Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible wit…
Combines generalized and graded geometry to explore new structures.
problem Exploring new structures on generalized tangent bundles of graded manifolds.
method Introduces canonical brackets, Dirac structures, and generalized complex structures.
result Canonical bracket on a generalized tangent bundle of a graded manifold.
Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
The paper explores the geometry of holomorphic flows and orbits.
problem Understanding the local geometry of holomorphic flows and their equilibria.
method Analyzing the local geometry of first-order equilibria and higher-order equilibria under holomorphic conditions.
result Holomorphic Poincaré-Bendixson theorem: bounded non-periodic orbits are homoclinic or heteroclinic.
The abstract explores analogues of Hodge theory in Lie algebroids.
problem Exploring analogues of Hodge theory in Lie algebroids.
method Establishing equivalence of conditions and applying algebraic theory to geometric setting.
result Equivalence of analogues of Hodge theory conditions in Lie algebroids.
The study explores weightings on submanifolds and their geometric properties.
problem Understanding weightings on submanifolds and their geometric implications.
method Detailed exploration of weighted normal bundles, weighted deformation spaces, and weighted blow-ups.
result A description of weightings in terms of subbundles of higher tangent bundles, leading to new concepts for Lie algebroids and groupoids.
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.
problem Exploring non-smooth geometries on Lie groups and their applications.
method Study of left-invariant metrics on Lie groups, focusing on nilpotent and Carnot groups.
result Illustrates the role of metric Lie groups, particularly Carnot groups, in various mathematical contexts.
New puzzles from geometry and topology.
problem Exploring puzzles from geometric and topological concepts.
method Construction from square-tiled shapes, discussion of underlying mathematics.
result Puzzles naturally associated to puzzle spaces.
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.
The geometry of the universal hyperKaehler implosion for SU(n) is explored. In particular, we show that the universal hyperKaehler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperKaehler implosion.
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
We explore the geometry of the Nahm-Schmid equations, a version of Nahm's equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang--Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral cu…
The paper explores quaternionic curves using differential geometry.
problem Understanding quaternionic curves.
method Differential geometry applied to quaternionic curves.
result Simpler formulations of quaternionic curves.
We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.
Explains how geometry and statistics intertwine, focusing on information geometry.
problem Understanding the interplay between geometry and statistics.
method Introduces differential topology, geometry, probability, and (pre-)Frobenius manifolds.
result Discovers connections between geometry and statistics, particularly in information geometry.
Notation for spin coefficients for metrics of neutral signature in four dimensions is introduced. The utility and interpretation of spin coefficients is explored through themes in null geometry familiar from (complex) general relativity. Four-dimensional Walker geometry is exploited to provide examples and the generali…
Survey explores geometric aspects of policy optimization in control systems.
problem Understanding the geometric relationships between control design and optimization.
method Geometric perspective on policy optimization, focusing on parameterization and topology.
result Implications of policy geometry on stability and performance of local search algorithms.
This paper reviews Douglas curvature in Finsler geometry.
problem Exploring Douglas curvature in Finsler spaces.
method Historical review, characterizations, generalizations, and applications.
result Significance and applications of Douglas curvature in Finsler geometry.
Projective geometry aids in analyzing fields near compact manifolds.
problem Analyzing fields near compact manifolds.
method Developed a projective exterior differential tractor calculus.
result Analogous calculus for projectively compact manifolds.
Paper explores Monge-Ampère in deep learning and quantum geometry.
problem Understanding the Monge-Ampère equation in deep learning.
method Review of Boltzmann learning, connection to optimal transport, insights from quantum geometry, renormalization group flow.
result Space of covariance matrices in learning dynamics coincides with the CAH cone.
Complementing the previous paper in the series, this paper classifies ∣2∣-graded parabolic geometries, listing their important properties: the group G0, the graded tangent bundle gr(T) and its algebraïc bracket, the relevant cohomology spaces and the standard Tractor bundle $\mc{T}$. Several of these geometries …
This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html
Article explores Thurston's circle packing theorem in 3-manifold geometry.
problem Understanding Thurston's circle packing theorem in 3-manifold geometry.
method Analyzes the Koebe-Andre'ev-Thurston Theorem and its relation to Thurston's circle packing theorem.
result Illustrates the significance of Thurston's circle packing theorem in 3-manifold geometry.
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.
Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.
problem Analyzing G2-structures on Calabi-Yau 7-manifolds. method Reduced Ansätze for the Laplacian coflow on various Calabi-Yau 7-manifolds.
result Obtained a modified Kähler-Ricci flow.
