April 22
- Jono
- Introduce us to Topological Entropy.
- Rob
- Tell us all about CW-complexes.
- Noah
- Introduce us to higher homotopy groups.
- Angela
- Talk to us about cubical homology.
April 29
- Sam
- Tell us about nonlinear dynamics.
- Elizabeth
- Introduce us to Knots.
- Markus
- Tell us about Space Filling Curves and all of their amazingness.
- Kyle
- Tell us about Fiber Bundles and the associated long exact sequence of homotopy groups.
Past Discussions
January 21
- Jonathan
- Euler's Theorem (section 1.1 of [A]).
- Markus
- Everything up to the preliminary version of the Classification Theorem in "Conway's ZIP Proof."
- Corey
- Lemmas 1, and 2 in "Conway's ZIP Proof."
- Noah
- Lemma 3 and The Classification Theorem in "Conway's ZIP Proof."
January 28
- Robert
- Definition of a topology. Compare the defintions in [A] and [M]. Discuss the examples in [A] and [M].
- Sam
- Definition of a basis and proofs of Lemmas 13.3 and 13.4
- Kyle
- Definition and examples of the order topology.
- Elizabeth
- Definition of product topology and proofs of Theorems 15.1 and 15.2.
- Angela
- Definition of subspace topology and Example 3. Also, show that if we consider the real numbers as a subset of the plane, the subspace topology is the usual one.
February 4
- Jono
- Theorem 17.4, being sure to explain exactly what the theorem is saying. Also, Theorem 17.10 and an example of a sequence in a non-Hausdorff space which has more than one limit point.
- Noah
- Give details of the pasting lemma (a.k.a. the glueing lemma) and an example of what can go wrong if the sets A and B are not closed in X.
- Corey
- Explain the main difference between the box and product topology. Give a proof of Theorem 19.6 and Example 2. Also discuss the function in Example 2 in the context where the product topology is used rather than the box topology.
- Markus
- Theorem 20.5. Be sure to explain exactly what it is that the theorem is getting at. Follow this up with a discussion of Examples 1 and 2 in § 21.
February 11
- Angela
- Topological Groups. Give a few examples and then discuss problem 7 on page 146 of [M].
- Elizabeth
- Orbit Spaces. Tell us what it means for a group to act on a topological space. Give some examples of familiar topological spaces presented as the orbit space of a group action. Also, you may want to consider problems 28, 33, and 34 in Chapter 4 of [A].
- Sam
- Define (locally) arc connected (see me for a definition) and show that (locally) arc connected implies (locally) path connected. Give an example (or two) to show that the converse is false.
- Kyle
- Define what it means for a space to be hyperconnected and ultraconnected (see me for a definition). Show that these ideas are independent, each imply connected, and that ultra connected implies path connected. What can you say about a real valued functions with a hyperconnected domain?
February 18
- Rob
- Discuss problems [M] §26 # 13 and §29 # 9.
- Noah
- Discuss problem [M] §28 # 7.
- Markus
- Find two non-homeomorphic spaces which have homeomorphic one-point compactifications. Also, discuss [A] Ch. 4 # 8.
- Jono
- Discuss [M] §30 # 18.
February 25
- Kyle
- Discuss [M] §51 # 3
- Angela
- Discuss [M] §52 # 7
- Elizabeth
- Discuss the path and path homotopy lifting lemmas ([M] Lemmas 54.1,2). Present the main steps of the proofs with out all the nitty gritty details.
- Sam
- Compute the fundamental group of the circle. Again, outline the key steps of the proof.
March 4
- Rob
- Discuss [M] §57 # 4.
- Noah
- Discuss [M] §58 # 9.
- Jono
- Present a solution to the following problem: Jono
- Markus
- Discuss [A] Ch. 5 # 27
March 18
- Markus and Elizabeth
- Present the statement and proof of the Seifert-van Kampen Theorem given in chapter 1 of Hatcher's book, pages 43-46.
- Kyle and Angela
- Present the computation of the fundamental group of the complement of a torus knot in R^3 (see Example 1.24 of Hatcher's book). As examples, compute the fundamental group of the comlements of the unknot and the trefoil. (The bits about mapping cylinders are in Chapter 0)
March 25
- Sam and Noah
- Digital Topology and Digital Image Processing. (See me for materials.)
- Angela
- Present [A] Theorem 5.13 and the examples that follow, and any others which you can come up with that you like. Be sure to discuss the Lens spaces.
- Jono
- Winding number and Cauchy Integral Formula.
April 1
- Rob, Elizabeth, and Sam
-
Present the Suplementary Exercises 1-5 on pages 499-500 of [M]. The point here is to show that when studying the relationship between covering spaces and the fundamental group, the hypotheses that the base space be path connected, locally path connected, and semi-locally simply connected are the right ones.
- Kyle and Markus
-
Give a proof of the Simplicial Approximation Theorem ([A] Theroem 6.7). Be sure to give us enough set up so that everything makes sense. In particular, it should be clear what simplicial complexes are, what the polyhedron of a simplicial complex is, and what it means for a map to be simplicial. Lots of pictures and examples would be wonderful.
April 8
- Angela and Noah
-
Define the groups E and G in [A] and show they are both isomorphic to the fundamental group. Compute some nice examples.
- Jono
- Define the homology groups of a simplicial complex and show us that the first homology group is the abelianization of the fundamental group. Compute some nice examples.
- Rob
- Explain how a simplicial map of simplicial complexes yields a map of simplicial homology. Explicitly compute the induced map on homology for the quotient map from the sphere to the projective plane described in [A] problem 27.
April 15
- Elizabeth and Kyle
- [A] problems 24-29.
- Sam and Markus
- [A] problems 30-32.