Normal forms have been used since Poincare. The problem of converting an element in a Lie algebra into its normal form can be a difficult calculation. In joint work with Rick Churchill we have applied the method of spectral sequences to this problem. The talk will be both an introduction to normal forms and (to a lesser degree) an introduction to spectral sequences. All are welcome Thursday at 4PM in 148 Hawthorn, after which there is traditionally a group going to a local restaurant for dinner.
A campus map (including GPS coordinates) is available here, and more information about the college is available there, including driving directions. On this campus map, 15 is the cafeteria building, 9 is the LRC, and 10 is the Hawthorn Building. Please try to park in the large student lot next to Hawthorn, labeled F on the map.
Central in toric geometry and topology are several important spaces which include moment-angle complexes, the Davis-Januszkiewicz space and toric manifolds. In any complex-oriented cohomology theory, the cohomology rings of many of these spaces have elegant descriptions in terms of the underlying combinatorics. For KO-theory however the situation is more complex. Even so, a surprising amount of the structure does survive from the complex-oriented case. A report of joint work in progress with: Luis Astey, Martin Bendersky, Fred Cohen, Don Davis, Matthias Franz, Sam Gitler, Mark Mahowald, Nigel Ray and Reg Wood.
The (stable) chromatic spectral sequence has had a significant impact on our understanding of the stable homotopy groups of the spheres. I will talk about preliminary attempts to construct an unstable version. I will describe some of the questions that arise in the unstable world (e.g. an unstable version of the Morava stabilizer algebra) and a chromatic interpretation of the Hopf invariant.
Quillen's axioms for homotopical algebra generalize some of the standard notions and constructions from algebraic topology to other categories. Methods of homotopy theory in the category of simplicial sets provided a major motivating example. One of our goals is to illustrate Quillen's ideas, at a simple level, in combinatorial categories related to the category of simplicial sets. As an example, I will sketch a Quillen model structure on a category of directed graphs, and describe the resulting homotopical algebra. It seems to fit well with traditional concepts in graph theory, and with zeta functions and spectra of graphs from algebraic graph theory and symbolic dynamics. Joint work with Aristide Tsemo.
A $\Pi$-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object $\Lambda$, one may ask if it can be realized topologically: Is there a space $X$ such that $\pi_*X$ is isomorphic to $\Lambda$ as a $\Pi$-algebra, and if so, can we classify them? Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a $\Pi$-algebra, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of $\Pi$-algebras. What do these groups look like, and can we compute them? We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated $\Pi$-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage and 3-stage homotopy types, and compare them to what is known from other approaches
Randy McCarthy and I developed a version of Tom Goodwillie's calculus of homotopy functors for functors from pointed categories with finite coproducts to abelian categories using cotriples and basic tools from homological algebra. Our construction requires that the domain category be "based", i.e., it has an object that is both initial and final. I will discuss recent work to develop, for the purpose of analyzing some functors of unbased categories, a cotriple version of the calculus of functors for functors whose domain categories are not based. This is joint work with Kristine Bauer and Randy McCarthy.
Orbifolds (originally introduced as V-manifolds by Satake) are paracompact spaces with an atlas which exhibits the local structure as the orbit space of the action of a finite group on Euclidean space. An orbifold is called representable if it can be presented as the orbit space of a manifold by the action of a compact Lie group. A large class of orbifolds, including all orbifolds for which the local groups act effectively, is known to be representable. In this talk we will discuss what one needs to do in order to generalize equivariant homotopy theory to obtain orbifold homotopy invariants. Orbifolds can be represented by smooth \'etale groupoids with a proper diagonal, where two such \'etale groupoids represent the same orbifold if and only if they are Morita equivalent. This presentation gives rise to a notion of orbifold morphism which is the right one to study orbifold homotopy theory. For this reason these maps have also been called "good maps". An orbifold is representable precisely when it can be represented by a smooth translation groupoid, which is unique up to Morita equivalence. Good maps between representable orbifolds can be represented as spans of equivariant maps between translation groupoids, where the left-hand side of the span is a Morita equivalence. This means that any generalization of constructions for G-spaces to orbifolds needs to be invariant under Morita equivalence. As an application we will discuss orbifold Bredon cohomology. This is joint work with Laura Scull (Fort Lewis College).
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