New knowls for the conjectures collection

Core idea

This index collects supporting definitions for a collection of formal conjectures and open problems.

Topology, actions, and homotopy

• Locally compact space
• Locally compact group
• Continuous group action
• Topological manifold
• Closed manifold
• Separable space
• Homotopy equivalence
• Homotopy sphere

Arithmetic and real algebraic geometry

• Divisibility
• Factorial
• $p$-adic valuation
• $p$-adic integers
• Semialgebraic set
• Piecewise-polynomial function

Normed and Hilbert-space geometry

• Unit sphere of a normed space
• Surjective linear isometry
• Orthonormal basis
• Closed linear subspace
• Orthogonal complement
• Orthogonal projection
• Invariant subspace of an operator

Quantum information

• Normalized state vector
• Rank-one projector
• Positive semidefinite operator
• Positive operator-valued measure
• Normal operator

Complex geometry and group theory

• Holomorphic map
• Complex manifold
• Almost-complex structure
• Integrable almost-complex structure
• Torsion-free group

Triangulated-category foundations

• Preadditive category
• Initial object
• Zero object
• Shift functor
• Triangle in a shifted category
• Distinguished triangle
• Pretriangulated category
• Octahedral axiom
• Triangulated category
• Mapping cone

Important pre-existing knowls reused

• Topology and manifolds: topological space, continuous map, homeomorphism, metric sphere, smooth manifold, diffeomorphism, and Lie group.
• Algebra and arithmetic: group, faithful action, Galois extension, Galois group, group algebra, and zero divisor.
• Analysis and quantum theory: Banach space, Hilbert space, bounded operator, and density operator.
• Categories: category, morphism, additive category, and short exact sequence.