Posts tagged with isomorphism

Mirror symmetry is an example of a duality, which occurs when two seemingly different systems are isomorphic in a non-trivial way. The non-triviality of mirror symmetry involves quantum corrections. It’s like the Fourier transform, where “local” in one domain translates to “global”—something requiring information from over the whole space—in the other domain.

Under a local/global isomorphism, complicated quantities get mapped to simple ones in the dual domain. For this reason the discovery of duality symmetries has revolutionized our understanding of quantum theories and string theory.

summer school on mirror symmetry (liberally edited)

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Thinking about local-global dualities gave me another idea about my model-sketch of knowledge, ignorance & expectation.

• Under physical limitations, at a fixed energy level, Fourier duality causes a complementary tradeoff between frequency and time domains—not both can be specific. Same with position & momentum, again at a fixed energy level.
• Under human limitations, at a fixed commitment of effort|time|concentration, you can either dive deep into a few areas of knowledge|skill, or swim broadly over many areas of knowledge|skill.

If I could come up with a specific implementation of that duality it would impose a boundary constraint on that model-sketch. Which would be great as optimal time|effort|concentration|energy could be computed from other parts of decision theory.

Monotone and antitone functions

(not over ℝ just the domain you see = `0<x<1⊂ℝ`)

These are examples of invertible functions.

(Source: talizmatik)

hi-res

one can basically describe each of the classical geometries (Euclideanaffineprojective,sphericalhyperbolicMinkowski, etc.) as a homogeneous space for its structure group.

The structure group (or gauge group) of the class of geometric objects arises from isomorphisms of one geometric object to the standard object of its class.

For example,

• • the structure group for lengths is `ℝ⁺`;
• • the structure group for angles is `ℤ/2ℤ`;
• • the structure group for lines is the affine group `Aff(ℝ)`;
• • the structure group for `n`-dimensional Euclidean geometry is the Euclidean group `E(n)`;
• • the structure group for oriented 2-spheres is the (special) orthogonal group `SO(3)`.

Terence Tao

(I rearranged his text freely.)

(Source: terrytao.wordpress.com)

homotopy

hi-res

## What is a “structure-preserving” map?

• "X does something whilst preserving a certain structure"
• "There exist deformations of Y that preserve certain properties"
• "∃ function ƒ such that P, whilst respecting Q"

This common mathematical turn of phrase sounds vague, even when the speaker has something quite clear in mind.

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Smeet Bhatt brought up this unclarity in a recent question on Quora. Following is my answer:

It depends on the category. The idea of isomorphism varies across categories. It’s like if I ask you if two things are “similar” or not.

• "Similar how? you ask.

Think about a children’s puzzle where they are shown wooden blocks in a variety of shapes & colours. All the blocks that have the same shape are `shape-isomorphic`. All the blocks of the same colour are `colour-isomorphic`. All the blocks are wooden so they’re `material-isomorphic`.

In common mathematical abstractions, you might want to preserve a property like

after some transformation `φ`. It’s the same idea: "The same in what way?"

As John Baez & James Dolan pointed out, when we say two things are "equal", we usually don’t mean they are literally the same. `x=x` is the most useless expression in mathematics, whereas more interesting formulæ express an isomorphism:

• Something is the same about the LHS and RHS”.
• "They are similar in the following sense".

Just what the something is that’s the same, is the structure to be preserved.

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A related idea is that of equivalence-class. If I make an equivalence class of all sets with cardinality 4, I’m talking about “their size is equivalent”.

Of course the set  is quite different to the set  in other respects. Again it’s about "What is the same?" and "What is different?" just like on Sesame Street.

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Two further comments: “structure” in mathematics usually refers to a tuple or a category, both of which mean “a space" in the sense that not only is there a set with objects in it, but also the space or tuple or category has mappings relating the things together or conveying information about the things. For example a metric space is a tuple . (And: having a definition of distance implies that you also have a definition of the topology (neighbourhood relationships) and geometry (angular relationships) of the space.)

In the case of a metric space, a structure-preserving map between metric spaces would not make it impossible to speak of distance in the target space. The output should still fulfill the metric-space criteria: distance should still be a meaningful thing to talk about after the mapping is done.

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I’ve got a couple drafts in my 1500-long queue of drafts expositing some more on this topic. If I’m not too lazy then at some point in the future I’ll share some drawings of structure-preserving maps (different “samenesses”) such as the ones Daniel McLaury mentioned, also on Quora:

• Category: Structure-preserving mapsInvertible, structure-preserving maps

• Groups: (group) homomorphism; (group) isomorphism
• Rings: (ring) homomorphism; (ring) isomorphism
• Vector Spaces: linear transformation, invertible linear transformation
• Topological Spaces: continuous map; homeomorphism
• Differentiable Manifolds: differentiable map; diffeomorphism
• Riemannian Manifolds: conformal map; conformal isometry

A mereology joke from my forthcoming dissertation.

hi-res

John Baez and James Dolan, From Finite Sets to Feynman Diagrams

• an explosion of ideas
• equality `x=x` is boring
• why is `6÷2=3` ?

(Source: arxiv.org)

• Why bicontinuity is the right condition for topological equivalence (homeomorphism): if continuity of the inverse isn’t required, then a circle could be equivalent to a line (.99999 and 0 would be neighbours) — Minute 8 or so.
• Geometric construction (no complex numbers) of the circle group.
• Pappos’ theorem. (Minute 31)
• Pascal’s theorem.
• Desargues’ theorem.
Hat tip to +Ozymandias Haynes.

[G]eometry and number[s]…are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa. …

[O]ne can view the length ❘AB❘ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments that are congruent to AB.

With this perspective, ❘AB❘ no longer lies in the standard semigroup ℝ⁺, but in a more abstract semigroup (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically.

A unit of length can now be viewed as just one of many different isomorphisms Φ: ℒ → ℝ⁺ between and ℝ⁺, but one can abandon … units and just work with directly. Many statements in Euclidean geometry … can be phrased in this manner.

(Indeed, this is basically how the ancient Greeks…viewed geometry, though of course without the assistance of such modern terminology as “semigroup” or “bilinear”.)
Terence Tao

(Source: terrytao.wordpress.com)