Revision as of 18:12, 8 April 2023

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Algebraic Geometry of Computing

Finite state machines appear in a variety of instantiations: mechanical, electronic, fluidic. The physical mechanisms involved necessitate that the design is described by differential equations, but ultimately the manipulation of abstracted "logical" states is the final goal. Thus we can describe the architecture of a general finite state machine with [math]\displaystyle{ \mathbb{Z}/2\mathbb{Z} }[/math] algebra (or other finite rings too).

Gates

Typically you will see a logic gate defined by its values on all combinations of inputs as a "truth table":

AND
x	y	x AND y = AND(x,y)
0	0	0
0	1	0
1	0	0
1	1	1

And statements written with logical connectives: [math]\displaystyle{ (x\and y)\or z = OR(AND(x,y),z) }[/math]
Along with distributive laws: [math]\displaystyle{ (x\and y)\or z = (x\or z)\and(y\or z) }[/math], [math]\displaystyle{ (x\or y)\and z = (x\and z)\or(y\and z) }[/math]
De Morgan's laws: [math]\displaystyle{ \neg(x\and y) = (\neg x)\or (\neg y) }[/math], [math]\displaystyle{ \neg(x\or y) = (\neg x)\and (\neg y) }[/math]
All of which apply to more complicated sentences rather than just individual variables. These laws along with commutative and associative laws are sufficient to evaluate and simplify any general logical expression, however we contend that this is the wrong language for computing and makes other important aspects - the dynamics and algebra - obscure.
There is one thing we can extract from logical connectives before moving on. The disjunctive normal form allows us to read truth tables and directly translate them into connective formulae which we can use later. Let us look at a different example which will help us escape the artificiality of AND and OR.

XOR
x	y	XOR(x,y)
0	0	0
0	1	1
1	0	1
1	1	0

XOR is only "true" or 1 when x or y but not both, are 1. Disjunctive normal form says that we can view the x, y entries as unary operators which return the input with no change, combine these as given on the lines which evaluate to 1, and take the OR of all of them for the total connective form of the truth table. Here is the third line: [math]\displaystyle{ x\and\neg y }[/math].
The total is: [math]\displaystyle{ \begin{align*} &(x\and\neg y)\or (\neg x\and y) \\ =&(x\or(\neg x\and y))\and(\neg y\or(\neg x\and y)) \\ =&(x\or\neg x)\and (x\or y)\and(\neg y\or\neg x)\and(\neg y\or y) \\ =&(x\or y)\and (\neg y\or\neg x) \\ =&(x\or y)\and\neg (y\and x) \end{align*} }[/math]
This process can be viewed as a sum of "elementary functions" which are only 1 on one line each, and building a general function/table.

Read prototype

Linear algebra, Mechanics, Relativity and Fields, Differential Geometry

The starter pack to physics and differential geometry

Our point of view is that the texts typically used in physics and especially mathematics degree tracks are window dressing for the real job of being a mathematical physicist or even an engineer. Excellent texts meet a certain standard we set here; Texts should be concise to respect the reader's time and occupations, interdisciplinary, at least relating mathematical tools between areas of mathematics:

Lang's algebra text contains examples and applications in geometry and number theory throughout
Vaisman emphasizes the typically algebro-geometric method of sheaves in a differential geometry setting and to develop the theory of multiple sorts of manifolds
We choose physics texts which connect to heavy mathematical machinery such as curvature and covariant derivatives in general relativity or symplectic/variational geometry in mechanics
The algebraic topology texts are not "pure" either - focusing on applications to differential or algebraic geometry, and many more.

Thus, the structure of this book list will be centered around core topics in theoretical physics which are already given direct connection to technology and reality, and the mathematics that follows from the theory rather than simply chasing popular formalisms. Future iterations will make an effort to connect with more computational content, such as that seen in representation theory or Olver's text on applications of Lie groups. This pertains to our last criterion that there should be some elementary aspects in a text - showing the translation of the abstract machinery into basic computations to make the relationships with other areas even more transparent.

Also see this list of video lectures, the lectures by Schuller concisely summarize various algebraic and geometric constructions commonly appearing in theoretical physics.

A related set of texts to this one, working with the same basics has been produced leading all the way up and through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory here.

List Structure

Calculus is not in the pictured starter pack because it is found more often in high school curricula, while linear algebra (despite being core to "applied mathematics" topics such as engineering, numerical computing, and statistics) is often missing at the required level of rigor. Thus, we suggest looking at any Basic Mathematics to quickly fill in your gaps and as a source of quick and dirty computational techniques used universally.

The texts by Landau are the list's core. While on that track, you should start dipping into the texts listed under the Landau volumes to enhance your perspective on repeated readings

The General Mathematics section covers the knowledge that would be acquired in standard (but basic) graduate courses on the different areas of mathematics that later develop into modern topics, and should be developed alongside Landau.

The Aspirational section contains some of the big ideas, which may be repeated from earlier but deserve emphasis. These are the triumphs of mathematics, peaks that everyone deserves to climb.

Basic Mathematics

Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.

Linear Algebra

Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.

Calculus

Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.

Landau

Mechanics

Classical mechanics of particles by Lev Landau.

Applications

Symplectic geometry and other mathematical Structures of Classical Mechanics

The Classical Theory of Fields

Classical field theory of electromagnetism and general relativity by Lev Landau.

Applications

Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory

Quantum Mechanics

Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz

Applications

Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics

Sets for Mathematics

Categorical approach to set theory by F. William Lawvere.
Backbone reference:

Tensor Analysis on Manifolds

Tensor analysis by Richard Bishop and Samuel Goldberg.
Prerequisite:

Linear Algebra

Backbone reference:

Lectures on Differential Geometry

Differential geometry by Shlomo Sternberg.
Prerequisite:

Linear Algebra

Backbone reference:

Cohomology & Differential Forms

Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.
Backbone reference: