Possible topics for the final project

• Nov 1: tell me what you're thinking about doing.
• Dec 6: draft due.
• Dec 13: final revision due.

• Algebraic topology and complex analysis. Homotopy invariance of path integrals and Cauchy's theorem. Computing degrees, homotopy theoretic proof of fundemental theorem of algebra.
• Homology/cohomology. Although we're not covering it in our class, you could write about it in your final project. It comes in many (related) flavors:
  • Simplicial or polygonal homology - the most intuitive version - based on triangulations.
  • Singular homology - the most functorial - a little non-intuitive.
  • Cellular homology - formulated for CW complexes - we'll talk about these later.
  • de Rham cohomology - a differential geometric version using differential forms.
  • Sheaf cohomology - measures cohomology by the failure of local information to give global information - uses "sheaves".
• Jordon curve theorem. Let C be a closed simple (nonintersecting) curve in the plane. Theorem: the complement of the curve consists of two components, the "inside" and the "outside". Surprisingly tricky to prove. A related problem is "invariance of domain" - is R^n homeomorphic to R^m if n is not equal to m?
• Handlebodies. We built surfaces using polygonal identification. There is another way to build surfaces and manifolds, by "attaching handles". This perspective is closely linked to "surgery theory".
• Morse theory. A beautiful theory that decomposes manifolds using critical points of functions on them. Milnor wrote a very nice account of this. Related to "handlebodies".
• Higher homotopy groups. The fundemental group is the study of maps of circles into your space. The higher homotopy groups study maps of n-spheres into your space. They are mysterious. Especially the higher homotopy groups of spheres.
• Chain complexes, and chain homotopy. There is an algebraic version of homotopy where you replace spaces with chain complexes, and the notion of homotopy becomes chain homotopy.
• Simplicial sets. Another combinatorial/algebraic way to model homotopy is the notion of a simplicial set. It is a triangulation, without the triangles!
• Poincare homology sphere. Suppose that an n-manifold X has the same homology as the n-sphere. Is X a sphere? Poincare showed this was false by producing his "exotic homology sphere". He conjectured that if you additionally assumed that the fundemental group was trivial, then X was a sphere. This famous problem is known as the poincare conjecture, has only recently been solved for n = 3. The cases of large n were handled by Smale.
• Galois theory and covering spaces. Galois theory feels like covering spaces, but the analogy goes farther than that. Two ideas
  • Read Andreas Dress's new proof of the fundemental theorem of Galois theory. Very slick.
  • Learn about the algebraic version of covering spaces, so called etale maps. This requires some commutative algebra background.
• Branched coverings. In complex analysis it is common to study things that are almost covering spaces, so called branched coverings. Riemann surfaces arise as branched coverings of projective space.
• The Gauss-Bonnet theorem. Learn about scaler curvature, and the Gauss-Bonnet theorem - which Adam mentioned in his Euler characteristic talk. The Euler characteristic of a surface is given by the total curvature of the surface.
• Hyperbolic space. Higher genus surfaces arise as quotients of group actions on negatively curved space, so-called hyperbolic space. Variants of this topic could involve learning about Teichmuller space, or Fuchsian groups.
• Classifying spaces, principle fibrations. Galois covering spaces are examples of principle fibrations - these are bundles of discrete groups that lie over the space. By replacing the discrete groups with topological groupsp, you get the more general notion of a principle fiber bundle. A related topic is that of the classifying space. Another related topic is the notion of a VECTOR BUNDLE.
• Universal covers for bad spaces. We will prove that universal covers exist for spaces which are "semi-locally simply connected". What if you have an arbitrary weird topological space. Daniel Biss wrote an amusing article that explains what you do.
• Homology/cohomology of groups. There is a way to take the homology and cohomology of a group. This records important algebraic information about the group.
• Seifert/Van Kampen using covering spaces. There apparently is a slick proof of the Seifert/Van Kampen theorem using covering spaces.
• Categories, and colimits. Learn about the abstract notions of categories and functors. The universal property expressed in the Seifert/Van Kampen theorem is an instance of the notion of a "colimit" of a diagram.
• Fundamental groupoid. This also ties into categories. The idea is rather than pick a basepoint, and define the fundamental group, use all points simultaneously. These pesky basepoint issues dissolve away, and the Seifert-Van Kampen theorem attains a cleaner statement.
• Lens spaces. By our classification of surfaces, and our computation of their homotopy groups, we saw that two surfaces are homotopy equivalent if and only if they are homeomorphic. Is this true for arbitrary manifolds? The answer is no, and is proved using Lens spaces and Whitehead torsion.
• Cobordism. A looser way to classify manifolds is up to cobordism: two manifolds are cobordant if they form the boundary of a manifold of one dimension higher.
• Kuratowski's Theorem. Which graphs can be embedded in the plane? The complete answer is known.
• Knot theory. There are a lot of possibilities here - fundemental groups of know complements, knot invariants, etc.
• Arithmetic groups and trees. Bass-Serre theory gives a presentation for any group acting on a tree. These examples occur in the study of arithmetic groups. Serre wrote a beautiful book on the subject - the english translation is entitled "Trees".