Notes from 7/21

Here are the notes I took from last Wednesday

Professor Sullivan discussed the two main problems we are faced with in mathematics

1. Recognition - Given an object, determine what we are looking at.

2. Classification - Given certain specifications, list all the possible objects.

When discussing these problems it is necessary to define what we mean by two maps being equivalent. This definition depends on what situation we are in and what we are studying.

I. Maps between two different spaces.

[Definition] Two maps f: X -> Y and g: X' -> Y' are said to be equivalent if there are structure preserving bijections Ixx': X->X' and Iyy':Y->Y' such that Iyy'*f = g*Ixx'.

What a structure preserving bijection is depends on the spaces we are dealing with. For example, bijections for sets, isomorphisms for vector spaces, groups, homeomorphisms for topological spaces, diffeomorphisms for smooth manifolds, etc.

Since from this point of view X,X' are equivalent and Y,Y' are equivalent it suffices to just consider g: X->Y

[Examples]

1) If X and Y are finite sets all the important information of a map are contained in the following data:

-Cardinality of domain

-Cardinality of range

-Cardinality of preimage of each point in range

2) If X and Y are finite dimensional vector spaces the problem is even simpler. All we need is:

-Dimension of domain

-Dimension of range

-Dimension of image

II. Maps between a space and itself.

[Definition] Similarly, we define maps f: X -> X and g: X' -> X' to be equivalent if there is a single structure preserving bijection Ixx': X->X' such that Ixx'*f = g*Ixx'.

Thus, in this case we are looking for solutions to g = I^(-1)*f*I. This is in contrast to case I where we are looking for solution to g = Ixx'^(-1)*f*Iyy'. In case II there are less unknowns, so the problem is such more difficult and its study is much more fruitful.

[Examples]

1) Finite Dimensional Vector Space

The problem of finding equivalent maps translates to determine conjugacy invariants of matrices. Some familiar conjugacy invariants are a matrix's determinant and trace (and in fact each coefficient of a matrix's characteristic polynomial).

Thus, just by considering this simple example, we can see the strong difference in cases I and II. In case I we had a finite number of criteria by which every map could be classified. In case II, given a map between an n-dimensional vector space and itself, we have an n-dimensional family of conjugacy invariants.

2) We can consider power of maps now and study how successive powers of a map behave.

3) X is a finite set and our map is a bijection.

(In terms of group theory, out problem would translate to determining the conjugacy classes of Sn - the group of permutation)

We know that one particular bijection between a set in itself is a a "rotation" of itself elements. Namely, if we label our elements 1 through n, the bijection would send 1->2, 2->3, ..., n-1->n, n->1. (There is only one such bijection up to isomorphism).

We can then study every bijection by breaking it up into what are called cycles, which is a group of elements that act as a rotation. To determine the cycles of a bijection we start with an element and see where this element goes under powers of our bijection. We continue looking at powers of the bijection until we come back to the element we started with. The elements that we cycled through, in order, are the member of the first cycle.

We continue this process with another element not in the first cycle to determine the second cycle. This process ends after a finite amount of time.

Looking at the cycles of a bijection gives us a "picture" of the bijection. For example, suppose our set has 3 elements. The possible bijections are as follows:

-3 cycles of length 1. This would be the identity map as each element remains fixed under the map.

-1 cycle of length 3. This would be what was described as a rotation above.

-1 cycle of length 1, 1 cycle of length 2. The map would fix one element and interchange the other two.

To get an idea of how big the class are that we are looking at the number of bijections is n!, which is approximately (n/e)^n. The number of different pictures (or in different language the number of conjugacy classes of Sn) is approximately e^(n/2).

Now the discussion turned to how the Greeks compared the length of two objects.

How do we compute the ratio of two objects such as the ones below?

|-----------------|

|---|

We put as many of the smaller pieces as we can in the bigger piece. For example, we can put 4little pieces inside the smaller piece

|-----------------|

|---|---|---|---|

We then take the remaining part of the bigger piece and see how many times it can fit inside the smaller piece. For example, we can put 1.

|----|

|--|

You then continue this process definitely or indefinitely to conclude that the ratio of the original lengths is (3 + (1/(1+(1/(x+ ...)))) where x would be the next step of the process.

This process could equivalently be discussed by studying rotations of an arc around a circle.

[Theorem] A continued fraction of a number X is eventually periodic iff X is the root of a quadratic equation with integer coefficients.

[Problem for homework] Given two curves on a surface, is there always a homeomorphism of a surface to itself that sends one curve to the other. You can assume that the two curves have minimal self intersection.

The undergraduate then began the present the progress on their projects.

