arXiv research

Study geometry and topology of manifolds with radial curvature bounds.

ICLR 2021 challenge in computational geometry and topology attracted 16 teams.

Perturbative string amplitudes are correctly derived from the string geometry theory, which is one of the candidates of a non-perturbative formulation of string theory. In order to derive non-perturbative effects rather easily, we formulate topological string geometry theory. We derive the perturbative partition functi…

Study shows curvature constraints force submanifolds to have specific topology or geometry.

Teaches Dirac operators for geometry and topology.

Survey of mass partition problems in geometry and topology.

New puzzles from geometry and topology.

The paper explores the geometry and topology of DNN decision boundaries.

These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.

Study distance functions on manifolds linking geometry to topology.

Simplified proof of a famous geometry result for students.

Tropical geometry aids in computing topological quantum field theories.

Survey of recent progress in Sasakian geometry.

Article explores Thurston's circle packing theorem in 3-manifold geometry.

The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set …

Survey on geometry and topology of maximal antipodal sets.

This is a survey article on the recent development of "stringy geometry and topology of orbifolds", a new subject of mathematics motivated by orbifold string theory.

This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, an…

The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.

We define some new invariants for 3-manifolds using the space of taut codim-1 foliations along with various techniques from noncommutative geometry. These invariants originate from our attempt to generalise Topological Quantum Field Theories in the Noncommutative geometry / topology realm.

We prove for the automorphism group of an arbitrary parabolic geometry that the $C^0$ and $C^{\infty}$ topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.

Study on topological order on fractal geometries, proving no-go theorem and fault-tolerant gates.

New method uses topological data analysis for better change point detection.

Survey on categorifying Jones polynomial.

Atiyah-Singer theorem links math fields, predicts topological insights.

TabPFN's internal geometry topology correlates with dataset reliability.

We survey results on compact Clifford-Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact forms.

Survey of Floer theories and their connections.

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

The paper proves metrizability and dynamics of Weil bundles.

BiHermitian geometry, discovered long ago by Gates, Hull and Roceck, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target spac…

This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a deta…

Surveying connections between algebraic geometry and surface topology.

Survey on GKM theory in low dimensions, highlighting combinatorics-geometry interplay.

BiKaehler geometry is characterized by a Riemannian metric g_{ab} and two covariantly constant generally non commuting complex structures K_+^a_b, K_-^a_b, with respect to which g_{ab} is Hermitian. It is a particular case of the biHermitian geometry of Gates, Hull and Roceck, the most general sigma model target space …

Complete classification of rod complements in 3-torus using topology.

Explains how knots relate to 4D shapes.

The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-…

We characterize functions which are growth types of Riemannian manifolds of bounded geometry.

This article reviews $\infty$-bundles and their applications in geometry and physics.

The paper proposes conjectures about moduli space rigidity.

Study of rod packings in 3-torus using 3-manifold geometry.

Constructs a topological cover of real line's multiplicative group.

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which…

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well …

Algorithm finds Dirichlet domains for hyperbolic surfaces.

Characterizes Anosov flows in 3D using symplectic and contact geometry.

It is known that the topological T-duality exchanges $H$ and $F$-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced in arXiv:1408.2649 to defi…