Notes from 06/23/10

The meeting started with undergraduate projects reports:

Anand's problem is that given a surface (the pair of pants or the punctured torus), find the minimum intersections of all the cyclic reduced words (which are composed of a, b, A, B, where A and B are the inverses of a and b respectively)of a given length with another fixed surface word (For example, Ab or aB). He started to run Chris's program of computing minimum intersections of a single word and tried to modify it for his problem.

Jingyu's project is similar. Given the fixed surface (the pair of pants or the punctured torus), she needs to study the maximal number of minimum crossings (including the self-intersections and mutual intersections) of a multi-word (which defined as a finite collection of cyclic reduced words whose combinatorial length add up to the total length of the multi-word).
Now she is trying to come up with a computer program to solve this problem.

Rob's problem is to study the realizations as geodesics of a give configuration in hyperbolic geometry. Now he is trying to show that if the given configuration wraps around one more time of one of the legs of the pair of pants, the configuration which passes the triple point is still not realizable as before. However, no obvious contradiction can be deduced from his classic argument (considering the sum of the interior angles of the front triangle and the sum of the exterior angles of the back hexagon and derive a contradiction).

The Agenda for the main discussion:
1.Construction of surfaces of constant negate curvature.
2.Discrete fixed point free surface of hyperbolic isometries.
3.Fundamental groups and covering spaces.

First we know that a torus can be obtained by the following procedure:


Now if we unwrap the torus, we get a "helix tube". We straighten the helix tube, we obtain an infinite cylinder. Then if we unwrap this cylinder again, we get R^2, the Euclidean plane, which is the universal covering space of a torus. The fundamental domain here is the standard square which is tilings the plane R^2.

On the other hand, we can also obtain a torus by gluing a hexagon as follow:


Then if we unwrap this torus twice as before we get its covering space is:


which is also a tiling of R^2 using hexagon as you can imagine. Magically, the fundamental domain of the torus here is the hexagon.

Now to construct surfaces of constant negate curvature, we can use the idea of covering space and the Poincare model for the hyperbolic geometry.

Similarly, if we use the octagon as the fundamental domain to glue a double torus which looks like:


However, there would be a "mass of paper" at the intersections where all the loops meet as you can imagine. To clean the mass, the idea is to give this intersections a lot of negative curvature to fit in all the paper shown as follow:



This uses the Poincare model for the hyperbolic geometry as follow:


This is the universal covering space of the double torus.(This was done by Thurston when he had his oral^_^)



This picture is obtained from the first one by performing a hyperbolic isometry sending the left vertex of the large bottom a to the center of the disk.

If you have time, take a look at this cool video to help you understand!!
part 1: http://www.youtube.com/watch?v=AGLPbSMxSUM
part 2: http://www.youtube.com/watch?v=MKwAS5omW_w&NR=1

Wed June 19

Fix a geometry. Define a tile as a convex figure bounded by geodesic arcs.
Problem: Make a list of tiles where each angle is either pi/n or 2pi/n with symmetry.


HOMEWORK:
1. Anand will work out the problem for plane geometry (that is, in R^2).

2. Jingyu will work out the problem for spherical geometry (that is, in the sphere). (The answer is an interesting finite list and a not-so-interesting infinite list)


3. Matt may work out this problem  for hyperbolic geometry LATER

4. Robert: Get a tiling of the plane by hexangons. Consider the group of symmetries with no fixed points. Determine which surface is obtained.

5. Google Russell's paradox. (Wikipedia will be a good source). Read the contents. It will be discussed next week.

6 Jingyu will try to formalize the argument proving that in a simply connected space a tiling will "fit".

The picture below is a tiling of the Poincare disk by pentagons with right angles.

Homework for June 16

What is the number of  surface words  on an alphabet of 2n letters  x1, x2.. xn, X1, X2.. Xn?

Notes from 6/2/10

Thought I'd jot down some of the notes I took from last week's meeting:

We started with some more formal definitions.

Definition: A subset U of Rn is called open if for each element x in U, there exists some n-ball centered at x and contained in U. U is an open set.

Definition: A neighborhood of a point x in Rn is a subset X of Rn such that X contains an open set, U, containing x.

Exercise: The collection of open sets in Rn is:

1. Closed under arbitrary union

2. Closed under finite intersection

3. The null set is open, Rn is open

Any neighborhood X of x is also a neighborhood for all y in V for some open set V which also contains x.

4. Hausdorff Property of Rn: If x is a distinct point from y, then there exist disjoint neighborhoods of x and y.

Fact: A function f which maps Rn to Rm is continuous if and only if fro all x in Rn and any neighborhood V of f(x) in Rm there is a neighborhood, U, of x in Rn so that f(U) (the restriction of the function to the subset U) is a subset of the neighborhood V in Rm.

Definition: A Hausdorff topological space X is a set of points provided with a family of subsets called "open sets" which satisfy properties 1,2,3,4 listed above.

