isomorphismes

“Ramified” means that around the point in question the projection map has the same behaviour as does…

Tue, 29 Oct 2024 08:36:37 -0600

“Ramified” means that around the point in question the projection map has the same behaviour as does the projection from the parabola to its abscissa.

C. Herbert Clemens, A Scrapbook of ℂ Curves

Test functions and [tempered] distributions require the notion of topological vector space ……

Tue, 29 Oct 2024 08:15:43 -0600

Test functions and [tempered] distributions require the notion of topological vector space … distributions can be traced back to Green’s functions in the 1830’s to solve ordinary differential equations … the 1936 work of Sergei Sobolev on hyperbolic PDE’s.

Laurent Schwartz introduced the term “distribution” by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on.

(^ if there is a dipole, there must be a notion of subtraction, hence the need for a vector, and to speak of this very conceptually, use a TVS)

“the Yang-Mills equations are nonlinear, therefore there is little hope of finding a closed-form…

Tue, 15 Oct 2024 07:36:45 -0600

“the Yang-Mills equations are nonlinear, therefore there is little hope of finding a closed-form solution.” Such a statement seems plausible. Linear differential equations with constant coefficients are the only differential equations for which a general solution is given in closed form.

As often occurs in life, however, the exceptions to the rule are sometimes more interesting than the rules themselves. Let us digress from quantum physics to the motion of water, where British shipbuilder John Scott Russell noticed a solitary wave in a canal in August 1834.

Neither Airy nor Stokes accepted this observation, yet in 1895 Korteweg and de Vries found an equation for a wave travelling in shallow water in one direction: u̇ + 6•u•uₓ + uₓₓₓ = 0. The KdV equation is easily solved by restricting from two independent space-time dimensions (x,t) to a single dimension x−λt — a frame matching the speed λ of a travelling wave.

Mikhail Ilʹich Monastyrskiĭ, Riemann, Topology, and Physics

Hadamard knew in 1898 that negative curvature and simply connectedness for surfaces embedded in…

Sun, 13 Oct 2024 08:20:49 -0600

Hadamard knew in 1898 that negative curvature and simply connectedness for surfaces embedded in 3-space force uniqueness of geodesics joining two points—implying that any segment of geodesic is also a shortest path.

But there is a long way toward the modern statement: “on any complete abstract Riemannian manifold of ≥0 curvature of any dimension, curvature is the quotient of its universal covering by a discrete group of isometries.”

Marcel Berger, Riemannian Geometry during the Second Half of the Twentieth Century

——-

Hadamard, 1898 being Les surfaces à courbure opposées et leurs lignes géodésiques

Hamiltonian mechanics is the feminine side of classical physics. Its masculine side is Lagrangian…

Sat, 04 Nov 2023 23:30:21 -0600

Hamiltonian mechanics is the feminine side of classical physics. Its masculine side is Lagrangian mechanics, formulated in terms of velocities (tangent vectors) rather than momenta (cotangent vectors).

Lagrangian mechanics focusses on the difference of kinetic – potential energies; Hamiltonian mechanics focusses on their sum.

Richard Montgomery, reviewing a book by Stephanie Frank Singer and recalling lectures by Shing-Shen Chern

the cotangent bundle (differential forms) is the feminine side of calculus-on-manifolds; the tangent…

Fri, 03 Nov 2023 16:30:21 -0600

the cotangent bundle (differential forms) is the feminine side of calculus-on-manifolds; the tangent bundle (vector-fields) is the masculine side.

Shing-Shen Chern, via Richard Montgomery

Multimaps

Fri, 13 Jan 2023 23:30:23 -0600

Cartesian functions send {A}→{B} with exactly one tail a↦ per a∈{A} connecting to each head ↦b∈{B}.

In other words B has to be equal size or smaller than A.

This is true mapping rings to rings, groups to groups, sets to sets, vector spaces to vector spaces,

… it’s just a property of arrows really.

When mathematicians want to talk about “one-to-many” (using the database lingo) or “multimaps” (some stupid word I heard on Wikipedia which absolutely nobody anywhere ever thought was a good term), though, they’re not left outside.

If you’ve got a bundle of arrows ⇶ with tails from {a₀, a₁, a₂, a₃} ⇶ {b₁₄}, then that’s a bundle of tails all heading to the same place. If you “grab them all by the head”

So when mathematicians want to talk about a multimap, they use a preimage ƒ⁻¹. Let’s say the kernel for example–it’s “everything that gets thrown in the trash”—so if multiple things get thrownin the trash,

(linear subspace / quotient / ring morphism kernel)

So this is how they can associate a bunch of stuff, to one point. For example every point on a manifold gets a tangent space. Maybe this is a vector space for example–which is a lot bigger than just one point.

That would be a problem for 1-to-≥1 functions, so the mathematicians need to turn the arrows around. That’s why they define the projection map π:E→B to send a ton of things e∈E onto that one point b∈B i.e. p∈M.

"A cylinder is the product of an interval and a circle because it is both an interval of circles and..."

Sat, 03 Sep 2022 14:30:27 -0600

“A cylinder is the product of an interval and a circle because it is both an interval of circles and a circle of intervals.”

- Jeff Weeks, The Shape of Space

"Betti numbers measure how many cuts, at most, a surface could sustain before being split into..."

Sat, 27 Aug 2022 19:00:41 -0600

“Betti numbers measure how many cuts, at most, a surface could sustain before being split into separate pieces.”

- >20 wikipedians

y³ = x • (x–2) • (x+2) rotated to show all three branch points...

Sun, 07 Aug 2022 13:01:06 -0600

y³ = x • (x–2) • (x+2)

rotated to show all three branch points at once

by Dan Piponi