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数学

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Andrius Kulikauskas

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数学笔记

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https://ncatlab.org/nlab/show/cellular+approximation+theorem#applications

Any functor {$F$} can be thought of as {$F:J\rightarrow C$} where {$J$} is the shape, the index set (how) and {$C$} is the image (what). And for any functor we can ask if it has a limit (an object {$L$} with maps {$\psi_X:L\rightarrow F(X)$} for all objects {$X$} of {$J$}, such that for all {$f:X\rightarrow Y$} in {$J$} we have the analogous morphisms commute in {$C$}, and that is universal as such). So that limit is Why. And is the colimit Whether? Or is whether simply the object with its identity morphism? Compare with the Yoneda lemma.

Relate walks on trees to covering groups. What do conjugates (paths) mean? What is the homotopy group?

https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics

https://www.amazon.com/Quantum-Challenge-Foundations-Mechanics-Astronomy/dp/076372470X

In what sense are Feynman diagrams relativistic given that they have directions for time and for space?

Instead of thinking of speed of light, think of a clock that doesn't tick, so that t=0 always. And this is the case for the quantum harmonic osciallator and for the particle-clocks with no steps.

One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}.

Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006

Gerald B. Folland

  • Quantum Field Theory: A Tourist Guide for Mathematicians 2021
  • Quantum Field Theory 2008

Bohm Pilot Wave, Thomas Spencer

Relative invariance - more global than another

Relate the three-cycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter.

Alan Turing, Cybernetics and the Secrets of Life

My approach to special relativity lets me work in units in my own frame.

https://en.wikipedia.org/wiki/Algorithmic_information_theory Gregory Chaitin = Shannon + Turing = Compression-Decompression as understanding.

https://en.wikipedia.org/wiki/Cristian_S._Calude Philosophy of computation

https://www.youtube.com/watch?v=xP5-iIeKXE8 Life in life

Thinking about the expansion of the universe as a reduction of density, by which the mass of particles becomes ever less important, by which we have an increase of entropy (becoming less deliberate). And we can reverse this by starting with an increase in entropy and arriving at the expansion of the universe.

Relate Ellerman's heteromorphism and comma category.

Consider how the understanding of Yoneda lemma in terms of a left Kan extension, and in particular, the factoring, relates to the push down automata.

San Francisco Meet Up interests: Dependently typed programming languages. Language aspects of category theory. Functional programming. Topos, lambda calculus. Is type theory advantageous? Modeling infinitesimals.

Kan extensions are a framework for universality. Consider example 6.1.3, the Yoneda Lemma. Think in which ways the universality of limits, colimits, adjunctions, etc. is captured by Kan extensions. All of these universal properties can be thought of in terms of initial or terminal objects in the appropriate categories, such as the category of cones, or the comma category for the universal mapping property for adjunctions. So consider the relevant categories. How do they relate to the classification of adjoint strings?

Dan Shiebler. Kan Extensions in Data Science and Machine Learning

Matematika išplaukia iš (poreikių tenkinimo) algoritmų taikymo, vedančio iš duotybių į bendrybes. O tos bendrybės įkūnija, išreiškia tam tikrus prieštaravimus, juos paverčia sąvokomis, kurias galima mąstyti toliau. Pavyzdžiui, apskritimas iškyla iš begalinės simetrijos visom kryptim, arba iš virve aprėpto ploto maksimalizavimo.

  • Vaizduotė (24 matai) ir Neįsivaizduojamieji (2 matai) yra iš viso 26 matai. Ar juos išreiškia stygų teorija?

Mathematics is described in terms of set theory. The category of graphs {$\textrm{Set}^{A}$} where {$A$} is the category with two objects, edges E and vertices V, and two nontrivial morphisms target {$t:E\rightarrow V$} and source {$s:E\rightarrow V$}. Similarly, all mathematical structures and their structure preserving morphisms should have a similar expression in terms of sets and their relationshps. Work out various examples. Then study the role of {$\textrm{Set}^{X}$} in the Yoneda Lemma.