Keren discussed her problem of studying the connections between hyperbolic length of curves and the combinatorial length of curves. She explained how the problem involves computing the eigenvalues and eigenvector of matrices representing the transformations of the surfaces. She successfully determined which matrices properly describe the transformation and calculated the eigenvalues and the slope of he eigenvectors.

Ren explained her project of examining surfaces transformation of the plane with 3 holes removed. She is studying two particular transformations: (Tab) interchanging points a and b, with 1 going over the top and (Tcb) interchanging points b and c, with 3 going over the top. She is then studying the composition Tab*Tcb*Tab*Tcb*... and how a curve changes under these compositions.

If we let the generators of the plane with 3 holes removed be {a,b,c} , then she explained how Tab changes b at a and changes a to b along with conjugation by a^(-1). Similarly, Tcb changes b to c and changes c to b along with conjugation by c.

Ren also said that under these compositions the combinatorial length of a curve increases like the Fibonacci numbers and there is never cancellation in the words.

Notes from 07/ 14/10

Here are some notes I took on the past Wednesday.
Professor Sullivan continued his topic about "the fundamental group". To construct a fundamental group using the universal covering space, we first need following definitions.
[Definition] A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, that is a function

d: M*M -> R

such that for any x, y, z in M
1. non-negativity: d(x,y) >= 0
2. identity of indiscernibles: d(x,y) =0 iff x = y
3. symmetry: d(x,y) = d (y,x)
4. triangle inequality: d(x,z) <= d(x,y) + d(y,z)

[Definition] Given a metric space (X, d), a subset U is called open iff for any element u in U, there exists a set B(u,r) = {vd(u,v)<=r}.

Now the metric space X is also a topological space.

We will assume that X is locally simply connected. Then we can construct a universal covering space and a covering map to study some important consequence of the fundamental group.
Step 1: Let X be a topological space and choose a base point x in X.
Step 2: For each point x' in X, consider all the continuous paths (i.e a continuous map from [0,1]to X) from the base point x to x'. We say that two such paths are equivalent if they can be deformed to one another without moving the start and end points and without breaking. These two paths are the same "rout".

[Definition:] The set of all such routs is the universal covering space X~. The covering map is the map from X~ to X that maps any rout, from x to x', to x'.

[Definition:] If X is a metric space with metric d, then the length the curve g: [a,b] -> X is
length(g)= sup{These sumnation from i=0 to n-1 of d(g(ti),g(t(i+1)): n is a natural number and a=t0 < t1<... < tn=b}.

[Definition:] A metric space is a length metric space if the intrinsic metric (i.e. the infimum of the length of all paths from one point to the other) agrees with the original metric of the space.

Unsurprisingly, the shortest paths are constant paths and they are nowhere differentiable. Also, since we mentioned last time, the covering map is a local isometry. An isometry is a distance-preserving map between two metric spaces so it is easy to understand that the covering map preserves the path lengths.

Next, we came to a theorem which allows us to do much with the covering spaces.

[Theorem (Homotopy Lifting):] Suppose p: Z -> X is a covering map for a space X. Let f : I^n -> X be a map from the unit n-cube to X, and F: I^ (n+1) -> X a homotopy of f to another map f' : I^n -> X. Suppose g : I^n -> Z is a map satisfying p* g =f. Then there exists a unique map L: I^(n+1) ->Z satisfying LI^n = g; p*L = F

It says that if z in Z is a point such that p(z)=x, and g is a path in X starting at x, then there is a unique path g' in Z starting at z such that p of g'=g. We say that a path in X has a unique lift to Z, once the starting point of the lift has been chosen.

[Definition:] The product of the two paths is called action.
More precisely, Let G be a topological group and X any topological set. If there is a continuous function f: G*X -> X, where G*X is given the product topology such that f(1,x) = x as well as f(g1g2,x) = f(g1, f(g2,x)), then the function is called the action of G on X.

[Theorem:]1. The fiber over the base point is a group.
2. This group Gamma_x acts on every fiber Gamma_x'.

Gamma_x * Gamma_x' -> Gamma_x'

3. The action is by left translation in the Gamma_x.
4. The action on each Gamma_x' is without fixed points unless acting with the identity.

After the lecture given by Professor Sullivan, the undergrad reported their progress during the week.

After having a good understanding of the problem, Keren started calculating the geometric length of a given geodesic in a given hyperbolic surface and study its distribution. She explained that in the upper-half of the Poincare disk model, in order to compute the geometric length, we needed to find out the eigenvalues of the transform matrices and then compute its trace. Actually she is now trying very hard to get these eigenvalues.

