Categorical lifting of the Jones polynomial: a survey
This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology....
Higgs bundles, harmonic maps, and pleated surfaces
This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the analytically...
Geometry lessons - School Yourself
Essential stuff for describing the world around you. 1. Lines and angles. 2. Related angles. What about angles bigger than 360 degrees? 3. Triangles. See if it's really true, and then prove it! They have two …
Parallel Write-Efficient Algorithms and Data Structures for Computational Geometry
In this paper, we design parallel write-efficient geometric algorithms that perform asymptotically fewer writes than standard algorithms for the same problem. This is motivated by emerging non-volatile memory technologies with read performance being...
Courant bracket - Wikipedia
In differential geometry, a field of mathematics, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to
Courant algebroid - Wikipedia
In differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more
Theodore James Courant - Wikipedia
Theodore James "Ted" Courant is an American mathematician who has conducted research in the fields of differential geometry and classical mechanics. In
A Geometric Action for the Courant Bracket
An important operation in generalized complex geometry is the Courant bracket which extends the Lie bracket that acts only on vectors to a pair given by a vector and a p-form. We explore the possibility of promoting the elements of the Courant brac...
Note on pre-Courant algebroid structures for parabolic geometries
This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebroïd structure. This structure is a Courant algebroïd if and only if the the curvature $κ$ of the Cartan connection vanishes. In all other cases,...
differential geometry - If a topological space is homeomorphic to a ...
Jan 22, 2018 · If a topological space is homeomorphic to a smooth manifold, then will it be a smooth manifold? Ask Question Asked 8 years, 1 month ago Modified 1 year, 3 months ago
elalish/manifold
Geometry library for topological robustness (⭐ 1868)
The Local Validity of Special Relativity, Part 1: Geometry
In this two-part essay, we distinguish several senses in which general relativity has been regarded as "locally special relativistic". Here, in Part 1, we focus on senses in which a relativistic spacetime has been said to be "locally (approximately)...
A spectral element method for meshes with skinny elements
When numerically solving partial differential equations (PDEs), the first step is often to discretize the geometry using a mesh and to solve a corresponding discretization of the PDE. Standard finite and spectral element methods require that the unde...
Christopher Clavius astronomer and mathematician
The Jesuit scientist Christopher Clavius (1538-1612) has been the most influential teacher of the renaissance. His contributions to algebra, geometry, astronomy and cartography are enormous. He paved the way, with his texts and his teaching for 40 ye...
LS-Galleries, the path model and MV-cycles
We give an interpretation of the path model of a representation \cite{Lit1} of a complex semisimple algebraic group $G$ in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS--galleries in the affine Coxe...
Monads for Instantons and Bows
Instantons on the Taub-NUT space are related to `bow solutions' via a generalization of the ADHM-Nahm transform. Both are related to complex geometry, either via the twistor transform or via the Kobayashi-Hitchin correspondence. We explore various as...
Interface doping and the effect of strain and oxygen stoichiometry on the transport and el…
The 5$d$ series semimetallic Dirac nodal-line perovskite SrIrO$_3$ presents a promising system to study the interplay between spin-orbit coupling and electron-electron interactions in the epitaxial thin film geometry. The competition between the band...
Polyhedral geometry of Phylogenetic Rogue Taxa
It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this "rogue taxon" effect. In this paper we...
From Prediction to Simulation: AlphaFold 3 as a Differentiable Framework for Structural Bi…
AlphaFold 3 represents a transformative advancement in computational biology, enhancing protein structure prediction through novel multi-scale transformer architectures, biologically informed cross-attention mechanisms, and geometry-aware optimizatio...
Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs
We analyze constraints for embedding local SU(5) F-theory GUTs into consistent compactifications and construct explicit three-generation models based on the geometry of arXiv:0904.3932. The key tool for studying constraints in this problem when the...
Farthest neighbor - Wikipedia
Farthest neighbor may refer to: Farthest neighbor graph in geometry The farthest neighbor method for calculating distances between clusters in hierarchical
ShapeMOD: Macro Operation Discovery for 3D Shape Programs
A popular way to create detailed yet easily controllable 3D shapes is via procedural modeling, i.e. generating geometry using programs. Such programs consist of a series of instructions along with their associated parameter values. To fully realize t...
