January 2014, Baltimore, MD

AMS Sessions: January 16 – 17

Satellite Conference: January 16, 8:00 – 11:00 am

Session Dinner: January 16, 5:30 pm

NEW (2014-01-13): Johns Hopkins University has provided transportation between the convention center and satellite session on Thursday morning (7:15 AM). More details on the local travel page.

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Dinner:
Brio Tuscan Grille

100 East Pratt Street

Baltimore, MD 21202

(410) 637-3440

brioitalian.com

Click titles to show/hide abstracts [Click here to show all abstracts]

Jan. 16, 2014

JHU bus departs 7:15 AM

8:00 AM – 11:00 AM

Krieger Hall 205, Johns Hopkins University

8:00 AM

Luke Wolcott [Download slides]

Telescope conjectures and Bousfield lattices for localized categories of spectra

We investigate several versions of the telescope conjecture on
localized categories of spectra, and implications between
them. Generalizing the “finite localization” construction, we show
that on such categories, localizing away from a set of strongly
dualizable objects is smashing. We classify all smashing localizations
on the harmonic category, \(H\mathbb{F}_p\)-local category and *I*-local category,
where *I* is the Brown-Comenetz dual of the sphere spectrum; all are
localizations away from strongly dualizable objects, although these
categories have no nonzero compact objects. The Bousfield lattices of
these categories are also computed.

8:30 AM

What should \(\pi_0\) of a stratified space be?

Stratified spaces occur in many contexts, e.g. in (mathematical models for) phase transitions in physics, as quotient spaces of proper Lie group actions, or more generally for proper Lie groupoids.

Applications to higher category theory suggest the interest of a version of \(\pi_0\) for topological groupoids which takes values in a category more general than discrete sets, eg in profinite topological spaces or perhaps databases.

Some references:

9:00 AM

Vitaly Lorman [Download slides]

Connective covers of Real Johnson-Wilson theories

Real Johnson-Wilson theories are fixed points of standard Johnson-Wilson theories under the action of an involution. They have nice properties that make them useful for computations and “see” more information than the *E(n)*'s. Their connective covers are fixed points of truncated Brown-Peterson theories. I will discuss work-in-progress to compute the cohomology of these connective covers.

9:30 AM

Jonathan Beardsley [Download slides]

Toward Descent Cohomology and Twisted Forms in Homotopy Theory

We discuss work in progress to extend the classical theory of descent cohomology and twisted forms to the homotopical setting. We will briefly review the classical case and also discuss a potential application of the theory to Galois and Hopf-Galois extensions of ring spectra.

10:00 AM

Orthogonal derivatives of spaces of link maps

I will discuss ongoing work with Arone on computing the orthogonal derivatives of spaces of link maps, making comparisons along the way with spaces of embeddings for motivation. Work of Goodwillie-Klein and Arone give some answers to questions about embeddings whose analogs for link maps are still unknown.

10:30 AM

Deb Vicinsky [Download slides]

Suspensions of graphs and categories

I will construct the suspension functor in the category of directed graphs as well as in the category of small categories with the canonical model structure. Then I will describe the stabilizations of these categories. This work is motivated by efforts to find examples which link Goodwillie derivatives of the identity functor to operads.

Jan. 16, 2014

1:00 PM – 3:50 PM

Room 329, Baltimore Convention Center

1:00 PM

Marcy Robertson [Download slides]

Models for infinity prop(erad)s

We give a brief introduction to colored props and their ilk. We then propose ways of encoding the notion of up-to-homotopy prop.

1:30 PM

Howard Marcum [Download slides]

On certain tertiary homotopy operations

Secondary and higher order homotopy operations (Toda brackets) were introduced by Toda in order to construct elements of the homotopy groups of spheres as part of his “composition method” for computing these groups. As early as the 9-stem a tertiary operation (quaternary Toda bracket) was needed to describe a generator. Unfortunately a coherence condition and considerable detail obscure the definiton of this operation. Our aim here is to define and study such operations both in the classical topological setting and, ideally, in the abstract setting of a 2-category with zeros that admits a suspension 2-functor. We focus in particular on a new tertiary operation called the box quaternary operation.

