Archive for May, 2010

III.21 Elliptic Curves by Jordan S. Ellenberg: Page 190

Wednesday, May 12th, 2010

Functional form and theoretic definition of an elliptic curve.

Relation to Abelian groups called E(K).

Abelian varieties.

Mordell-Weil group.

Rank of E(K) through the Birch-Swinnerton-Dyer conjectural formula.

Given a prime p and a subgroup of elements, P, such that pP = 0 determines a subgroup called E(K)[p].  Using the algebraic closure of K, say K-bar, we can get the subgroup E(K-bar)[p].  Under suitable conditions this group is isomorphic to (Z/pZ) squared.  The Galois group, Gal(K-bar/K), permutes this group.  Which gives rise to Galois representations.  These ideas were used by Andrew Wiles to prove Fermat’s Last Theorem.  Related to Langlands program, automorphic forms and modular forms.

If E is an elliptic curve over C, the complex field, then we have a complex manifold.  Induces transformations which are simply translations.  This gives rise to Hodge theory.  Elliptic functions with periods.

Moduli space gives rise to an object called an orbifold and modular curves.

III.19 Duality: Abelian Groups, Homology and Cohomology, Differential Forms, Distributions, Mirror Symmetry and Representation Theory; III.20 Dynamical Systems and Chaos : Pages 189 – 190

Tuesday, May 11th, 2010

Abelian Groups

Homology and Cohomology

Differential Forms

Distributions

Mirror Symmetry

Representation Theory

Dynamical Systems and Chaos

III.19 Duality: Sets and Their Complements, Dual Vector Spaces and Polar Bodies: Page 188

Monday, May 10th, 2010

In set theory the theorems remain true if one replaces the sets by their complements and interchanges union and intersection.  There is a duality between sets and their complements.

The set of all linear functionals of a vectors space is a dual space to the vector space.  If we have a Banach space then the space of all continuous linear functionals is the dual space.   The duality induces pairings.

Polar bodies are defined using the inner product on Euclidean n space.  A Platonic solid centered at the origin has a dual of polar bodies which are multiples of the dual Platonic solid.

III.19 Duality: Page 187

Friday, May 7th, 2010

An object and a corresponding dual object.

No single definition for dual.

The five Platonic solids are place in three groups according to their duals.

Points and lines in a projective plane have duals.  Lines as pairs of antipodal points on a sphere of radius one.

III.18 Distributions, Terence Tao: Page 186

Thursday, May 6th, 2010

Analytic Functions – nice but sparse.  Examples include: exp, sin, polynomials.

Test functions.  Infinitely differentiable on closed interval [0,1].

Continuous functions.  Good for evaluation defined on [0,1].

Square integrable functions.

Borel measures.

Distributions.

Hyperfunctions.

Distributions as a weak limit of test functions.

Distributions are useful in the study of partial differential equations.

III.18 Distributions, Terence Tao: Page 185.

Wednesday, May 5th, 2010

Evaluation of points using set-theoretic ideas of domain, range and sets.

Bundles, sections, schemes and sheaves.

Resulting in measure and distribution.

Function spaces, rough and smooth functions.

Distributions or generalized functions tend to be “rough”.

Linear functional.

Dual space is the space of continuous linear functional, these tend to be “smooth”.

III.17 Dimension, Self Similarity, Fractional Dimension, Hausdorff Measure, Koch Snowflake: Pages 183 – 184.

Tuesday, May 4th, 2010

Self Similarity

Fractional Dimension

Hausdorff Measure

Koch Snowflake

III.17 Dimension: Pages 181 – 182.

Monday, May 3rd, 2010

Dimension is not a single concept.  There are many conflicting possibilities for the definition of dimension.

1.  The number of coordinates.

2.  If a d-1 dimensional closed barrier can be placed between any two points of X then we say X is d-dimensional.  This is the inductive dimensional of a set, Brouwer.

3.  A finite cover of open sets exists which satisfy three coverage properties.

4.  There are homological and cohomological definitions.  The homological definition is the largest d for which some substructure of X has a nontrivial d-th homology group.  This definition can be extended to groups and rings.

5.  The number that corresponds to the best measure of size, such as, length, area, volume and non-integer extensions of these.

The Cantor set is defined by removing middle thirds of the closed interval from zero to one. 

A typographical mistake was found.  1/3 can be written as 0.1 or 0.02222… in ternary expansions.  Not as “0.22222…” which is equal to 1.0.

III.16 Differential Forms and Integration, Terence Tao: Pages 179 – 180.

Saturday, May 1st, 2010

A continuously differentiable k-form has a derivative that is a (k+1)-form.  An example of a the shell of a sphere and a solid sphere is given.  In the shell case the flux through the surface is used.  In the solid sphere case small compartments are used and the net flux between opposing faces is aggregated.

This may be thought of as generalizing the fundamental theorem of calculus.  This can be further generalized to yield Stokes’s theorem.  On an oriented manifold the integral of dw on the entire set S is equal to the integral of w on the oriented boundary of S.  Differentiation is said to be the adjoint of the boundary operation.

Right hand rule.

Hodge duality.

Gradient

Curl

The divergence theorem, Green’s theorem and Stokes’s theorem.

Connection between usual integrals and integration of differential forms.

“Pushes forward” and “pullback” manifolds.  Change of variables.

Pullback operation respects wedge product and the derivative.