State Hall 134

Wayne State University

Detroit, Michigan, USA

Speaker: **Kristen Mazur**, Lafayette College

Time: Saturday October 5, 10:00-11:00

Title: Additional structure on the category of Mackey functors

Abstract: The stable homotopy groups of a G-spectrum are Mackey functors, and the zeroth stable homotopy group of a commutative G-ring spectrum has the extra structure of a Tambara functor. However, while Mackey functors and Tambara functors make frequent appearances in equivariant stable homotopy theory, much of their underlying algebra remains mysterious. I will discuss a new structure on the category of Mackey functors such that Tambara functors are commutative algebra-like objects. Moreover, the advantage to this new structure is that it is concrete and computable.

Speaker: **Agnes Beaudry**, University of Chicago

Time: Saturday October 5, 11:30-12:30

Title: An algebraic finite resolution for K(2)-local computations at the prime 2

Abstract: Using their finite resolution of the K(2)-local sphere, Goerss, Henn, Mahowald and Rezk have solved a number of open problems at the prime 3. Among other applications, they have confirmed the chromatic splitting conjecture and computed the Brown-Comentez dual of L_K(2) S^0. In this talk, I present the algebraic analogue of the finite resolution for the prime 2. I explain how we can use the geometry of elliptic curves to simplify computations. I also explain how preliminary computations for the mod-2 Moore spectrum lead us to expect a different decomposition of L_1 L_K(2) S^0 than that predicted by the chromatic splitting conjecture.

Speaker: **Craig Westerland**, University of Minnesota

Time: Saturday October 5, 2:30-3:30

Title: Distributional problems in arithmetic and the homology of moduli spaces

Abstract: I'll review a number of open conjectures in number theory due to Linnik, Malle, Cohen-Lenstra, and others on the distribution of number fields with specified properties (e.g. Galois group, class group, discriminant). Reformulating these as questions over function fields in positive characteristic, they can be posed as problems about the enumeration of points (defined over finite fields) in certain moduli spaces. This may then be translated into a question on the homology of the complex points of these moduli spaces. In joint work with Ellenberg and Venkatesh, we've resolved one of these conjectures -- the Cohen-Lenstra heuristics for function fields. The solution is closely related to classical results in homotopy theory, namely the approximation theorem for iterated loop spaces, and homological stability for configuration spaces.

Speaker: **Marcy Robertson**, Western University

Time: Saturday October 5, 4:00-5:00

Title: Schematic homotopy types of operads

Abstract: The rational homotopy type X_Q of an arbitrary space X has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space X are not accessible through the space X_Q. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces, Toen introduced the notion of a pointed schematic homotopy type (X \times k)^sch over a field k.

In his recent study of the pro-nilpotent Grothendieck-Teichmuller group via operads, Fresse makes use of the rational homotopy type of the little 2-disks operad E_2. As a first step in the extension of Fresse's program to the pro-algebraic case we discuss the existence of a schematization of the little 2-disks operad.

Speaker: **John Klein**, Wayne State University

Time: Sunday October 6, 9:30-10:30

Title: Disjunction and intersection

Abstract: Suppose N is a compact manifold that is equipped with a finite collection Q_1, ..., Q_j of pairwise disjoint submanifolds in N. Given a map f: P --> N, where P is compact manifold, I shall consider the problem of deforming f off of each of the Q_i simultaneously. The purpose of this talk is to explain when this is possible and what the obstructions are when it's not.

Speaker: **David Gepner**, Purdue University

time: Sunday October 6, 11:00-12:00

Title: To be announced

Abstract: To be announced

Email: isaksen at math.wayne.edu