I3: Projective, Lorentz, Manifolds and Differential Geometry, Riemannian Metrics, I4: The General Goals of Mathematical Research: Solving Equations, Linear, Polynomial, Diophantine, Differential, Classifying: Identifying Building Blocks and Families, Equivalence, Non-equivalence and Invariants, Generalizing,Weakening Hypothesis and Strengthening Conclusions, Page 43 – 55

One view of projective geometry is all sets of lines in R3 that go through the origin with both points of the units sphere regarded as the same.    A transformation takes any invertible linear map and apply it to R3.  Linear maps may be multiples of one another.  The resulting group of transformations is like GL3(R).  Regarding all non-zero multiples of any given matrix are equivalent gives the projective special linear group PSL3(R).  It is the three dimensional equivalent of PSL2(R).  Because PSL3(R) is bigger than PSL2(R) the projective plane comes with a richer set of transformations than the hyperbolic plane.  So fewer geometric properties are preserved and, for example, there is no obvious notion of projective distance.

Lorentz geometry is used in the special theory of reletivity to model four-dimensional spacetime.  This space time is also known as Minkowski space.  The usual notion of distance betwen two points (s,x,y,z) and (t,a,b,c) is replaced by the generalized distance -(s-t)(s-t) + (x-a)(x-a) + (y-b)(y-b) + (z-c)(z-c).   The minus sign on the first term reflects the fundamental difference between space and time.  A Lorentz transformation is a linear map from R4 to R4 that preserves the generalized distance.  Let g be the linear map that sends (t,x,y,z) to (s,a,b,c) and G the corresponding 4 by 4 matrix with elements (-1,1,1,1,) on the diagonal and off diagonal elements equal to zero.   A Lorentz transformation is one whose matrix A satifies AGAT = I, I the 4 by 4 identity matrix and superscript T the transpose of the matrix.  The transpose of matrix A is matrix B with elements Bij = Aji.   

A point is said to be spacelike if -tt+xx+yy+zz > 0, timelike if tt + xx +yy + zz < 0 and in the light cone if tt + xx + yy + zz =0.  Lorentz geography is used in general relativity which may also be thought of as the study of Lorentzian manifolds.

A d-dimensional manifold is any geometrical object M with the property that every point x in M is surrounded by what may appear to be a portion of d-dimensional Euclidean space.  For example, small parts of a sphere, toris or projective plane are very close to plane and thus are 2-manifolds.   One can use an atlas to formally define a manifold.  Calculus is possible for functions defined on manifolds. 

Differentiability may be defined for manifolds.  Manifolds with transitions functions that are differentiable, and equal on different charts, are said to be differentiable manifolds.  Manifolds for which transition functions are continuous but not necessarily differentiable are called topological manifolds.

The manifold ideas generalize where the Euclidean spaces are replaced by manifolds.  The domain of the linear map is called a tangent space.

On any given manifold there may be a more than one metric defined.  The Riemannian Metric allows one to calculate distances of paths on a manifold.

The general goals of mathematical research are listed.  These goals include: solving equations, classifying, generalizing, discovering patterns, explaining apparent coincidences, counting and measuring determining whether different mathematical properties are compatible, working with arguments that are not fully rigorous, finding explicit proofs and algorithms, and addressing what you find in a mathematical paper.

Solving equations involvers three issues: does a given equation have any solutions, if an equation has solutions, then does it have exactly one solution and what is the set in which solutions are required to exist?  The first two questions are known as existence and uniqueness.  The third may provide a generalization.

Linear eauations may be written in matrix notation as A x = b.  Here A is a matrix, x and b are column vectors.  A and b are known and the problem is to solve for x.

Polynomial equations are of the form: anxn +  an-1xn-1 + … + a2x2 +  a1x1 + a0 = 0.  Solving quadratic requires the intoduction of irrational numbers and complex numbers.  There are fomulas for finding solutions of polynomials for degrees n = 1, 2, 3 and 4.  There cannot be a formula for polynomials of degree 5 or higher.  Abel and Galois showed that no formula exists for degress 5 or higher.

The field of algebraic geometry is concerned with solutions of polynomial equations in more than one variables. 

Diophantine equations are those where the solutions are restricted to the field of integers.  The most famous diophantine equation is the Fermat equation xn + yn = zn.  Andrew Wiles proved that this equation has no integer solutions for n > 2.  It does however, have an infinite number of solutions for n = 2.  There is no systematic solution to Diophantine equations.   Diophantine equations are solved in subsets, since there is no systematic approach to solve them all at once.

Differential equations include the simple harmonic motion equation and the heat equation.  The simple harmonic motion equation is:  d2x/dt2 + k2x = 0.  The solutions to these equations are functions.  The general solution to the simple hamonic motion equation is:  x(t) = A sin(kt) + B cos( kt).  These differential equations are called linear because, for example, if we write f(f) = d2x/dt2 + k2x, then f is  alinear map in the sense f(f + g) = f(f) + f(g) and f(af) = af(f).  The heat equation has this same property.  Such differential equations are called linear.  A few differential equations including some nonlinear ones can be solved exactly.

The three body problem is an example where the solution may not be written down explicitly, and the solution may be in fact chaotic.

Regular polytopes in dimensions three and higher fall into three families: the n-dimensional versions of the tetrahedron, the cube and the octahedron,  five exceptional examples, the dododecahedron, the icosahedron and three four dimensional polytopes, which have 120 three dimensional faces, 600 tetrahedra faces and one with verticies of the forms with plus and minus 1’s and plus and minus 2’s with zeroes.

Primes are used as multiplicative building blocks of integers.  In a similar way finite groups are all products of basic groups that are called simple groups.

Two objects are called equivalent if we are not concerned about their differences.  A topologist might consider a sphere and a cube equivalent, as well as a donut and a teacup.  The sphere and the torus are examples of compact orientable surfaces.  Given a compact orientable surface on can find a an equivalent surface built out of triangles that is topologically equivalent. 

One defines the Euler characteristic of one of the surfaces built out of triangles as the number of verticies – the number of edges plus the number of faces.  This gives us a way of showing that a sphere is topologically different from a torus.  One has Euler character istic of 2 and the other of 0, so they are topologically different.  This is because the Euler characteristic is the same for all triangulations of a surface and if two surfaces are continuous deformations of one another then the have the same Euler characteristic. 

The Euler characteristic is an example of an invariant.  Ifg one has a general invariant f, then if two objects X and Y are equivalent then f(X) =  f(Y).  To show that two objects are not equivalent one can show that f(X) and f(Y) are different.

In mathematics onetakes specific cases and tries to generalize them.   This may help in understanding the specific case.

Sometimes one can waken a hypothesis and then prove the conclusion.  For example, is there a number (integer) that can be written as the sum of four cubes in ten different ways?  Attacking this problem diectly is very hard.  However, one can show that there must be such a number by looking at how many ways there are of summing ten different cubes for all number sless than 1,000,000,000.  A combinatoric argument show that there must be such a number.  This argument does not supply the number but only guarantees its existance.

Instead of weakening the hypothesis one may think of this method as strenthening the conclusion.  This uses the augument that P \Rightarrow  Q may also be written v\Rightarrow vQ.

Leave a Reply

You must be logged in to post a comment.