SL(2,C) character variety related to hyperbolic geometry. SL2(C) character varieties

Universal enveloping algebra is an abstraction where the generators are free and thus yield infinite generators. Whereas the Lie algebra may be in terms of concrete matrices and the underlying generators, when understood not in terms of the Lie bracket but in terms of matrix multiplication, may have relations such as {$x^2=0$}, {$h^2=1$}.

Information is what you learn. What you learn grows at the boundary, has the shape of the boundary. A shape can be thought of as being created by integrating over these boundaries as they increase.

Tai-Danae Bradley: Information is on the Boundary

Prove that the matrix made up of eigenvectors diagonalizes a matrix.

In special relativity, think of distance squared over time as surface area per time, the difference beween the surface areas of two spheres, one expanding with velocity v, and the other with velocity c.

For John: How could we get negative energy? Consider how to get imaginary square roots. For example, if a speed is greater than the speed of light, then the relationship between time and position is multiplied by an imaginary number.

Quaternions, Dirac equation: Pauli matrices are the three-cycle for learning and they are extended by a fourth dimension of non-learning (what is absolutely true or false) for the foursome.

https://en.wikipedia.org/wiki/Lebesgue_covering_dimension

Covering spaces with repetition yield the spaces they cover.

Enveloping algebra (important for adjunctions) is related to Hochschild cohomology.

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

Shmuel Weinberger. Algebraic theory of topology. Manifolds and K-theory: the legacy of Andrew Ranicki

Lifting a path is like inverting a functor. How is that related to adjunctions? Adjunction is conditional inversion.

Unclear whether the empty space is path connected.

Think of a universal covering space as expressing the unfolding of a space, thus expressing eternal life.

Wikipedia Library

Peace conference

Relate triangulated categories (with squiggles {$X\rightsquigarrow W = X\rightarrow TW$}) to monads with likewise squiggles.

Samwel Kongere vaizdo įrašai

Nafsi Afrika Acrobats - Pyramid of Peace

Research/Notes

  • Monads deal with scopes: none, some, and so on. The logic of the sevensome.

https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/motion-and-special-relativity/

Relate {$F_1$} with the basis element 1 in a Clifford algebra.

The house of knowledge for mathematics describes 4 representations (properties) of everything (onesome, totality), which through their unity establish, define space as algebraic, consisting of enumerated dimensions:

  • center (nullsome)
  • balance
  • set of roots of a polynomial
  • list of basis vectors

(Relate this to the binomial theorem.) And it describes 4 representation of the nullsome (center), which through their unity establish, define a point as analytic. This describes four choices:

  • induction (adding a vertex, converting the center to a vertex, recursively)
  • max or min (adding an axis, as with cross polytopes)
  • least upper or greatest lower bound (making a division, a separation on one side or the other)
  • limit (center?)

Meanings are variously related by adjunctions. They enrich the meaning and extend the context.

  • Think of my understanding of my three grandfathers as changing with context.

4 levels of knowledge is sufficient (in the chain complexes). The house of knowledge describes those 4 levels. It relates the analytical view of a point with the algebraic view of a space. Consider the Zig Zag Lemma as applying the three-cycle to set up four levels of knowledge, 4 x 3 = 12 circumstances.

Counterquestions

  • Consider them as a subset of the utility graph {$K_{3,3}$} which describes the three utilities problem and arises in the proof of Kuratowski's theorem characterizing planar graphs.
  • The utility graph can be drawn as a hexagon, in which case only one graph can cross the center if it is to be a planar graph. In that case the center line goes from God's perspective to the world's situation. And this arrangement makes person-in-general and person-in-particular equal in status. Thus it provides a context for such equality of status. And it defines a division of everything into two: general (not knowing) and particular (knowing). It supports the equality of gender.
  • Consider how the counterquestions define divisions of everything and relate to Bott periodicity.
  • Consider how the counterquestions arise in Jesus's house of knowledge and how that relates to the house of knowledge for mathematics.
  • Consider how the counterquestions express visualization and paradox.

Enveloping algebra (important for adjunctions) is related to Hochschild cohomology which is a special case of the functor Ext.

  • {$\operatorname{Hom}_{A^e}(A,M)$} (where {$A^e:=A\otimes_k A^{op}$} is the enveloping algebra of A and A is considered an A-bimodule via the usual left and right multiplication)

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

Counterquestions

Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (half-link)

A chain complex is loose and has slack, which is the basis for homology. An exact sequence is tight and has no slack. A division of everything is tight and has no slack.

  • Santykis su Dievu yra atgarsis, kaip kad dalelytė turi santykį su savo lauku.
  • Fizikos dėsnių raida yra pavyzdys Dievo įsakymo patobulinimo.

Fivesome

John Baez, Michael Shulman. Lectures on n-Categories and Cohomology.

  • (-1)-categories are hom(x,y) sets where x and y are parallel 0-morphisms in a 0-category, which is to say, a set. But the only 0-morphisms in a set are the identity morphisms. Thus hom(x,y) is either an identity morphism (when x=y) or the empty set (otherwise). These are the two possible (-1)-categories.
  • (-2)-categories are hom(x,y) sets where x and y are -1-morphisms in a -1-category. But there is only one non-empty (-1)-category and it has only one morphism. Thus there is only one (-2)-category and it consists of this unique morphism. This category expresses necessary equality when there is only one choice. That is reminiscent of the choice from a single choice which is modeled by {$F_1$}, the field with one element.

Note that there is only one empty set. But there could also be many empty sets. And all can be thought of as an empty set. Can the search for constancy be considered a search for emptiness?

Emily Riehl, Dominic Verity. Elements of ∞-Category Theory

Foursome

For C and D categories we have

  • f is (essentially) 0-surjective {$⇔$} f is (essentially) surjective on objects;
  • f is (essentially) 1-surjective {$⇔$} f is full;
  • f is (essentially) 2-surjective {$⇔$} f is faithful;
  • f is always 3-surjective.

Foursome

A functor between ordinary categories (1-categories) can be:

  • essentially surjective ≃ essentially 0-surjective
  • full ≃ essentially 1-surjective
  • faithful ≃ essentially 2-surjective
  • Every 1-functor is essentially k-surjective for all k≥3.

A functor {$F:C→D$} is essentially surjective if it is surjective on objects “up to isomorphism”: If for every object {$y$} of {$D$}, there exists an object {$x$} of {$C$} and an isomorphism {$F(x)≅y$} in D.

A functor F:C→D can be:

essentially (k≥0)-surjectiveforgets nothingremembers everything
essentially (k≥1)-surjectiveforgets only propertiesremembers at least stuff and structure
essentially (k≥2)-surjectiveforgets at most structureremembers at least stuff
essentially (k≥3)-surjectivemay forget everythingmay remember nothing

This formalism captures the intuition of how “stuff”, “structure”, and “properties” are expected to be related:

  • stuff may be equipped with structure;
  • structure may have (be equipped with) properties.
  • Install: OBS Studio

Yoneda lemma lets you go from natural isomorphism of homsets to natural isomorphism of functors.

Are the doubts and counterquestions related to electromagnetism, U(1) and the related gauge theory?

Observing symmetry requires breaking symmetry.

https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking

Is the associativity diagram for monoidal categories an example of the fivesome?

Involution is square root of permutation. Compare with spin as square root of geometry.

Math Discovery

  • How is gravity related to the argument by continuity?

Local and global quantum are linked by experiments, by "the complicated interplay between infrared and ultraviolet affects", by a conspiracy of IR/UV mixing.

Walks

  • Independent entries vs. Rotational invariance yield {$P[X]\propto e^{-\frac{1}{2}\textrm{Tr}X^2}$}.
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Šis puslapis paskutinį kartą keistas May 22, 2022, at 09:41 PM