January 22 |
Charmaine Sia (Harvard) |
Structures on forms of K-theory |

January 29 |
Greg Arone (UVA) |
A branching rule for Lie(n) and the partition poset |

February 5 |
Lennart Meier (UVA) |
The homotopy theory of homotopy theories |

February 12 |
Greg Arone (UVA) |
The topological Tverberg conjecture, again.
The topological Tverberg conjecture has been a prominent open question in combinatorial topology. Recently Florian Frick announced a counterexample. We will review the conjecture and try to understand the counterexample. |

February 19 |
Viktoriya Ozornova (Bremen) |
Fibrancy of partial model categories |

February 26 |
Emanuele Dotto (MIT) |
Equivariant calculus of functors
Let $G$ be a finite group. I will define "$J$-excision" of functors on pointed $G$-spaces, for every finite $G$-set $J$. When $J$ is the trivial $G$-set with $n$-elements we recover Goodwillie's definition of $n$-excision. When $J=G$ we recover Blumberg's notion of equivariant excision. There are $J$-excisive approximations of homotopy functors which fit together into a "Taylor tree", as well as "$J$-homogeneous" approximations classified by suitably equivariant spectra. We will finish with a curious splitting of the "horizontal layers" of the tree of a homotopy functor, which recovers the tom Dieck splitting in the case of the identity functor. |

March 5 |
Mona Merling (JHU) |
* CANCELED DUE TO SNOW * Equivariant algebraic K-theory
The first definitions of equivariant algebraic K-theory were given in the early 1980�s by Fiedorowicz, Hauschild and May, and by Dress and Kuku; however these early space-level definitions only allowed trivial action on the input ring or category. Equivariant infinite loop space theory allows us to define spectrum level generalizations of the early definitions: we can encode a G-action (not necessarily trivial) on the input as a genuine G-spectrum.
I will discuss some of the subtleties involved in turning a ring or space with G-action into the right input for equivariant algebraic K-theory or A-theory, and some of the properties of the resulting equivariant algebraic K-theory G-spectrum. For example, our construction recovers as particular cases equivariant topological real and complex K-theory, Atiyah�s Real K-theory and statements previously formulated in terms of naive G-spectra for Galois extensions.
I will also briefly discuss recent developments in equivariant infinite loop space theory from joint work with Guillou, May and Osorno (e.g., multiplicative structures) that should have long-range applications to equivariant algebraic K-theory. |

March 12 |
No Seminar |
Spring Break |

March 19 |
Vitaly Lorman (JHU) |
Computing with Real Johnson-Wilson Theories
Complex cobordism and its relatives, the Johnson-Wilson theories, $E(n)$, carry an action of $C_2$ by complex conjugation. Taking homotopy fixed points of the latter yields Real Johnson-Wilson theories, $ER(n)$. These can be seen as generalizations of real K-theory and are similarly amenable to computations. We will outline their properties, describe a generalization of the $\eta$-fibration, and discuss recent computations of the $ER(n)$-cohomology of some well-known spaces, including $CP^\infty$. |

March 26 |
Agnes Beaudry (UChicago) |
The Chromatic Splitting Conjecture at $n=p=2$
Understanding the homotopy groups of the sphere spectrum $S$ is one of the great challenges of homotopy theory. The ring $\pi_*S$ is extremely complex; there is no hope of computing it completely. However, it carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that, after localizing at a prime, the sphere spectrum is filtered by ``simpler" spectra called the chromatic layers, which we denote by $L_nS$. How these layers interact with each other is a mystery. A conjecture of Hopkins, the chromatic splitting conjecture, suggests an answer to the problem. The difficulty of the problem grows fast with $n$, and varies with the choice of prime at which we localize. The chromatic splitting conjecture is known to hold in its strongest form at all primes $p$ when $n=1$, and at all odd primes when $n=2$. However, it does not hold when $p=n=2$. In this talk, I explain why it fails in this case. |

April 2 |
Michael Andrews |
Non-nilpotent elements in motivic homotopy theory
Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP-homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and while $\eta:S^{1,1,}\to S^{0,0}$ induces zero on BP-homology, it is non-nilpotent. Work with Haynes Miller has led to a calculation of $\eta^{-1}\pi_{*,*}(S^{0,0})$, proving a conjecture of Guillou and Isaksen.
I�ll introduce the motivic homotopy category and the motivic Adams-Novikov spectral sequence before describing this theorem. Then I�ll show that there are more periodicity operators in chromatic motivic homotopy theory than in the classical story. In particular, I will describe a new non-nilpotent self map. |

April 16 |
Irina Bobkova (Rochester) |
Resolutions of the $K(2)$-local sphere spectrum
Computing the stable homotopy groups of spheres is one of the main problems in algebraic topology. In this talk, I will introduce chromatic homotopy theory which describes the homotopy of the $p$-local sphere spectrum $S$ through a family of localizations $L_{K(n)}S$ with respect to Morava K-theories $K(n)$. Considerable information here can be derived from the action of the Morava stabilizer group on the Lubin-Tate theory. Then I will specialize to the $K(2)$-local category and talk about finite resolutions of the $K(2)$-local sphere spectrum by a sequence of spectra. |

April 23 |
Andrew Putman (Rice) |
Representation-theoretic patterns in the stable cohomology of congruence subgroups
I will explain how to use representation-theoretic tools arising from the mathematics surrounding the Artinian conjecture to understand patterns in the stable cohomology of congruence subgroups (for simplicity, mostly congruence subgroups of SL(n,Z)). This is joint work with Steven Sam. |