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...