Aspects of Orbifold Cohomology and K-theory:In this lecture we will describe invariants of orbifolds which extend the well-known formulas for orbifold Euler characteristics. This involves a suitable definition of orbifold K-theory, its comparison with orbifold cohomology and a decomposition theorem. We also define a version which incorporates twisting by discrete torsion. A number of examples will be discussed, including twisted symmetric products.

Mock Reflection Groups:This is a report on some joint work with Tadeusz Januszkiewicz and Rick Scott. It turns out that there is a rich class of examples of nonpositively curved closed manifolds which are tiled by either permutohedra or associahedra. Such examples arise as certain blow-ups of $\bf RP^n$ of projective hyperplane arrangements associated to finite reflection groups. The universal covers of such examples yield tilings of $\bf R^n$ by permutohedra or associahedra. The group of symmetries $A$ of such a tiling of the universal cover is generated by involutions, but in general it is not a reflection group, rather it is a "mock reflection group". I will explain these examples, give a presentation for the groups $A$ and discuss some of their properties.

The Best of all Possible Maps is Sometimes Not Good Enough:I will talk on a example due to Ontaneda, Raghunathan and myself of a diffeomorphism between closed negatively curved Riemannian manifolds such that the homotopic harmonic map is not univalent.

The Integer Valued SU(3) Casson Invariant for Homology 3-Spheres:This talk will discuss a recently defined SU(3) Casson invariant, defined in joint work with Hans Boden and Paul Kirk. It involves a different correction term than the earlier gauge theory definition (joint with Boden). The new correction term is easier to compute and, best of all, is "nonzero only when absolutely necessary." I'll discuss the invariant's behavior under connected sums and surgeries on torus knots.

The Splitting Problem for Topological Hochschild Homology:Topological Hochschild homology, as its name suggests, is a topological version of an algebraic invariant of rings. It is interesting as an approximation of algebraic K-theory, through a variant of the Dennis trace map. But the procedure of topological ``fattening'' affects the construction very differently for different rings---I plan to discuss the known cases and mention an application to the structure of Eilenberg-Mac Lane spectra.

Cosi fan tutte

It is a case of mistaken identity - or is it? - when two officers make a bet on the faithfulness of their lovers. With the help of a cynical bachelor and a feisty lady's maid, they disguise themselves and attempt to seduce each other's sweethearts. What follows is Mozart's most mischievous comic opera, Cosi fan tutte - with a plot that caused a scandal in its day. It makes audiences marvel at the twists and turns of love; and charms, as ever, with delicious music, general good fun, and an unexpected happy ending. Originally written as a commentary on women in eighteenth-century Vienna, Cosi endures by proving that, in matters of love, men and women "All Act the Same."

Grope Cobordism in Dimensions 3 and 4:We explain how the Goussarov-Habiro approach to finite type nvariants of knots is related to the geometric notion of a grope cobordism in 3-space. This leads to natural refinements of the notion of finite type which can be related via grope cobordism in 4-space to the Cochran-Orr-Teichner filtration of the knot concordance group.