lnq 👨🏿‍🦱: geometry/topology
Showing posts with label geometry/topology. Show all posts
Showing posts with label geometry/topology. Show all posts


Hodge conjecture

Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.



Presented here are wrong/incorrect approaches to tackling the problem. These are offered as case studies of what to avoid.

/ brane

A brane (from 'membrane') :== some mathematical generalization of a category from zero (0) or more higher dimensions. A combination of branes is called a mesh.
My portfolio is a twenty-six dimensional (26-dim) toy [🧩puzzleware] with braneworld structure according to pitch class:
0-brane = point particle
1-brane = string
2-brane = manifold/sheet/tape
3-brane = cell
25-brane = Starbureiy automaton
(see Egglepple, lnq🧑🏿, patch, walk, mathemusic)

/ fold

Folding :== differential steps in the process of a brane's animation (ie. symmetry-breaking) as it assumes its final topology. Attainment of this (solution) - the closed🔒 string - should result in a 0-bubble twistor [yesegalo🍄meshroom0b(u,u)].

For the most part, "to fold" an object is to bend, compress, twist, or otherwise permute it so that it becomes self-integrated according to its calculus. By-and-large, we accomplish this using fruut as part of stew choreography.

A fold would be an intermediate state within the walk process (preimageimage). Mathemusically, a fold is conguent to some movement.

Essential to stereotyping, our interest in folds and folding*, in general, is so that we can (more easily) synthesize middlegame makeup.** An act of hyperlinking (see The Origamic Symphony🎶, Gallivan fold, manifold, hyperlink, route, tip, knot🪢)
/// An inbetween will not always qualify for being one of these states.

/ coverage

A spread's coverage [cover-patch identity]* :== the twistorspace posture (knot🪢 formation) for statistically graphing a [supersymmetric] zone on(to) some pitch (ie. fibor-bundling a set of BIG + SMALL branes). Since there are 100 different covers (0-99) that can be patched 26 different ways (a-z), there exists a total of 2,600 discrete coverages.** An example would be something like [(cover-34,patch.g) = coverage 34g].

As cryptographic shape schemes [puzzles🧩], coverages are fundamentally combinatorial; spanning the entire range of branes(SMALL + BIG). A coverage can be (strategically) both preventive and responsive. (see #coverage, toonlight, 🔋e:e statistics, patchwork, cassette, Honne Bay🪱)

/ twistorspace

Ludologically, twistorspace or "slot" :== a complex configuration space (eg. geotop) of solutions to the twistor identity ||f(u,u)=0||.

Because a fibor is encoded to close🔒 its loopstring➿ the exact same way every time (ascertained from the uniqueness property), there must exist an acoustic bound in place enforcing (secured🔐 with crypto) these conformations (bubble or vacuum); without such a (force) field, the signal (as measured in ¢ents🪙) would not bifurcate (into MONEY or bubblegum) as a function of symmetry-breaking.
/// +We assume that each twistor field is unique to some fibor. Essentially, this is a mapped spinor. The 'spin' (an object's spinning metric signature) here is identifiable with the quantum fields of arbitrary spin.
+Everything in twistorspace is dyadic (u,u).

Twistorspace' complexity is curvilinear (T)*.** More accurately, sesquilinear. (see handicap, juke field, coverage)

/ zero-bubble

The zero-bubble (also written as "0-bubble" or "0b") or "lock" :== a bubble shapeframe (stereogram*) with no inflatons (scalar fields responsible for blowup from a single point [0-brane]).** Greater than or equal to three (3) spatial dimensions.

A 0b is a type of homeomorphism [map reduction] as a totally twisted loop➰ (ie. integration in twistorspace down to one ¢ent🪙), and formally marks the end of a proof.🔲→🎯 (see goal, open string, identity, font, crypto, MONEY)
/// The supposition for 'zero' (+/-) is that when tabulating the yield, all spots on the mesh which would allow for an opening, vanish (0,0).

Locks are the ideal [most secure🔐] ¢ent🪙 structures [signed certificates] for a closed🔒 string, as measured by absolute compliance to its handicap. Ludologically, they are part of the endgame [statistical inference to egg(+) or epp(-)]. Taken by themselves as 'flags'🏁, zero-bubbles (frictionless assets = spread(0)) are considered solved puzzles🧩 (ie. completely mapped/encrypted/melodic/folded) when it comes to juking. {input = output, /return = [0]}
Function map: open stringstew choreographylockfibor

/ loop➰

Mathematically, a loop➰ :== a closed🔒 string with no knots or tanglements whose path initiates and terminates at the same point.

Ludologically, a loop is a simple infinite walk (ie. circuit); such that it encloses leaves (reading frames) on a loopstring➿. In both cases, a loop is continuous with either no nuts, or has homologically conjoined endpoints.
/// For sake of stew choreography, a loop is just a completed walk that has become unavailable for recycling.

/ Hodge conjecture

Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

/ manifold

/ knot🪢

A knot🪢 is a closed string🔒 as a link with one (1) component. It can also be defined as a tangled string in Euclidean space (R3).

/ tip

A thread-inflection point (tip) is an inflection point (ie. maxima/minima) on a graph. (see also ear, double-u economics, UUallet)
/// The exercise of tipping usually comes in the form of pay-per-post.

/ braid

/ Poincaré conjecture

/ noncommutative geometry

/ topology

/ geometry

/ geometry and topology

/ supergeometry