August 30 |
Everyone |
Organizational Meeting |

September 6 |
Nick Kuhn (UVA) |
Differential Graded Hopf Algebras |

September 13 |
Carolyn Yarnall (UVA) |
Determining Slice Towers
I will begin by recalling the definition of the slice filtration along with some of its basic properties. Then I will discuss some computational methods for determining the slice towers of suspensions of the Eilenberg-MacLane spectrum associated to the constant Mackey functor for a cyclic p-group and provide some specific examples. |

September 20 |
Mike Hill (UVA) |
$[G]-E_\infty$ ring spectra
The talk will be in two parts: I. General discussion of "What does $E_\infty$ mean?" II. How does this apply in an equivariant context? |

September 27 |
Kate Ponto (UKY) |
Multiplicativity of fixed point invariants
For a fibration (with a connected base space) the Euler characteristic of the total space is the product of the Euler characteristics of the base and fiber. If the fibration satisfies restrictive additional hypotheses this extends to generalizations of the Euler characteristic such as the Lefschetz number and Nielsen number. Recently Mike Shulman and I have extended these results to the Reidemeister trace and eliminated many of the classical hypotheses. The key to our approach is to think of the Euler characteristic as an endomorphism rather an integer. With this change in perspective, the product of integers becomes a composite of functions. |

October 4 |
Sean Tilson (Wayne State) |
Power operations and the Kunneth spectral sequence
Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, the relevant algebraic operations are zero (other than the square). Despite the negative results we are able to use old computations of Steinberger's with our current work to compute operations in the homotopy of some relative smash products. |

October 11 |
Dustin Clausen (MIT) |
p-adic J-homomorphisms
I'll start by recalling the real J-homomorphism. Then I'll introduce its p-adic analogs, which more-or-less replace the use of real numbers in the real J-homomomrphism with p-adic numbers. Then I'll explain two results that show the analogy of these p-adic J-homomorphisms with the real J-homomorphism: first, they produce essentially the same elements of the stable stem (the classical "image of J" elements); and second, there is a "product formula" connecting all the J-homomorphisms (real and p-adic). This product formula is a generalization of the quadratic reciprocity law of number theory. |

October 18 |
Claire Tomesch (UChicago) |
Categories of Cohesion and 'Discretized' Model Categories
The purpose of this talk is to describe a notion of category of cohesion -- a concept of Lawvere introduced to describe 'relative discreteness' -- and its role in defining and understanding a model structure on Simpson-Tamsamani style versions of weak n-categories. The main payoff of this approach is an iterable construction of a model structure which takes into account the special role of 'discrete' objects. |

October 25 |
Vesna Stojanoska (MIT) |
A study of tmf cooperations
We combine three strategies to studying cooperations in connective topological modular forms: the Adams spectral sequence and its relation to Brown-Gitler modules following Mahowald's approach to cooperations in connective real K theory, Laures's theory of q-expansions of multi-variable modular forms, as well as level structure approximations. As a result, we obtain an algorithmic procedure for determining the structure of the smash product of tmf with itself.
This is a report on joint work in progress with Behrens, Ormsby, and Stapleton. |

October 31 |
Costumes! |
Halloween Party (6-9PM)
Come prepared to hand out candy, carve pumpkins, and socialize. Prizes / good karma awarded for coming in costume! |

November 1 |
Emily Riehl (Harvard) |
Homotopy coherent adjunctions of quasi-categories
We show that an adjoint functor between quasi-categories may be extended to a simplicially enriched functor whose domain is an explicitly presented "homotopy coherent adjunction". This enriched functor encapsulates both the coherent monad and the coherent comonad generated by the adjunction. Furthermore, because its domain is cofibrant, this data can be used to construct explicit quasi-categories of (co)algebras. Given time, we describe how our techniques can also be used to re-prove many of the foundational results about the category theory of quasi-categories. This is joint work with Dominic Verity. |

November 8 |
John Lind (JHU) |
Higher geometry and algebraic K-theory
A cohomology theory E is particularly useful when we can understand its cocycles \(E^*(X)\) in terms of geometric objects associated to the space X. A basic example is the description of topological K-theory in terms of complex vector bundles. I will give an analogous interpretation of cocycles for \(E=K(R),\) the algebraic K-theory of an associative ring spectrum, in terms of bundles of R-modules over X.
The main technical development is a symmetric monoidal model for the category of spaces in which \(A_{\infty}\) spaces are strict monoids. |

November 14 |
Kristen Mazur (UVA) |
A G-Symmetric Monoidal Structure on the Category of G-Mackey Functors
I will describe a new G-symmetric monoidal structure on the category of G-Mackey functors. The current G-symmetric monoidal structure in incredibly complicated and is almost impossible to compute. Thus, I build a new structure that has the following advantages. First, it is both easy to define and easy to compute. Second, under this new structure we can show that Tambara functors are the G-commutative algebras. Finally, it is known that Tambara functors exist as the zeroeth stable homotopy groups of commutative G-ring spectra but there are very few examples of them available. Therefore, the new G-symmetric monoidal structure will provide a means of creating examples of Tambara functors. |

November 22 |
Thanksgiving |
No Seminar |

December 6 |
Nathaniel Stapleton (MIT) |
An Application of Transchromatic Generalized Character Theory
The generalized character theory of Hopkins, Kuhn, and Ravenel has proved to be a very useful tool in the study of Morava E_n. In this talk, I will outline a compact construction of the transchromatic generalized character maps. The Morava E-theory of cyclic groups and symmetric groups have well known algebro-geometric interpretations. Using the relationship between the character maps and the transfer maps for Morava E-theory, I will provide algebro-geometric interpretations of the cohomology of some finite groups other than symmetric groups and cyclic groups. |