2:00 PM

Recent work in Homotopy Type Theory

Homotopy type theory is a homotopical interpretation of a system of formal logic, providing a system of foundations with intrinsic homotopical content and a computational implementation. It forms the basis of the Univalent foundations program, which was the subject of a recent special year at IAS. In this survey talk, I will show how to compute some homotopy groups of spheres in homotopy type theory, including \(\pi_3(S^2)\). These new logical proofs of classical theorems from algebraic topology make essential use of the new ideas of higher inductive types and the Univalence axiom.

2:30 PM

Homotopy coherent adjunctions

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 adjunction data encapsulates both the homotopy coherent monad and comonad defined by the adjunction. Using this result, we construct the quasi-category of algebras associated to a homotopy coherent monad and give a formal re-proof of the classical monadicity theorem. This is joint work with Dominic Verity.

3:00 PM

Bousfield Localization and Commutative Monoids

Localization is a fundamentally important tool in mathematics. Constructing the localization of a category at a given class of maps leads naturally to the notion of a model category. Bousfield localization is a method of localizing further by turning a given class of maps into weak equivalences. In this talk we will give conditions on a monoidal model category and on the class of maps being localized so that the Bousfield localization preserves strict commutative monoids.

This problem was motivated by an example due to Mike Hill which demonstrates that for the model category of equivariant spectra, even very nice localizations can fail to preserve strict commutative monoids. A recent theorem of Hill and Hopkins gives conditions on the localization to prohibit this behavior. When we specialize our general machinery to the equivariant spectra we recover this theorem. En route to solving the localization problem we introduce an axiom which guarantees us that commutative monoids inherit a model structure. If there is time we will discuss a generalization which allows preservation of structure over arbitrary operads, and relate this to the situation of algebras in equivariant spectra over equivariant operads.

3:30 PM

Kristen Mazur [Download slides]

Additional Structure on the Category of Mackey Functors

The stable homotopy groups of a *G*-spectrum are Mackey
functors, and the zeroeth stable homotopy group of a com- mutative
*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.

Jan. 16, 2014

5:30 PM

100 East Pratt Street

Note: The organizers will not handle money; checks will be separated for individual diners.

Jan. 17, 2014

8:00 AM – 10:50 AM

Room 329, Baltimore Convention Center

8:00 AM

Jonathan Campbell [Download slides]

Topological Hochschild Homology and Koszul Duality

Topological Hochschild Homology (*THH*) is an important
invariant that comes up in both algebraic *K*-theory
computations and topological quantum field theories. In this talk
I’ll present a duality for *THH* of Koszul dual
*E*_{1}-algebras. This duality has possible
applications for computations, and is also the shadow of a
conjectured richer structure for field theories.

8:30 AM

The Kervaire invariant one problem

This talk will be about the Kervaire invariant one problem in algebraic topology.

9:00 AM

GRT-equivariance of Tamarkin's construction of formality morphisms

Given two homotopy algebras and an infinity-morphism between them, it is natural to ask that, if we can modify the two homotopy algebras in some structured way, can we modify the infinity-morphism in some similar way, so as to preserve the new structures? In this talk we describe a situation in which the answer is yes, and indicate how it is possible. We will also give an application of these results, to show that Tamarkin's construction of formality morphisms is equivariant with respect to the action of the Grothendieck-Teichmuller group.

9:30 AM

Nicholas Nguyen [Download slides]

The adjoint action of a homotopy-associative *H*-space on its loop space

We define the adjoint action of a homotopy associative
*H*-space on its loop space, generalizing the definition by
Kono and Kozima for a Lie group (from Kono, Kozima, 1993). We
proceed to use this generalized adjoint action to characterize
homotopy associative *H*-spaces whose homology over
\(\mathbb{F}_p\) is a commutative ring, generalizing a result of
Iwase (from Iwase, 1997).

