Geometrically connects adjoint and coadjoint orbits of semidirect products.
problem Establishing a bijection between adjoint and coadjoint orbits for semidirect products.
method Proving a geometric bijection using specific subgroup conditions and homotopy types.
result Homotopy types of orbits in bijection are the same under certain conditions.
We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs…
This paper proves a bijection between complex and symplectic categories of tori.
problem Ambiguities in transition functions cause difficulties in constructing a functor.
method SYZ construction and transform to solve ambiguities and prove bijection.
result Proves the existence of a bijection between complex and symplectic categories of tori.
Method flattens complex surfaces with consistent density and shape.
problem Shape deformations and local geometric distortions in density-equalizing maps for multiply-connected surfaces.
method Formulates density diffusion as a quasiconformal flow, solving an energy minimization problem involving the Beltrami coefficient to ensure bijectivity and control distortion.
result Achieves optimal parameterization of multiply-connected surfaces with bijective and controlled geometric distortions.
Geometric bijection and homotopy equivalence between Lie group orbits.
problem Isomorphic adjoint and coadjoint representations of Lie groups.
method Geometric bijection and homotopy equivalence of orbits.
result Geometrically defined bijection and homotopy equivalence between adjoint and coadjoint orbits.
We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes.
Homotopy actions of Lie algebroids defined as L∞-algebra morphisms.
problem Defining and studying homotopy actions of Lie algebroids.
method Constructing homological vector fields on the semi-direct product and proving bijection.
result The construction is a bijection between homotopy actions and homological vector fields.
Classifies geodesic-preserving bijections in Thurston geometries.
problem Identifying bijections that preserve geodesics in different geometries.
method Comprehensive classification and proof for various geometries.
result Complete classification of geodesic-preserving bijections in Thurston geometries.
Novel mathematical approach using resurgent analysis reveals new structures in complex Chern-Simons theory.
problem Curious bijection in vertex algebras and SCFTs.
method Resurgent analysis, numerical algorithms, singularity elimination.
result New structures and patterns in complex Chern-Simons theory on hyperbolic 3-manifolds.
CIFs replace single bijections with continuous families to avoid topological limitations.
problem Normalising flows struggle with targets with complex topologies.
method Propose Continuously Indexed Flows (CIFs) replacing single bijections with a continuous family.
result CIFs avoid topological limitations and perform better empirically.
Paper extends tree bijection for hyperbolic surfaces without requiring cusps.
problem Computing volumes and distances on hyperbolic surfaces without cusps.
method Extend tree bijection to half-tight cylinders, using Busemann function.
result Tree bijection can now be applied to surfaces without cusps.
Study of J-Hermitian matrices and geometric mean definition.
problem Understanding the cone of J-Hermitian matrices and its geometric mean.
method Analysis of the cone structure, Riemannian structure, and definition of J-geometric mean.
result Uniquely characterized J-geometric mean defined as a solution to a Riccati-type equation.
Geometric model predicts unique, smooth ooid shapes.
problem Understanding the unique shapes of ooids formed by carbonate particles.
method Investigating planar curve evolution under a nonlocal geometric equation, demonstrating unique, time-invariant shapes.
result The model predicts a unique, time-invariant, smooth shape for ooids, agreeing with natural observations.
A bijection proves a polynomial volume for genus-0 hyperbolic surfaces with boundaries.
problem Proving the Weil-Petersson volume polynomial in boundary lengths for genus-0 surfaces.
method Generalizing a tree bijection to handle geodesic boundaries, extending spine construction.
result Explicit formula for three-point function in Weil-Petersson random surfaces.
New framework for better mapping of surfaces onto ellipsoids.
problem Mapping genus-0 closed surfaces onto spheres results in large distortion.
method Combining conformal and quasi-conformal mappings onto ellipsoids.
result Achieved a variety of ellipsoidal parameterizations with bijectivity.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
The sl_3 spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl_3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection v…
We identify causal models with unobserved confounding using bijective generation mechanisms.
problem Identifying causal relationships with unobserved confounders.
method Establish counterfactual identifiability for BGMs and propose a learning method.
result Learned BGMs enable efficient counterfactual estimation.
We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface S with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between …
We show that any multiplicative bijection between the algebras of differentiable functions, defined on differentiable manifolds of positive dimension, is an algebra isomorphism, given by composition with a unique diffeomorphism.
Paper proves bijection of periodic instantons to singular monopoles.
problem Establishing bijection between spatially periodic instantons and singular monopoles.
method Uses Nahm transform and Fourier-Mukai transform, intertwining with Kobayashi-Hitchin correspondences.
result Nahm transform is a bijection as suggested by heuristic.
Improved ANN-based Monte Carlo simulation for Higgs decay events.
problem Accurate simulation of Higgs boson decay events.
method Monte Carlo simulation using an Artificial Neural Network (ANN) with improved training algorithm.
result The ANN simulation of Higgs decay is within 0.7% of the true value and achieves 26% unweighting efficiency.
