Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …
The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.
problem Understanding the asymptotic dimensions of manifolds and spaces with geometric decompositions.
method Analyzing the fundamental groups and using geometric decompositions to derive asymptotic dimension bounds.
result Asymptotic dimensions of certain manifolds and spaces are bounded and equal to specific values.
We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
Three geometric analysis results on curve flows and Lie groups.
problem Analyzing geometric flows and Lie groups.
method Curve-shortening flow, point-wise curvature preserving flow, Lie group analysis.
result Interpolation between Sol and hyperbolic space in Lie groups.
Geometric study of thermodynamics using cotangent bundles.
problem Formulating classical thermodynamics using geometric structures.
method Contact geometry on cotangent bundles, homogeneous coordinates for intensive variables.
result Geometric formulation of thermodynamics with homogeneity properties.
In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth result of Morita. One motivation of this work is to facilitate the application of …
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof…
Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…
Study geometric step options with jumps, deriving pricing equations and characterizations.
problem Pricing geometric step options in markets with jumps.
method Symmetry and parity relations, partial integro-differential equations, ordinary integro-differential equations.
result Derive semi-analytical pricing results for geometric step options.
We say that a germ G of a geometric structure can be transplanted into a manifold M if there is a suitable geometric structure on M which agrees with G on a neighborhood of some point P of M. We show for a wide variety of geometric structures that this transplantation is always possible provided that M does in fact adm…
Geometrization Theorem solves complex geometry problems.
problem Complex geometry problems in differential geometry.
method Based on Hamilton's program, proved by Grigory Perelman.
result Generalized Poincaré's Conjecture.
We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic 3 manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…
Geometric GNNs improve graph discrimination through GWL.
problem Discriminating geometric graphs embedded in Euclidean space.
method Proposed a geometric version of the Weisfeiler-Leman test (GWL) for geometric graphs.
result Characterized the expressive power of geometric GNNs based on physical symmetries.
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
Revisits geometric PDE uniqueness in Riemannian and CR geometry.
problem Uniqueness of solutions to geometric PDEs in Riemannian and CR geometry.
method New proofs and reconstruction of Jerison-Lee identity.
result Stronger uniqueness result in CR geometry.
In this note, we show that there exist cusped hyperbolic 3-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds, and…
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…
New geometric examples of involutions on Hilbert square K3 surfaces.
problem Existence of non-symplectic involutions on Hilbert square K3 surfaces.
method Geometric constructions of involutions on IHS manifolds.
result New examples of involutions proven to exist.
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic submanifold theorem due to Jaco, Shalen and Johannson.
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…
Karigiannis discusses geometric flows of G2-structures, focusing on existence and uniqueness.
problem Existence and uniqueness of geometric flows of G2-structures.
method Introduced geometric structures, geometric flows, and discussed qualitative features. Focused on Ricci flow and DeTurck trick, then extended to G2-structures.
result Clarified conditions for short-time existence and uniqueness of G2-Laplacian flow.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. 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.
Geometric AD framework simplifies derivative computation in JAX.
problem Efficient and accurate automatic differentiation.
method Jet functors and Weil algebras for geometric analysis.
result Unified view of derivative propagation with algebraic exactness.
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.
Geometrically transforms word embeddings into a common space for better comparison.
problem Comparing embeddings from different sources is challenging.
method Applies orthogonal rotations and Mahalanobis scaling to transform embeddings into a shared latent space.
result The method improves word similarity and analogy tasks.
This paper develops a general method for constructing Poisson integrators.
problem Lack of a general theory for Poisson integrators due to geometric challenges.
method Adapting structural results about symplectic realizations to create geometric approximations.
result Developed a general approach for constructing geometric integrators on Poisson manifolds.
Flexible approach for normal approximations in geometric and topological statistics.
problem Normal approximation for complex statistics not expressible as sums of score functions.
method Flexible add-one cost operator combined with strong stabilization theory.
result Established normal approximation results for geometric and topological statistics.
Soft geometric bias improves physical dynamics predictions.
problem Learning physical dynamics with exact group equivariance can degrade performance.
method Object-centric world models using geometric algebra neural networks.
result Soft geometric inductive bias leads to better physical fidelity predictions.
The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.
problem Understanding geometric finiteness in mapping class groups and constructing new examples.
method Examined several constructions of subgroups and determined conditions for geometric finiteness.
result Provides new examples of parabolically geometrically finite and reducibly geometrically finite subgroups.
Generative Adversarial Nets (GANs) represent an important milestone for effective generative models, which has inspired numerous variants seemingly different from each other. One of the main contributions of this paper is to reveal a unified geometric structure in GAN and its variants. Specifically, we show that the ad…
Enhances quantum circuit synthesis using deep learning and geometric methods.
problem Optimizing quantum circuits for time efficiency.
method Combining deep learning with geometric control techniques.
result Improved time-optimal control in quantum circuit synthesis.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
Study geometric properties of generalized vacuum static spaces.
problem Estimating geometric properties of generalized φ-vacuum static spaces. method Proving estimates for φ-scalar curvature and first eigenvalue of the Jacobi operator, and rigidity under various geometric assumptions. result Proved a result related to the Cosmic no-hair conjecture.
We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stac…
In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…
Combines Gaussian process and Geometric Harmonics for better uncertainty estimation.
problem Uncertainty estimation in kernel-based methods.
method Combines Gaussian process and Geometric Harmonics.
result Alternative interpretations of uncertainty and accelerated Bayesian Optimization.
Uniform small energy regularity for fractional geometric problems proved.
problem Proving regularity for fractional geometric problems.
method Analyzing parabolic boundary reaction Ginzburg-Landau problems and fractional harmonic maps to spheres.
result Uniform small energy regularity results for s∈(0,1), answering a posed question. In high dimensions, the mean and geometric median are nearly identical.
problem Understanding the relationship between mean and geometric median in high-dimensional spaces.
method Analytical derivation and simulation of the distance between mean and geometric median.
result The distance between mean and geometric median vanishes with dimensionality in high dimensions.
Study on electrostatic systems with boundary, proving new geometric inequalities.
problem Electrostatic systems with boundary in higher dimensions.
method Investigation of electrostatic systems on compact manifolds with boundary, establishing new geometric properties.
result Proved sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds.