In modern terms this formulation leads to many questions: can we define volume in such a way? If not, what are the obstructions? What about in higher dimensions? What if instead we looked at oriented polygons? In my talk we will explore the classical solutions to these problems, some modern reformulations, and the new directions that these reformulations lead us.

Elementary features of these spaces as well as their connections to spaces of knots, links, and homotopy groups of spheres will be addressed. These topics are based on joint work with R. Budney as well as J. Berrick, Y. Wong and J. Wu.

This conjecture, useful for studying the image of J and hence the stable homotopy groups of spheres, was first proved by Friedlander following a sketch of Quillen. One can speciously summarize the argument as follows: reduce to the case where X is a variety over a finite field, and then use that Ψ^p identifies with pullback along the Frobenius map X → X.

The talk will be devoted to explaining this proof, centering on etale homotopy theory, an algebraic-geometric way of capturing much of the homotopy type of complex algebraic varieties. Basically no algebraic geometry will be assumed, but the price may be a lot of "moral" arguments based on topological intuition. I'll try to keep the statements precise, though.

I will explain how Goodwillie's cutting method enables us to understand the homotopy type of this space of emeddings. I will then prove that the rational homology of that space is actually an invariant of the rational homotopy type of M. The proof is based on Kontsevich's theorem on the formality of the little cube operad and Arone's description of the layers of Weiss' orthogonal tower for the space of embeddings. This is a joint work with Greg Arone and Ismar Volic.

In this talk, I will present the simplicial resolution of a space by wedges of spheres. We will look at Stover's original construction, and then at the Dwyer-Kan-Stover E² model structure on pointed simplicial spaces, where a resolution by spheres is a cofibrant replacement. Some applications will be discussed.

Bousfield later expanded the theory, but we won't get to that.

Lazard's theorem implies that the fibered category that classifies formal group laws and isomorphisms between them is a prestack over affine schemes in the fpqc topology. The associated stack is known as the moduli stack of formal groups, M_fg. There is an equivalence of categories between quasi-coherent sheaves over M_fg and comodules over an appropriate Hopf algebroid. Sheaf cohomology over this stack can be identified with the E₂-term of the Adams-Novikov spectral sequence.

Such observations suggest that the perspective of stacks is of interest in this area. The purpose of this talk will be to introduce basic concepts of stacks, formal Lie varieties over schemes, and formal groups. There will be some discussion about the properties and geometry of M_fg, and allusions to a stack/sheaf-theoretic proof of LEFT.

Sources include works by Goerss, Hopkins, and Vistoli.

The goal of this talk is to give an overview of what sorts of cobordism categories such results exist for, of what these results look like, and of what sorts of things go into these results.

In this talk, I'll introduce these concepts and summarize various recent works of Etingof, Gelaki, Nikschych, and Ostrik, which deal with graded fusion categories, and which allow us to completely classify fusion categories of Frobenius-Perron dimension pⁿ, pq, pqr, and pq². Along the way we'll see a 3-category, a classifying space, some split orthogonal groups, and some applications of group cohomology.

Disclaimer: So far as we can tell, these notions are essentially disjoint from those discussed last week by Matt Gelvin, despite a coincidence of buzzwords.

Instead, we will discuss the work of Broto, Levi, and Oliver, replacing "fusion data" with another algebraic structure, called the p-centric linking system. The advantage of this approach is that it is possible to prove, hopefully within an hour, that the resulting conjecture is true. The disadvantage is that p-centric linking systems are much harder to work with in their own right than fusion systems, and it is in this sense that the resulting conjecture is weaker.

Be that as it may, some result is better than none, so that's what we'll talk about. What's more, the talk will keep in mind the possibility that someone might be foolish enough to try to give a talk about the stronger MP conjecture at some future date. If such a talk were to occur, the current talk might be considered as a sort of introduction.

There is a generalization of the universal coefficient spectral sequence or the E-Adams spectral sequence resulting in a Goerss-Hopkins spectral sequence computing the homotopy of a such a mapping space.  The goal is to identify the E₂ term of this spectral sequence as the André-Quillen cohomology of a simplicial algebra over a simplicial operad.  In order to do this, we will talk about the category of simplicial algebras over a simplicial operad and how to make sense of derivations and thus the right derived functors of derivations.

In this talk, I will present a paper of C.T.C. Wall from 1965 exploring such questions. We will look for necessary and sufficient algebraic conditions corresponding to different kinds of finiteness for a CW-complex. In particular, there are conditions to know if a complex X is finitely dominated. When that happens, the obstruction to finiteness turns out to be a class in the reduced projective class group of the integral group ring of $\pi_1(X)$.