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.