topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
In the course of providing a geometric proof of the spin-statistics theorem, Berry & Robbins 1997 asked, at each natural number $n \in \mathbb{N}$, for a continuous and $Sym(n)$-equivariant function
from the configuration space of $n$ points (ordered and unlabeled) in Euclidean space $\mathbb{R}^3$
to the coset space of the unitary group $\mathrm{U}(n)$ by its maximal torus, hence the complete flag manifold of flags in $\mathbb{C}^n$,
both equipped with the evident group action by the symmetric group $Sym(n)$.
For the first non-empty case $n = 2$ this readily reduces to asking for a continuous map of the form $\mathbb{R}^3 \setminus \{0\} \xrightarrow{\;\;} \mathbb{C}P^1 \simeq S^2$ which is equivariant with respect to passage to antipodal points. This is immediately seen to be given by the radial projection. But this special case turns out not to be representative of the general case, as this simple construction idea does not generalize to $n \gt 2$.
That a continuous and $Sym(n)$-equivariant Berry-Robbins map (1) indeed exists for all $n$ was proven in Atiyah 2000.
In this article, Atiyah turned attention to the stronger question asking for a function (1) which is smooth and $Sym(n) \times$$SO(3)$-equivariant and provided an elegant proof strategy for this stronger statement, which however hinges on some conjectural positivity properties of a certain determinant (discussed in more detail and with first numerical evidence in Atiyah 2001), interpreted as the electrostatic energy of $n$-particles in $\mathbb{R}^3$.
Extensive numerical checks of this stronger but conjectural construction was recorded, up to $n \lt 30$ , in Atiyah & Sutcliffe 2002, together with a refined formulation of the conjecture, whence it came to be known as the Atiyah-Sutcliffe conjecture.
The Atiyah-Sutcliffe conjecture has been proven for $n = 3$ in Atiyah 2000/01 and for $n = 4$ by Eastwood & Norbury 01.
The origin of the question in investigation of the spin-statistics theorem:
First form and first checks of the conjecture:
Michael F. Atiyah, The geometry of classical particles, in Surveys in Differential Geometry, Surv. Differ. Geom. 7, Int. Press 2000, 1–15 (doi:10.4310/SDG.2002.v7.n1.a1)
Michael F. Atiyah, Configurations of points, R. Soc. Lond. Philos. Trans. Ser. A, Math. Phys. Eng. Sci. 359 (2001) 1375–1387 (doi:10.1098/rsta.2001.0840)
Generalization of the codomain to flag manifolds of other compact Lie groups:
Michael F. Atiyah, Roger Bielawski, Nahm’s equations, configuration spaces and flag manifolds, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 157–176 (math.RT/0110112, doi:10.1007/s005740200007)
(using Nahm's equation)
Full formulation of the Atiyah-Sutcliffe conjecture:
Proof for $n = 4$:
Further discussion:
Dragutin Svrtan, Igor Urbiha, Atiyah-Sutcliffe conjectures for almost collinear configurations and some new conjectures for symmetric functions, (math.AG/0406386)
Dragutin Svrtan, Igor Urbiha, Verification and strengthening of the Atiyah–Sutcliffe conjectures for several types of configurations (math.MG/0609174)
Marcin Mazur, Bogdan V. Petrenko, On the conjectures of Atiyah and Sutcliffe, Geom Dedicata 158 (2012) 329–342 (doi:10.1007/s10711-011-9636-6, arxiv:1102.4662)
Joseph Malkoun, Root Systems and the Atiyah-Sutcliffe Problem, Journal of Mathematical Physics 60, 101702 (2019) (arXiv:1903.00325)
Joseph Malkoun, The Atiyah-Sutcliffe determinant (arXiv:1903.05957)
Last revised on September 13, 2021 at 17:58:38. See the history of this page for a list of all contributions to it.