I've been known to read/think/learn/talk about knots, braids, surgery theory, K(G,1) spaces, Heegaard-Floer theory, Combinatorial Heegaard-Floer theory, and 4-manifolds.
Abstract: I'll be giving an introduction to spectral sequences from the (co)homological standpoint, showing how they are useful for computing homology from a chain complex, especially in the case where our chain complex perhaps carries extra structure besides just a boundary map. Such cases arise with filtered chain complexes, double complexes, exact couples, etc. Once we go through the introduction and purpose of spectral sequences I hope to use the theory with double complexes to give a cool proof of the Snake Lemma, and I promise no diagram chasing will be necessary! I think this will be especially accessible to first year graduate students, since only knowledge of chain complexes and exact sequences are really necessary.
Abstract: John Berge wrote an unpublished paper on knots with surgeries yielding lens spaces in which he characterizes these knots in a genus 2 Heegaard splitting of S^3. He conjectured that all knots that have surgeries yielding lens spaces are of this characterization. He also gives a method for finding the exact lens space resulting from surgery of a particular knot, given that we know the homology class the knot represents in the Heegaard splitting. I'll give some introductory definitions/explanations on handlebody decompositions, Heegaard splittings, 0- and 1-bridge knots, and Dehn surgery before I start to talk about the theorems in the paper.
Abstract: I'll give a brief introduction to the braid group as the fundamental group of a configuration space (which normally is taken to be D^2 with n punctures), but will generalize it to braids on an arbitrary orientable surface M of genus g. Fadell and Neuwirth form a short exact sequence from the fibration of the n^th configuration space to the (n-1) configuration space, and the existence of sections leading to bijective maps from the pure braid group to a semi-direct product of free groups depends on the surface M, genus, and number of punctures n. From this we can see that the pure braid groups are residually torsion-free nilpotent.