Ren's project is about exploring surface transformation. Start with three holes in the plane with a circle enclosing the right two. Then switch the points: left over the middle and right over the middle and keep doing those two steps. The numbers of the times that the loops pass under each hole are always the Fibonacci numbers. The sequence of lengths of the iterates of the curve grows exponentially with rate the Golden ratio.

Also, one of the properties of this surface transformation can be represented by the pattern of the curve words which are composed of A,a,B,b,C,c. The pattern is the following: In all the even steps, we change all the A's (a's) to ABa's (Aba's) and change all the B's (b's) to A's (a's). In all the odd steps, we change all the B's (b's) to C's (c's) and change all the C's (c's) to cBC's (cbC's).

Anand continued his project on finding the minimum intersections of all the cyclic reduced words of a given lengths with another fixed surface word. In the beginning of this project, Anand started with some simple surfaces. Next, he was collecting data under each condition step by step and tried to modify his problem with the help of computer program.

Notes From 07/07/10

The meeting is mainly about undergrad progress reports.


Robert continued his project on configurations on hyperbolic surfaces (a pair of pants) that are realizable as geodesics. He attempted to show that by adding one more round in the configuration to one of the legs, similar results exist regarding realizable configurations.


However, the sums of exterior and interior angles are insufficient to produce a contradiction on the configuration suspected to be unrealizable. By taking into account the areas of various shapes as well, and using the formula

-Area+Exterior Angles=2pi*Euler Characteristic

It may be possible to find a contradiction.


Jingyu formalized the relevant definitions and suggested an approach to her project about self-intersection and mutual-intersection of multiwords. She defined cyclic reduced words (abbreviated as c.r.w.), intersection of c.r.w. as the number of minimal self-intersection of a representative of the homotopy class described by c.r.w., multi-words as a list of c.r.w., as well as self- and mutual-intersection of multiwords similar to the definition of intersection of c.r.w. She was to find a representative of a multiword such that the sum of its self- and mutual intersection is minimum. She realized that it is attained when both the self- and mutual-intersection are minimum, according to some existing theorems, and proposed an algorithm to find the representative.


She defined bigon as a shape of two vertices and two arcs connected to the vertices. A proper bigon is a bigon that cannot be eliminated through surgery, and a bigon that is not proper is called an improper bigon. The related theorem then states that the total intersection is minimal when all improper bigons are eliminated through surgery. This theorem implies that minimal total intersection is attained when self- and mutual-intersection are both minimum. It then follows that for a give configuration, we can reduce bigons from the outmost to the inmost, and end up with the desired configuration.


Keren's project is about finding the distribution of geometric length of a geodesic for a certain combinatorial length in a given hyperbolic surface, and the range of the geometric length to combinatorial length ratio. A hyperbolic surface can be projected to a Poincare disk model or an upper half-plane. In the Ppincare model, a surface is represented by a surface word, and the combinatorial length of a geodesic is the number of letters in the word of the curve. In the upper half-plane model, the actual geometric length can be computed from the trace of the transform matrices.

In the upper half-plane model, the x-coordinates of an axis of transformation of a geodesic can be determined by the slopes of eigenvectors of that transform matrix, while the length of the axis be related to the trace of the matrix. For a certain word of a geodesic, its length has the following relationship with the multiplication of all transform matices (denoted by M):

cosh(l/2)=(1/2)Tr(M)

Therefore we can calculate the geometric length of any given geodesic and study its distribution.

Also, Dr. Sullivan has talked about universal covering space.

He started with the fact that two simply connected geodesically complete surfaces of constant curvature -1 are isometric. This is also true for zero and curvature +1. Simply connected means the surface can shrink to a point, and geodesically complete means a curve never leaves the surface. This fact can be proved by the theory of covering spaces.

A corollary of this fact suggests that for any connected surface of -1 curvature and "complete", then its universal covering space is isometric to the noneuclidian plane, and its covering symmetry group becomes a group of noneuclidian motions. This group is called the Poincare group or fundamental group.

For any topological space X (with tiny assumption that X is path connected with locally "unique" paths up to deformation), choose a point x in X as the base point, then all the ways of going to a point from the base point can be drawn as a stack of points above the space. If you slide a pint locally, there is a cononical bijection between the ways. Moving the base point along a closed path produces a permutation of the stack of of the base point. This is the original definition of the fundamental group by Poincare.

Group of transformations of universal covering space is given as follows:

Given (X,x), we construct (X',x') with a map Pi: (X',x')->(X,x). The group of continuous bijections to fixed points Gamma= Pi_1(X,x) such that X'/Gamma=X and X' is simply connected.

(BTW X' and x' should actually be X- and x-tilde, I cannot type it)