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois co…
This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group L as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebr…
For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that corresp…
Geometric correspondence between spinors and horospheres in hyperbolic space.
problem Understanding the relationship between spinors and horospheres in hyperbolic geometry.
method Detailed exposition and step-by-step construction of the spinor--horosphere correspondence.
result Spinor--horosphere correspondence is a smooth, SL(2,C)-equivariant bijection. Abstract: Bijection strengthened to Morita equivalence integrating Poisson and Cartan-Dirac structures.
problem Bijection between coadjoint orbits and conjugacy classes.
method Morita equivalence of quasi-symplectic groupoids integrating Poisson and Cartan-Dirac structures.
result Strengthened bijection to Morita equivalence.
Bijective proof of map enumeration recursion formulae.
problem Counting maps of arbitrary topology.
method Iterating Tutte's algorithm and pair-of-pants decomposition.
result Combinatorial meaning for all terms of topological recursion.
We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are able to prove that there is a natural geometric correspondence between ribbon n-k…
The logarithmic Riemann surface Sigma_{log} is a classical holomorphic 1-manifold. It lives into R^4 and induces a covering space of C - 0 defined by exp. This paper suggests a geometric construction of it, derived as the limit of a sequence of vector fields extending exp suitably to embeddings of C into R^3, which tur…
The paper proposes methods for volumetric parameterization of 3D solid manifolds.
problem Complex structure of solid manifolds makes conventional approaches ineffective.
method Incorporates models to preserve geometric structure, achieve density equalization, and balance distortions.
result Various 3D manifold parameterizations with different properties can be achieved.
We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f:M→Rn is bijective if and only if Hn−1(M)=0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometr…
Proves bijection between smooth conformal immersions and immersions.
problem Finding conformal immersions of closed Riemannian surfaces.
method Reformulated using h-principle and proved bijection on path connected components. result Induces a bijection between smooth conformal immersions and immersions.
For all k∈]0,1[, we construct a canonical bijection between the space of ramified coverings of the sphere and the space of complete immersed surfaces in 3-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to k. We show, furthermore, that this bijection restricts to a homeomor…
New metrics found on toric LCS manifolds.
problem Finding compatible complex structures on toric LCS manifolds.
method Proved a bijective correspondence between toric LCS manifolds and pairs (C,a). result Compact toric LCS manifolds have a positive potential.
Let X be a topological space and f:X→X a bijection. Let C(X,f) be a set of integers such that an integer n is an element of C(X,f) if and only if the bijection fn:X→X is continuous. A subset S of the set of integers Z is said to be realizable if there is a topologi…
We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and l…
New model separates images into independent factors quickly and easily.
problem Separating high-dimensional data like images into independent latent factors.
method Combines bijective feature maps with linear ICA model on the Stiefel manifold.
result Models converge quickly and achieve better unsupervised latent factor discovery.
Affine maps on tori preserve lines.
problem Characterizing affine automorphisms on tori.
method Analogous to Euclidean space, established for tori.
result Affine maps on tori preserve lines.
Study invariant Poisson structures on homogeneous manifolds, algebraically and geometrically.
problem Characterize and understand invariant Poisson structures on homogeneous manifolds.
method Algebraic characterization and bijective correspondence with Lie subalgebras, symplectic foliation, and invariant contravariant connections.
result Established a connection between invariant Poisson tensors and Lie subalgebras with a 2-cocycle.
Normalizing flows simplify complex distributions through bijective transformations.
problem Defining expressive probability distributions efficiently.
method Bijective transformations on a base distribution.
result Unified perspective on normalizing flows for modeling and inference.
We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. Our bijections a…
Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.
problem Modeling continuous martingales with prescribed initial and terminal distributions.
method Developed geometric Bass martingales and established their properties.
result Explicit bijection and representation of geometric Bass martingales.
This paper presents a method to obtain geometric registrations between high-genus (g≥1) surfaces. Surface registration between simple surfaces, such as simply-connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus t…
Transformers mimic Bayesian reasoning in controlled settings, revealing geometric mechanisms.
problem Verifying if transformers perform Bayesian reasoning rigorously in natural data.
method Constructing Bayesian wind tunnels with known posteriors and proving memorization impossibility.
result Transformers achieve 10−3-10−4 bit accuracy in Bayesian posteriors, while MLPs fail. We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map f:X=⨿Xℓ→Y between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of X. We furthermore characterize the metric structure on Y with re…
Maps with a single face converge to hyperbolic surfaces in large genus.
problem Understanding geometric properties of high genus maps.
method Analyzing uniformly random maps and their convergence to hyperbolic surfaces.
result Lengths of simple cycles converge to a Poisson process.
New equations connect unit Killing vectors to initial data.
problem Characterizing initial data for Einstein vacuum with unit Killing vectors.
method Developed new equations (uKID) by eliminating scaling and using propagation identity.
result Found equations that are finite type and characterize unit normalized Killing vectors.