Archive for April, 2010

III.16 Differential Forms and Integration, Terence Tao: Page 178.

Friday, April 30th, 2010

0-forms, 1-forms, 2-forms, etc. are called differential forms.

Fundamental scalar function operations are addition, pointwise product and differentiation.  These can be generalized to differential forms.

An example of a solid or surface of a sphere is used to show the relationship between 3-forms and 2-forms.

There are sign changes related to antisymmetry in the derivation rule for differentiation.  The differentiation operator is nilpotent, d(dw) = 0.

III.16 Differential Forms and Integration, Terence Tao: Page 177.

Thursday, April 29th, 2010

The fundamental theorem of calculus generalizes to oriented curves.

Exact and closed 1-form’s.

Example of computation of flux – of for example a magnetic field.

Generalization of 1-dimensional linear dependence to multiple dimensions.

Antisymmetry and bilinearity of 2-form’s.

III.16 Differential Forms and Integration Continued, Terence Tao: Page 176.

Wednesday, April 28th, 2010

Ambient space.

Continuously differentiable path or “oriented rectifiable curve”.

Parametrized line segment as an example.

Section of a cotangent bundle as a 1-form.

Integration is a binary operation taking a curve and a form as input and returns a scalar.

III.16 Differential Forms and Integration, Terence Tao: Page 175.

Tuesday, April 27th, 2010

Three kinds of integration indefinite integrals, signed definite integrals and unsigned integrals.  In one dimension the signed and unsigned integrals have a simple relationship.

III.11 Countable and Uncountable Sets, III.12 C*-Algebras, III.13 Curvature, III.14 Designs, III.15 Determinants: Pages 170 – 174.

Monday, April 26th, 2010

Continued Fractions

III.1 1  Countable and Uncountable Sets

III.1 2 C*-Algebras

III.1 3 Curvature

III.1 4 Designs

III.1 5 Determinants

III.10 Computation Complexity Classes: Page 169.

Wednesday, April 21st, 2010

Computational Resources

Time

Memory

P and NP

PSPACE

Computational Complexity

http://qwiki.stanford.edu/wiki/Complexity_Zoo

Continued Fractions under The Euclidean Algorithm and Continued Fractions

III.9 Compactness and Compactification, Terence Tao: Pages 167-168.

Tuesday, April 20th, 2010

Compactness and Compactification

Finite and Infinite Set domains

Three statements about functions

Compact sets

Three assertions

A fourth assertion

Compact topological space

Tychonoff’s theorem

The extended real line

One point compactification

Minimal and maximal compactifications

Divergence

Limits

III.8 Categories, Eugenia Cheng, Class Field Theory, Cohomology: Page 166.

Monday, April 19th, 2010

I finished up the Category description in the end of the prior blog post.

III.8 Categories, Eugenia Cheng, Class Field Theory, Cohomology: Pages 165-166.

Friday, April 9th, 2010

Eugenia Cheng authored the section on mathematical categories.  The important maps between groups are group homomorphisms and between vector spaces are the linear maps.  These maps preserve structure.  A group homomorphism f from group G to group H preserves multiplication in the sense that f(g1g2) = f(g1) f(g2) for any pair of elements g1 and g2 in G.  Similarly linear maps of vector fields preserve addition and scalar multiplication. 

The above structure preserving map can be generalized to mathematical structures, A, B and C, and f and g structure preserving maps from A to B and B to C respectively.  Then the composition map gëf is structure preserving (if the range of one equals the domain of the other).  Structure preserving maps can be composed.  A and B are said to be isomorphic if there is a structure preserving map from A to B and an inverse that also preserves structure. 

A category allows us to discuss the above in the abstract.  A category consists of a collection of objects, and a collection of morphisms between these objects.  There is also a notion of composition of morphisms.  Compositions must be associative.  There must also be an identity morphism.

I addition to the group and vector space examples of categories there are also others, such as:

i.  Natural numbers as objects, morphisms from n to m are real n x m matrices, composition is matrix multiplication, this is an unusual example of a map between natural numbers m and n.

ii.  Objects may be the elements of a set, morphisms from x to y are “x = y”, and ordered set can use instead the inequalities, “x    y”, as the morphisms.

iii.  Any group can be the object and the morphisms from the group to itself are the elements of the group, group multiplication defines the composition of two morphisms.

iv.  Objects can be topological spaces, morphisms are continuous functions or the morphisms could be homotopy classes of continuous functions.

Morphisms are also called maps.  The maps can also be called arrows.  Arrows are used to emphasise the abstraction and in pictorial representations.

Shape refers to morphisms and equations the morphisms satisfy.

