Schedule

Computing differentials in the Adams spectral sequence

I will review classical methods computing differentials in the Adams spectral sequence, and then discuss some recent progress in joint work with Weinan Lin and Guozhen Wang.

Higher Lie Theory

L_\infty algebras, i.e. Lie algebras up to homotopy coherent homotopy, appear in a variety of contexts, including string theory and deformation theory. Over the last several decades, the outlines of a Lie theory for such objects has appeared in work of Sullivan, Getzler, Henriques and others.  In this talk, we'll present joint work with Chris Rogers (UNR) establishing Lie's second and third theorems for L_\infty-algebras, with a focus on Lie's third theorem as an interplay of homotopical algebra, differential topology and Lie theory.

Max Johnson - Quantitative Estimates on DHS Nilpotence

One of the key results of early chromatic homotopy is the Devinatz-Hopkins-Smith nilpotence theorem. Using a bit of modern technology but largely following their original approach, it is possible to provide numerical bounds on which orders of elements in the kernel of the MU-Hurewicz must be 0. I will provide a sketch of work-in-progress on producing such an explicit bound.

Jordan Benson - The C-Motivic Adams 5-Line

One input to the methods of Isaksen-Wang-Xu is knowledge of the C-motivic Adams spectral sequence. The E_2 page of this sequence is known to be a "tau-adjoined" version of the classical E_2 page through the 3-line, but low-dimensional calculations reveal the existence of tau^n-torsion elements starting in the 4-line. Using the techniques of Burklund-Xu, we compute the entire 4- and 5-lines of this E_2 page as a module over F_2[tau]. We also show that the first tau^2-torsion classes appear in the 7-line.

Shangjie Zhang - Equivariant $v_1$-self maps and the generalized Bredon-Landweber theorem

Let $G$ be a cyclic $p$-group or generalized quaternionic group, $X \in \pi_0 S_G$ be a virtual $G$-set, and $V$ be a fixed-point free complex $G$-representation. Under conditions depending only on the sizes of $G$, $X$, and $V$, we construct a self-map $\Sigma^V C(X)_{(p)}\rightarrow C(X)_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These equivariant $v_1$-self maps are transchromatic, in the sense that the telescope $C(X)_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points. As an application, we give a $C_{2^n}$-equivariant generalization of the Bredon-Landweber theorem.

Daniel Epelbaum - Algorithmic decidability of lifting-extension problems

In principle it would be useful to have an algorithm that does the following: on input a map from a cofibration to a fibration (a commutative square), it decides whether there is a lift of the map on the bottom that extends the map on the top. It is known that in general such an algorithm cannot exist. We discuss this result and investigate some restricted cases in which the existence of lifting-extensions can be algorithmically decided.

The motivic Freudenthal suspension theorem and beyond

The motivic Freudenthal suspension theorem gives information about the stable range for P^1-suspension in motivic homotopy theory.  I will describe this theorem and some improvements that give information about the P^1-meta-stable range after localization at an odd prime.  This is based on joint work, some in progress, with T. Bachmann and M.J. Hopkins.