SAD: Segment Any RGBD
The Segment Anything Model (SAM) has demonstrated its effectiveness in segmenting any part of 2D RGB images. However, SAM exhibits a stronger emphasis on texture information while paying less attention to geometry information when segmenting RGB imag...
Improving Cramér-Rao Bound With Multivariate Parameters: An Extrinsic Geometry Perspective
We derive a vector generalization of the square root embedding-based curvature-corrected Cramér--Rao bound (CRB) previously considered by the same author in \cite{srk} with scalar parameters. A \emph{directional} curvature correction is established...
Internal boundaries of the loop amplituhedron
The strict definition of positive geometry implies that all maximal residues of its canonical form are $\pm 1$. We observe, however, that the loop integrand of the amplitude in planar $\mathcal{N}=4$ super Yang-Mills has maximal residues not equal to...
Geometrical aspects of Amplitudes and Correlators in N=4 SYM
This thesis describes progresses made by the author and collaborators in the positive geometry description of superamplitudes and supercorrelators in planar N = 4 SYM....
Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous P…
We consider multicomponent local Poisson structures of the form $\mathcal P_3 + \mathcal P_1$, under the assumption that the third order term $\mathcal P_3$ is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such struct...
Planar velocity analysis of diesel spray shadow images
The focus of this work is to demonstrate how spatially resolved image information from diesel fuel injection events can be obtained using a forward-scatter imaging geometry, and used to calculate the velocities of liquid structures on the periphery o...
Feed-Forward SceneDINO for Unsupervised Semantic Scene Completion
Semantic scene completion (SSC) aims to infer both the 3D geometry and semantics of a scene from single images. In contrast to prior work on SSC that heavily relies on expensive ground-truth annotations, we approach SSC in an unsupervised setting. Ou...
What are the fundamental structures of concurrency? We still don't know!
Process algebra has been successful in many ways; but we don't yet see the lineaments of a fundamental theory. Some fleeting glimpses are sought from Petri Nets, physics and geometry....
Optimized interpolations and nonlinearity in numerical studies of woodwind instruments
We study the impedance spectra of woodwind instruments with arbitrary axisymmetric geometry. We perform piecewise interpolations of the instruments' profile, using interpolating functions amenable to analytic solutions of the Webster equation. Our...
Effect of Noncommutative Geometry on Accretion Disks around RGI-Schwarzschild Black Hole
In this study, we explore the combined effects of quantum gravity induced by non-commutativity and scale-dependent gravitational coupling on the thermal properties of the thin accretion disks around a Schwarzschild black hole. We consider a $κ$-defo...
erpas/rgis
RiverGIS is a QGIS plugin for creating HEC-RAS flow model geometry from spatial data. The functionality is similar to that of HEC-GeoRAS (⭐ 54)
Field induced cell proliferation and death in a thick epithelium
We study the dynamics of a thick polar epithelium subjected to the action of both an electric and a flow field in a planar geometry. We develop a generalized continuum hydrodynamic description and describe the tissue as a two component fluid system....
Degree (angle) - Wikipedia
In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler …
Geometry-Aware Learning of Maps for Camera Localization
Maps are a key component in image-based camera localization and visual SLAM systems: they are used to establish geometric constraints between images, correct drift in relative pose estimation, and relocalize cameras after lost tracking. The exact def...
Perspective-Equivariance for Unsupervised Imaging with Camera Geometry
Ill-posed image reconstruction problems appear in many scenarios such as remote sensing, where obtaining high quality images is crucial for environmental monitoring, disaster management and urban planning. Deep learning has seen great success in over...
Examples of austere orbits of the isotropy representations for semisimple pseudo-Riemannia…
Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere subm...
Dynamical Tropicalisation
We analyse the dynamics of the pullback of the map $z \longmapsto z^m$ on the complex tori and toric varieties. We will observe that tropical objects naturally appear in the limit, and review several theorems in tropical geometry....
Normal (geometry) - Wikipedia
transform to a surface it is often useful to derive normals for the resulting surface from the original normals. Specifically, given a 3×3 transformation matrix