A morphism f from a to b is written pictorially as: a  f™ b.  The f should be above the arrow but I couldn’t do that with the font I am using.  The composition of two morphisms f and g may be written: a  f™ g™ c.  g ë f = t ës is an equality between morphisms.  It may be pictorially displayed as a square with top and bottom sides a  f™ b  and d  t™ c.  And sides down-pointing arrows of a  s™ d and b  g™ c and asserting that the diagram commutes in the sense that either path from a to c yields the same composite, as in the equality. 

Commutative diagrams, generalizations of this simple square, can be used to describe free groups, free rings, free algebras, quotients, products, disjoint unions, function spaces, direct and inverse limits, completion, compactification, and geometric realization, among others.

A pictorial diagram that looks like an arrowhead may be used to describe a dijoint union of sets.  The universal property of a disjoint union reflects the “most free” way of having two sets may into another set.  Universal properties are central to how category theory describes canonical structures. 

An isomorphism is a morphism with a two-sided inverse.

Categories form a category – look out for a Russell type paradox.  Functors are the structure preserving maps for categories.  A functor F from a category X to a category Y tales the objects of X to the objects of Y and the morphisms of X to the morphisms of Y so that the identity of a is taken to the identity of Fa and the composite of f and g is taken to the composite of Ff anf Fg.  An important example of this is a functor that takes a “marked point” topological space S to its fundamental group p1(S,s).  And a continuous map between two topological (marked point) spaces gives rise to a homomorphism between their fundamental groups.

There is a concept of “natural transformation” a morphism between functors.  This is analogous to a homotopy between maps of topological spaces.  The analogy includes the topological idea of holes.  Avoidance of holes by paths is reflected in the commutativity of certain commuting squares.  This is known a s the naturality condition.

One example is that every infinite dimensional vector space is cononically isomorphic to its double dual.  this is not rue for finite dimensional vector spaces.  In the presence of natural transformations categories actually form a 2-category – a two dimensional generalization of a category.  An n-category has morphisms for each dimension up to n.

Category theory can be thought of as foundational and may replace set membership as the foundation of mathematics.

“Class Field Theory” is discussed under “From Quadratic Reciprocity to Class Field Theory”.

“Cohomology” is discussed under “Homology and Cohomology”.

III.6 Calabi-Yau Manifolds: Conjecture, Manifolds in Physics, Eric Zaslow, The Calculus of Variations and III.7 Cardinality: Page 164.

Thursday, April 8th, 2010

The Calabi-Yau Manifold Calabi conjecture is that the metric notion amounts to a nonlinear partial differential equation in u, namely: det(gab’ + 2u/zab) = |f|2

Calabi proved the uniqueness and Yau proved the existence of the solution to this equation.  The metric definition of a Calabi-Yau manifold is uniquely determined by its Kähler structure and its complex orientation.

The space of metrics with holonomy group SU(n) on a manifold with complex orientation is in correspondence with the space of inequivalent Kähler structures.  The requirement of conformality and a vanishing ricci tensor are essentially equivalent. 

A Calabi_Yau manifold with its SU(n)-holonomy metric has vanishing Ricci tensor and is of interest in general relativity.  Calabi-Yau manifolds figure prominently in the leading theory of quantum gravity, string theory.  Quantum gravity is the incorporation of Einstein’s theory into the quantum theory of particles.

In string theory the fundamental objects are “strings”.   Worldsheets are used to describe the two dimensional trajectory of the strings.  String theory is is constructed from the two dimensional Riemann surfaces of worldsheets to a space-time manifold, M.  Although the number of possible metrics is infinite under the assumption of invariance under local changes of scale the dimensionality reduces to a finite number of conformally inequivalent metrics and the theory is well defined.  Adding the idea of supersymmetry means we have the space-time manifold M to be a complex manifold.

These two conditions together mean that M is a complex manifold with holonomy group SU(n) or a Calabi-Yau manifold.  The requirement of conformality and a vanishing Ricci tensor are essentially equivalent.  This means the theory matches Einsteins although without matter. 

Calabi-Yau manifolds are both symplectic and complex in the topological string theory.  This leads to two types of topological strings labeled A and B.  Mirror symmetry is the phenomenon that the A version of one Calabi-Yau manifold is related to the B version of another different Calabi-Yau manifold.  The B version is a mirror partner.  The mathematical consequences of this mirror symmetry are being investigated.

“The Calculus of Variations” is discussed under “Variational Methods”.

A finite set has cardinality equal to the number of elements in the set.  A countably infinite set is said to have cardinality  \aleph_0.  The reals are uncountable and have cardinality 2^{\aleph_0}=\aleph_1.   These numbers are called cardinals and they can be added, multiplied and raised to powers.  This will be discussed further under “set theory”.