank kernel ridge regression.
problem Loose bounds on test error for finite-rank kernels in machine learning.
method Sharp non-asymptotic upper and lower bounds for KRR test error.
result Tighter bounds on finite-rank KRR test error, valid for any regularization parameters.
Proves Tits alternative for finite rank median spaces, restricting group actions.
problem Understanding group actions on median spaces.
method Extending Caprace-Sageev machinery and Hagen's theory to median spaces.
result Groups acting on finite rank median spaces are more restricted.
Convex cores found for group actions on median spaces.
problem Understanding group actions on median spaces without metric or topology.
method Introduced convex cores for actions on finite-rank median algebras.
result Actions on median spaces have nonempty convex cores.
Study finite rank bundle over J-Holomorphic map moduli spaces with exponential decay.
problem Understanding the structure of J-Holomorphic map moduli spaces.
method Prove exponential decay of derivative of gluing maps for a finite rank bundle.
result Exponential decay of the derivative of the gluing maps for the finite rank bundle.
Study shows how certain spaces can be mapped to R^n with specific properties.
problem Characterizing and mapping locally compact median spaces of finite rank.
method Analyzing the combinatorics of halfspaces and properties of isometric actions.
result Spaces with certain actions are isometric to R^n, and orbits are discrete.
The study optimizes Gaussian process approximations for finite-rank models.
problem Posterior behavior of finite-rank approximations differs from parent GP priors.
method Locally supported basis expansions with dependent Gaussian coefficients.
result Finite-rank expansions inherit the same posterior contraction rate as parent GP priors.
In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.
The article constructs Fuchsian Schottky groups with conformal boundaries.
problem Creating generalized Schottky groups with specific properties.
method Developed Fuchsian Schottky groups by including orientation-reversing isometries.
result Decomposed compact core of conformally compact Riemann surfaces into pairs of pants.
Optimal proof of finite small eigenvalues for specific geometric manifolds.
problem Proving finiteness of small eigenvalues for geometrically finite manifolds.
method Analyzing the spectrum of the Laplace operator on geometrically finite rank one locally symmetric manifolds.
result Optimal proof of finite small eigenvalues in a specific interval.
We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-c…
New examples show some convex-cocompact subgroups are separable.
problem Whether all convex-cocompact subgroups are separable.
method Using Manning-Mj-Sageev construction, examples of separable subgroups of arbitrary finite rank are given.
result Examples of separable convex-cocompact subgroups of arbitrary finite rank exist.
This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…
We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…
The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators on H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. We take the Jordan-Banach triple theory approach which allows us to define a natural Levi-Civita connection on M. We identi…
Constructs compactifications for median spaces with compact intervals.
problem Compactifications for median spaces with compact intervals.
method Generalises Roller boundaries of mCAT(0) cube complexes. result Recover zero-completions of Bandelt and Meletiou for median algebras.
Many 2D Artin groups are residually finite.
problem Residual finiteness of 2D Artin groups.
method Showed these groups split as free products with amalgamation or HNN extensions of finite rank free groups.
result Many 2D Artin groups are residually finite.
Study on complexity of random polynomials with deterministic spikes, identifying phase transitions.
problem Complexity of random Gaussian polynomials with deterministic spikes on a sphere.
method Variational formulas, Kac-Rice formula, determinant asymptotics of finite-rank perturbation of Gaussian Wigner matrices.
result Identification of a topological phase transition in the complexity function.
This paper is devoted to the computation of the space Hb2(Γ,H;R), where Γ is a free group of finite rank n≥2 and H is a subgroup of finite rank. More precisely we prove that H has infinite index in Γ if and only if Hb2(Γ,H;R) is not trivial, and furthermore, if and only if there …
Superrigidity theorem for actions on median spaces.
problem Actions of lattices on finite rank median spaces.
method Superrigidity and fixed point properties.
result Actions of irreducible lattices on finite rank median spaces have global fixed points.
We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.
We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.
We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.
The Serre-Swan theorem provides the link between projective modules of finite rank and vector bundles over compact manifolds, and plays a prominent role in non-commutative geometry. Its extension to non-compact manifolds is discussed.
Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V:=L(H). Using the algebraic structure of V, a torsionfree affine connection ∇ (that is invariant under the group of automorphisms of V) is defined on every connected component of M, which in this way beco…
We compute the categorified sl(N) link invariants as defined by Khovanov and Rozansky, for various links and values of N. This is made tractable by an algorithm for reducing tensor products of matrix factorisations to finite rank, which we implement in the computer algebra package Singular.
Algorithm estimates tensors from sparse observations with robust error bounds.
problem Estimating tensors from sparse noisy observations.
method Similarity-based collaborative filtering algorithm for tensor estimation.
result Achieves sample complexity nearly matching conjectured lower bound.
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
For word-equations in groups, we find a logarithmic bound on non-solutions.
problem Finding the length of non-solutions to word-equations in groups.
method Analyzing finite-rank free groups and applying results to broader classes of groups.
result Logarithmic bound on non-solutions for word-equations in groups.
New constraints found for algebro-geometric subgroups of mapping class groups.
problem Constraints for algebro-geometric subgroups of mapping class groups.
method Using deep work of Gibney, Keel, and Morrison, constraints on the Shafarevich morphism are derived to prove the infinite restriction of certain representations.
result Most Reshetikhin-Turaev representations of the mapping class group restrict to infinite representations on algebro-geometric subgroups when the genus is at least 3.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
Injective and surjective neural operators for function spaces.
problem Tackles injective and surjective neural operators in function spaces.
method Combines prior work in ReLU and operator learning, uses Fredholm theory and Leray-Schauder degree theory.
result Injective and surjective neural operators are universal approximators and maintain their properties in finite-rank implementations.
Let G be a word hyperbolic group. We prove that the algebraic K-theory groups of $\dbZ [G]$, $K_n(\dbZ[G])$, have finite rank for all $n\in \dbZ$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic K-theory groups of their group rings.
Study variance-optimal hedging of forward curve derivatives under stochastic volatility.
problem Variance-optimal hedging of forward curve derivatives with stochastic volatility.
method Assumes HJM-Musiela dynamics modulated by stochastic covariance, uses Galtchouk-Kunita-Watanabe projection.
result Density of finite-maturity strategies, convergence of finite-rank projections, decomposition of hedging error.
The paper finds obstructions to Lie algebroid representations up to homotopy.
problem Obstacles to representations up to homotopy of Lie algebroids.
method Analyzes Pontryagin characters and vector bundles over Lie algebroids.
result Vanishing of Pontryagin characters implies no representation up to homotopy.
We show that every automorphism α of a free group Fk of finite rank k has {\it asymptotically periodic} dynamics on Fk and its boundary ∂Fk: there exists a positive power αq such that every element of the compactum Fk∪∂Fk converges to a fixed point under iteration of αq.
We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Broué, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group G32, of order 1…
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applicat…
The paper defines infinite Schottky groups and their applications to infinite type surfaces.
problem Understanding group actions on infinite type surfaces.
method Definition and analysis of infinite Schottky groups and their properties.
result Every infinite type Riemann surface can be obtained as a quotient of a region of discontinuity of an infinite Schottky group.
Extends strong comparison principle for p-harmonic functions in Carnot-Caratheodory spaces.
problem Proving strong comparison principle for p-harmonic functions in specific geometric settings.
method Extends Bony's propagation of support argument to C^1 solutions of sub-elliptic p-Laplacian.
result Proves strong maximum and comparison principles for p-harmonic functions.
Hyperbolicity proven for a specific type of group extension.
problem Proving hyperbolicity of a specific group extension.
method Analyzing ascending HNN extension of groups with a free factor system and an injective endomorphism.
result Ascending HNN extension of a group is hyperbolic relative to a collection of maximal parabolic subgroups.
Defines map from skein module to Habiro ring for quantum modularity.
problem Quantum modularity conjecture and its arithmetic aspects.
method Defining a map from Kauffman bracket skein module to Habiro ring using Witten's conjecture.
result Image is an effectively computable module of finite rank.
If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X). In the present work we establish a topological classification of such bundles in…
We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broué, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequence…
Optimal scheme minimizes deviation in federated transfer learning for kernel regression.
problem Minimizing cumulative deviation in federated transfer learning across multiple datasets.
method Regret-optimal iterative scheme for continual communication between nodes and server.
result Explicit updates for the regret-optimal algorithm in finite-rank kernel regression.
Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.
problem Understanding the automorphisms of sphere complexes associated with graphs.
method Analyzing the group of proper homotopy equivalences and constructing an exhaustion of the sphere complex.
result The automorphism group of the sphere complex is isomorphic to the mapping group of the graph.