10:00 AM

Irina Bobkova [Download slides]

Towards a resolution of the spectrum \(E^{h\mathbb{S}^1_2}\) at the prime 2

Chromatic homotopy theory uses the algebraic geometry of formal
groups to organize calculations. In particular, at each prime
*p* there exists a series of homology theories *K(n)*,
called Morava *K*-theories and we can reconstruct the
homotopy type of p-local spectra from their Morava
*K*-theories localizations. When *n* = 2 a lot of
information can be derived from the action of a certain profinite
group, called the Morava stabilizer group, on the Lubin-Tate
theory. We can form homotopy fixed points spectra with respect to
this action and compute their homotopy groups using continuous group
cohomology. We discuss a generalization to prime 2 of work of
Goerss-Henn-Mahowald-Rezk on constructing a tower of fibrations,
whose inverse limit is the spectrum \(E^{h\mathbb{S}^1_2}\), a
“half” of the *K(2)*-local sphere. The successive
fibers of the tower 2 are homotopy fixed points spectra with respect
to specific finite subgroups of the Morava stabilizer group. This
makes the computations accessible as it is possible to make very
detailed calculations with finite subgroups using the theory of
elliptic curves.

10:30 AM

Donald Larson [Download slides]

The Adams-Novikov *E*_{2} term for *Q(2)* at the prime 3

In this talk we will discuss a computation of the Adams-Novikov
*E*_{2} term for the spectrum *Q(2)*. This
spectrum is built using degree 2 isogenies of elliptic curves, and
is closely tied to the 3-primary *K(2)*-local sphere. We will
also examine potential connections between our computation and
algebraic Greek letter families in the Adams-Novikov spectral
sequence for the 3-local sphere.

Jan. 17, 2014

1:00 PM – 5:50 PM

Room 329, Baltimore Convention Center

1:00 PM

Gorenstein homological algebra

Gorenstein homological algebra is essentially the study of
modules after sending certain modules to zero. In the simplest case
of modular representation theory, projective and injective modules
coincide and sending them to zero gives a triangulated category
called the stable module category. Such a simple plan will not work
for a general ring. We show, however, that by changing one's notion
of a “finite” module from finitely generated or presented to modules
of type *FP*_{∞}, we get good analogues of flat
and injective modules that are well-behaved for any ring. This
enables us to develop Gorenstein homological algebra and an
associated triangulated stable module category in full generality.

1:30 PM

Anna Marie Bohmann [Download slides]

An equivariant infinite loop space machine

An equivariant infinite loop space machine should turn
categorical or algebraic data into genuine *G*-spectra. While
infinite loop space machines have been a crucial part of homotopy
theory for decades, equivariant versions are in early stages of
development. I will describe joint work with A. Osorno in which we
build an equivariant infinite loop space machine that starts with
diagrams of categories on the Burnside category and produces a
genuine *G*-spectrum via the work of Guillou–May. This
machine readily applies to produce Eilenberg–MacLane spectra for
Mackey functors and topological *K*-theory.

2:00 PM

Mona Merling [Download slides]

Equivariant algebraic *K*-theory

In the early 1980's, Dress and Kuku, and Fiedorowicz, Hauschild
and May introduced space level equivariant versions of the plus and
*Q* constructions in algebraic *K*-theory. However,
back then, the methods did not allow for nontrivial group action on
the input ring or category. We generalize these definitions to the
case in which a finite group *G* acts nontrivially on a ring
(or an exact or Waldhausen category) and we show how to construct a
genuine equivariant *K*-theory spectrum with good properties
from a *G*-ring. An example of interest is that of a Galois
extension.

The equivariant constructions rely on finding categorical models
for classifying spaces of equivariant bundles (a joint project with
Guillou and May) and the use of equivariant infinite loop space
machines such as the one developed by Guillou and May, or the
equivariant version of Segal's machine. The comparison of these
machines, which will allow their interchangeable use in algebraic
*K*-theory constructions, is a joint project with May and
Osorno. New ideas are needed since, among other things, the
comparison theorem of May and Thomason fails equivariantly.

2:30 PM

Mehdi Khorami [Download slides]

Higher chromatic analogues of twisted *K*-theory

Let \(R_n\) denote the homotopy fixed point spectrum \(E_n^{hS\mathbb G_n}\), where \(S\mathbb G_n\) is the kernel of the determinant homomorphism \(\text{det}:\mathbb G_n\to \mathbb Z_p^\times\) with \(\mathbb G_n\) being the Morava group. Here \(E_n\) denotes the \(n\)-th Morava \(E\)-theory. We show that for a \(K(n)\)-local space \(X\) equipped with a \(K(\mathbb Z_p, n+1)\)-bundle \(P\to X\), the \(P\)-twisted \(R_n\)-theory of \(X\), \(R_{n*}(X, P)\), is defined and there exist a “universal coefficient” isomorphism \[R_{n*}(X, P)\cong R_{n*}(P)\otimes _{R_{n*}(K(\mathbb Z_p, n+1))} \mathbb R_{n*}.\]

This extends an analogous result on twisted \(K\)-theory in the \(K(n)\)-local category.