Study uses crochet to visualize non-Euclidean geometry.
problem Understanding non-Euclidean surfaces through physical models.
method Parametrization of crochet models to represent Lobachevskian surface.
result Crochet models reflect non-Euclidean geometry characteristics.
The paper explores deep learning through algebra and geometry, highlighting geometric structures and differential processes.
problem Understanding the geometric and algebraic foundations of deep learning.
method Investigates neural networks from perceptron to transformer, emphasizing geometric structures and differential processes.
result A coordinate-free formulation of backpropagation equations using canonical scalar products on matrix spaces.
The paper explores similarities in even and odd-dimensional geometry.
problem Understanding the Lagrangian Grassmannian in cosymplectic geometry.
method Study of compatible co-complex structures, Moser's trick, and Weinstein 1-form derivation.
result The de Rham class of the Weinstein 1-form is a co-flux.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.
problem Understanding the Hamiltonian stationarity of twisted Lagrangian tori in C^2.
method Investigation of differential geometry of twisted tori, including product and Chekanov's exotic tori.
result Only product tori are minimal under Hamiltonian deformations, indicating Chekanov's exotic tori are not area minimal.
The study explores conformal planes with finite areas.
problem Geometry of conformal planes with finite areas.
method Analyzes several questions about conformal planes.
result Exploration of conformal planes with finite areas.
Study on G2-type flag manifolds, focusing on invariant metrics and Ricci flow.
problem Characterizing and analyzing metrics on G2-type flag manifolds. method Investigation of invariant metrics, analysis of g.o. metrics, and Ricci flow techniques.
result Characterization of metrics invariant under maximal compact subgroups.
The article explores the fundamental gap in Bakry-Emery geometry.
problem The fundamental gap in Bakry-Emery geometry.
method Recalled Bakry-Emery geometry and connected eigenvalues with boundary conditions. Showed a connection between fundamental gap and Bakry-Emery geometry.
result Presented key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture.
Study explores geometric structure and prior for beta-logistic distribution.
problem Understanding the geometric structure and prior distributions of the beta-logistic distribution.
method Exploring dual geometric structure and uncovering α-parallel prior. result The beta-logistic distribution admits an α-parallel prior for any real number α. The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.
problem Understanding the connections between autonomous systems and geometric structures.
method Investigation of the Darboux-Halphen-Ramanujan system, contact geometry, and Frobenius manifolds.
result Highlighting the role of contact geometry in autonomous systems.
The paper explores symplectic geometry of Cartan-Hartogs domains.
problem Understanding the symplectic geometry of Cartan-Hartogs domains.
method Constructing a dual counterpart and computing symplectic capacity.
result A Cartan-Hartogs domain admits symplectic duality if and only if it reduces to a complex hyperbolic space.
The paper explores geometry and positive scalar curvature on non-compact manifolds.
problem Understanding the relationship between geometry and positive scalar curvature on non-compact manifolds.
method Analysis of volume growth, scalar curvature integral, and width in different dimensions.
result Proves minimal volume growth and integral of scalar curvature in three dimensions, and volume growth with stronger conditions in higher dimensions.
The paper explores geometric decompositions for Ricci tensors and their applications.
problem Understanding Ricci tensors on compact Riemannian manifolds.
method Utilizes Berger-Ebin and York L2-orthogonal decompositions. result New insights into Ricci almost solitons and harmonic maps.
Survey of linking information geometry and optimal transport.
problem Connecting two geometric frameworks for probability measures.
method Exploration of interactions and links between information geometry and optimal transport.
result Outstanding questions for both disciplines.
The paper explores how data geometry influences generalization in neural networks.
problem Understanding generalization in overparameterized neural networks.
method Theoretical exploration of overparametrized two-layer ReLU networks trained below the edge of stability.
result Generalization bounds adapt to the intrinsic dimension of data distributions and deteriorate as data concentrates towards the unit sphere.
Study circular tractrices and pseudospheres in 3D space.
problem Geometric properties of circular tractrices and pseudospheres.
method Exploration of geometric properties through analysis.
result Characterization of circular tractrices and pseudospheres.
The Funk metric connects billiards, projective geometry, and convex geometry.
problem Exploring the Funk metric's invariants and inequalities.
method Using the Funk metric, extending results from projective geometry and convex geometry.
result General affine inequalities and volume maximizers in Funk geometry.