We say two spaces X1 and X2 are homeomorphic if there is a bijection carrying the open sets of X1 to the open sets of X2.

Definition: An n-manifold is a Hausdorff topological space so that each point has a neighborhood homeomorphic to a neighborhood of a point in Rn.

There is a deep result that n is well-defined because of invariance of domain.

Definition: A closed manifold is a manifold that can be covered by finitely many finite neighborhoods.

After these series of definitions, we moved into the problem of classification of surfaces and curves by words.

Given a surface with boundary given by a surface word (which is obtained from constructing the surface by gluing a polygon), there exists a bijection between the free homotopy classes of curves of S and the cyclic reduced words in the alphabet of the surface word.

Let us take our surface word to be aAbB. Some curves are a, aB, AB, etc.

We are also interested in reduced words, which do not have a letter and its associated gluing letter next to each other in any cyclic permutation of the word.

For example, abA is not reduced, since it has a and A next to each other in another cyclic permutation of the word (say Aab).

Definition: A primitive word is a word that is not a power of other words.

We see for instance that the word abab is not a primitive word, as it is (ab)^2.

We can then ask questions about minimal self-intersection. For instance, fixing a particular surface, what are the possible self-intersections of a word of length 1? We then created a small chart characterizing some of the possibilities for curves.

We can ask a series of similar questions. For example, instead of considering just self-intersections, we can take a pair of curves and consider how many total intersections occur.

We also consider the idea of multiwords. Multiwords have a length that is the sum of the length of their component words. For example, some multiwords of length 2 might be:

{ab}, {a,b}, {bA}, {a,a}.

As with other words, we can attempt to classify these multiwords too by self-intersections.

Later on we shifted to a discussion on isometries associated with hyperbolic geometry. We discussed two different models, the Poincare disk model and the upper half-plane model.

Please let me know if anyone has any corrections/things I should re-word to these notes.

Notes from First Meeting

Here are some of the things I jotted down from last week's meeting: (Feel free to add/correct things)

Topology

What are some of the motivations for the study of topology?
There are some problems that do not depend upon lengths, angles, etc. but on more basic properties (such as connectedness etc.)
Example: Seven Bridges of Konigsberg problem.

Notion of a "surface"

Division into two broad categories:
I.) Configurations of closed curves and geodesics on surfaces
II.) Self-transformations of surfaces and their effects on curves

Surfaces we study are characterized by whether they are orientable (do not contain Moebius strips), the number of boundary components, and the Euler characteristic.

We wish to find invariants that determine the structure of surfaces.

A minimal surface is one where the mean curvature is zero (the radii of curvature for positive and negative sections sums to zero)

An example in real-life is soap bubbles.

For topology, we consider two surfaces the same if we can construct a continuous bijection from one to the other. This continuous bijection between surfaces is called a homemorphism; the two surfaces are said to be homeomorphic.

Manifolds are surfaces which locally take the appearance of Euclidean space. (Tangency of planes at every point of the manifold.)

We can consider curves on these manifolds. While there are an infinite number of curves up to deformation, we are interested in minimal complexity curves, which minimize intersection points.

We call the number of handles in a surface the genus of the surface. For example, a torus (a donut) has genus 1.

We can consider the behavior of curves upon cutting part of our surface.

So broadly, we will study:
surfaces, closed curves of surfaces, classes of closed curves of surfaces, and minimal intersection/self-intersection

We can transition into geometry by endowing our surfaces with further structure, such as a metric (notion of distance).
By adding a metric, we can discuss geodesics, which are the shortest paths between two points on a manifold. Unlike their counterparts in the plane (lines), geodesics can self-intersect.

We can now consider configurations that minimize self-intersection and the distances based on our metric.

We can use geometric arguments to demonstrate that certain configurations are not realizable as geodesics for a given metric, while others can be realized for the metric. These arguments make use of geometric properties though (such as the curvature of the surface).
We studied a particular example (the pair of pants) that was a hyperbolic surface (negative curvature).

Finally, we discussed a method of constructing bounded surfaces from polygons. We take an n-alphabet and consider a 2n-gon. Each letter of the alphabet must appear once. Furthermore, each letter has an associated term. So the letter, a, also has an associated term, a'. To construct the surface, we glue together each letter with its associated term.
For example, we can have our alphabet be {a,b}. (So n=2.) Depending on how we label the sides of our polygon, we can obtain different shapes. So we could have different "surface words" that yield different surfaces. Some surface words are aba'b' and aa'bb'.

We can also consider a different type of word. Given a surface, characterized by some surface word), we can consider different curves on the surface. We can characterize these curves based upon what sides of the polygon (after it has been glued together) they pass through in what order. We can reduce these words by noting that no letter can appear next to its associated letter: a should never appear next to a' in these reduced words.

Generally, we obtain the following statement:

Every free homotopy class of orientable closed curves on a surface with boundary can be labeled by a unique (up to cyclic permutation) reduced word in the alphabet of the surface word.