3:00 PM

Equivariantly Twisted Cohomology Theories

Twisted *K*-theory is a cohomology theory whose cocycles
are like vector bundles but with locally twisted transition
functions. If we instead consider twisted vector bundles with a
symmetry encoded by the action of a compact Lie group, the resulting
theory is equivariant twisted *K*-theory. This subject has
garnered much attention for its connections to conformal field
theory and representations of loop groups. While twisted
*K*-theory can be defined entirely in terms of the geometry
of vector bundles, there is a homotopy-theoretic formulation using
the language of parametrized spectra. In fact, from this point of
view we can define twists of any multiplicative generalized
cohomology theory, not just *K*-theory. The aim of this talk
is to explain how this works, and then to propose a definition of
equivariant twisted cohomology theories using a similar
framework. The main ingredient is a structured approach to
multiplicative homotopy theory that allows for the notion of a
*G*-torsor where *G* is a grouplike
*A*_{∞} space.

3:30 PM

Martin Frankland [Download slides]

Completed power operations for Morava *E*-theory

Morava *E*-theory is an important cohomology theory in
chromatic homotopy theory. Using work of Ando, Hopkins, and
Strickland, Rezk described the algebraic structure found in the
homotopy of *K(n)*-local commutative *E*-algebras via
a monad on *E*_{∗}-modules that encodes all power
operations. However, the construction does not see that the homotopy
of a *K(n)*-local spectrum is *L*-complete (in the
sense of Greenlees-May and Hovey-Strickland). We improve the
construction to a monad on *L*-complete
*E*_{∗}-modules, and discuss some applications.

4:00 PM

Power Operations and Commutative Ring Spectra

We will compute the action of the Dyer-Lashof algebra on
relative smash products using the multiplicative structure of the
Künneth spectral sequence. We will then interpret such
operations in terms of different possible
*E*_{∞}-structures. We will end with an
application of these computations to give a non-existence result for
*E*_{∞}-complex orientations of certain ring
spectra.

4:30 PM

Power operation calculations in elliptic cohomology

One question in homotopy theory is to construct and compute
stable power operations on elliptic cohomology theories. As a
particular instance, for *E* a Morava *E*-theory
spectrum of height 2, its algebra of power operations has the
structure of a graded twisted bialgebra satisfying a Frobenius
congruence. For the *K(1)*-localization of *E*, its
algebra of power operations has a single generator over the
coefficient ring. We illustrate this structure and provide explicit
formulas by doing calculations, at the prime 3, with moduli of
elliptic curves.

5:00 PM

Carolyn Yarnall [Download slides]

The Slice Tower and Suspensions

The slice filtration is a filtration of equivariant spectra
developed by Hill, Hopkins, and Ravenel in their solution to the
Kervaire Invariant One Problem. After briefly discussing some
properties of the filtration, I will present the slice towers of
\(\Sigma^nH\mathbb{Z}\) where the group *G* is a cyclic
*p*-group. I will highlight the patterns we see in this context by
displaying a few specific examples.

5:30 PM

Mark Behrens [Download slides]

tmf cooperations

The groups *bo***bo* have a been long
understood by the work of Adams, Mahowald, Milgram, and others. The
Adams perspective is based on numerical polynomials, wheras the
Mahowald perspective is based on a splitting of
*bo***bo* into the *bo*-homology of
integral Brown–Gitler spectra.

Following a program initiated by Mahowald, we will discuss the
analogous approach at odd primes. We will explain how to relate
Mahowald's *bo*-Brown-Gitler spectrum approach to Laures'
2-variable modular forms, and a third geometric approach involving
isogenies of elliptic curves.

For scheduling or session questions, contact Donald.

For local questions, contact Nitu.

For website questions, contact Niles.