Mayday 2019

Talks

All talks will be in Kent 120 (map)

Friday 10/4

4:00pm Kirsten Wickelgren (Duke)

An arithmetic count of rational plane curves

There are finitely many degree \(d\) rational plane curves passing through \(3d-1\) points, and over the complex numbers, this number is independent of (generically) chosen points. For example, there are 12 degree 3 rational curves through 8 points, one conic passing through 5, and one line passing through 2. Over the real numbers, one can obtain a fixed number by weighting real rational curves by their Welschinger invariant, and work of Jake Solomon identifies this invariant with a local degree. It is a feature of \(\mathbb{A}^1\)-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. We develop and compute an \(\mathbb{A}^1\)-degree, following Morel, of the evaluation map on Kontsevich moduli space to obtain an arithmetic count of rational plane curves, which is valid for any field \(k\) of characteristic not 2 or 3. This shows independence of the count on the choice of generically chosen points with fixed residue fields, strengthening a count of Marc Levine. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.

Saturday 10/5

9:00am Akhil Mathew

The Beilinson fiber square and topological cyclic homology

A theorem of Goodwillie identifies relative \(K\)-theory and cyclic homology for nilpotent thickenings of \(\mathbb{Q}\)-algebras. The extension of this result to arbitrary rings (or connective ring spectra) given by Dundas-Goodwillie-McCarthy relies on topological cyclic homology. More recently, Beilinson gave a version of Goodwillie's original result in a specific \(p\)-adic context. I will explain how recent advantages in topological cyclic homology reprove and refine Beilinson's result, and describe some applications to \(p\)-adic deformations of cycle classes. Joint with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus.

10:30am Angélica Osorno

Equivariant infinite loop space machines (or "Everything I learned from Peter")

An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalizations of the operadic approach and the Gamma-space approach respectively. In this talk I will report on joint work with Bert Guillou, Peter May and Mona Merling on adapting these machines to work multiplicatively and on understanding their categorical input.

1:30pm Charles Rezk

To be announced

2:45pm Agnès Beaudry

Invertibility, \(K(n)\)-locally

In this talk, I will discuss some of the tools that have been used recently to compute Picard groups of \(K(n)\)-local \(E_n\)-modules in \(G\)-spectra. I will also describe various phenomena that have been observed in computations. Here, \(E_n\) is Morava \(E\)-theory at height \(n\) and \(G\) is a finite subgroup of the Morava stabilizer group, acting on \(E\) via the action obtained from deformation theory. I will then explain how some of these ideas are being adapted to study the Picard group of the \(K(2)\)-local category of spectra.

4:00pm Tyler Lawson

Calculations in Hopf rings

In this talk I will discuss work of May and coauthors on power operations for \(E_\infty\) ring spaces. In particular, I will discuss techniques for calculating the effect of these operations for the complex cobordism spectrum \(MU\).

Sunday 10/6

9:00am Inna Zakharevich

The Dehn complex: scissors congruence, \(K\)-theory, and regulators

Hilbert's third problem asks: do there exist two polyhedra with the same volume which are not scissors congruent? In other words, if \(P\) and \(Q\) are polyhedra with the same volume, is it always possible to write finite unions \(P = \bigcup_{i} P_i\) and \(Q = \bigcup_{i} Q_i\) such that the \(P\)'s and \(Q\)'s intersect only on the boundaries and such that \(P_i \cong Q_i\)? In 1901 Dehn answered this question in the negative by constructing a second scissors congruence invariant now called the ``Dehn invariant,'' and showing that a cube and a regular tetrahedron never have equal Dehn invariants, regardless of their volumes. We can then restate Hilbert's third problem: do the volume and Dehn invariant separate the scissors congruence classes? In 1965 Sydler showed that the answer is yes; in 1968 Jessen showed that this result extends to dimension 4, and in 1982 Dupont and Sah constructed analogs of such results in spherical and hyperbolic geometries. However, the problem remains open past dimension 4. By iterating Dehn invariants Goncharov constructed a chain complex, and conjectured that the homology of this chain complex is related to certain graded portions of the algebraic \(K\)-theory of the complex numbers, with the volume appearing as a regulator. In joint work with Jonathan Campbell, we have constructed a new analysis of this chain complex which illuminates the connection between the Dehn complex and algebraic \(K\)-theory, and which opens new routes for extending Dehn's results to higher dimensions. In this talk we will discuss this construction and its connections to both algebraic and Hermitian \(K\)-theory, and discuss the new avenues of attack that this presents for the generalized Hilbert's third problem.

10:30am Mike Hill

\(N_{\infty}\) ring spaces

In equivariant homotopy theory, there are many generalizations of the classical notion of an \(E_\infty\) operad. These \(N_\infty\) operads record a coherently commutative multiplication, together with various kinds of twisted products where the group acts by permuting factors. Additively, this gives various kinds of transfers and models of spectra interpolating between naive and genuine \(G\)-spectra. Multiplicatively, we get ring spectra with coherently commutative multiplications and norm maps. In this talk, I'll discuss the kinds of combinatorial compatibility that arises when both the additive and the multiplicative stories are governed by \(N_\infty\) operads, focusing on the purely algebraic questions before describing a model in spaces due to Berman and Krstic.