— Fall 2014 —
8 Sept 2014
Localizing the Adams and Adams-Novikov Spectral Sequences
Haynes Miller proved the
n=1 case of the telescope conjecture at odd primes by computing
π_{∗}(v_{1}^{-1}S/p) explicitly. As a result of his work we understand the Adams spectral sequence for the Moore spectrum above a line of slope
1/(p^{2}–p–1). We will describe the analogue of Miller’s result for the sphere spectrum.
When we try to set
p=2 in our results we encounter problems. However, we can instead compute the η-localized Adams-
Novikov spectral sequence. Of course,
η^{-1}π_{∗}(S^{0})=0 but motivically there is an interesting question. We will compute
η^{-1}π_{∗,∗}(S^{0,0}) resolving a conjecture of
Guillou and
Isaksen, and understand the Adams-Novikov spectral sequence above a line of slope 1/5.
15 Sept 2014
Calculating the Adams Spectral Sequence for a Simplicial Algebra Sphere
While in the homotopy theory of simplicial algebras, the homotopy of ‘spheres’ is known, the unstable Adams spectral sequence is very far from degenerate. We’ll give some background on the setting, and discuss a method of calculating the E^{2}-page of this spectral sequence.
22 Sept 2014
Homotopy Fixed Points of Landweber Exact Spectra
Let
E be a Landweber exact spectrum (like K-theory or elliptic homology) with an action by a finite group
G. The talk is concerned with the following two questions:
- When is the norm map from the homotopy orbits to the homotopy fixed points an equivalence?
- When is the ∞-category of G-equivariant E-modules equivalent to that of E^{hG}-modules?
At the end, I plan to generalize these questions (and answers) to the context of certain derived stacks.
29 Sept 2014
Goerss-Hopkins Obstruction Theory for ∞-Categories
Goerss-
Hopkins obstruction theory is a tool for obtaining structured ring spectra from algebraic data. It was originally conceived as the main ingredient in the construction of
tmf, although it’s since become useful in a number of other settings, for instance in setting up a “naive” theory of spectral algebraic geometry and in
Rognes’s Galois correspondence for commutative ring spectra. In this talk, I’ll give some background, explain in broad strokes how the obstruction theory is built, and then indicate how one might go about generalizing it to an arbitrary presentable ∞-category. This last part relies on the notion of a
model ∞-category – that is, of an ∞-category equipped with a “model structure” – which provides a theory of resolutions internal to ∞-categories and which will hopefully prove to be of independent interest.
6 Oct 2014
Higher Associativity of Moore Spectra
Not much is known about homotopy coherent ring structures of the Moore spectrum M_{p}(i) (the cofiber of the p^{i} self-map on the sphere spectrum S^{0}), especially when i>1. Stasheff developed a hierarchy of coherence for homotopy associative multiplications called A_{n} structures. The only known results are that M_{p}(1) is A_{p-1} and not A_{p} and that M_{2}(i) are at least A_{3} for i>1. In this talk, techniques will be developed to get estimates of ‘higher associativity’ structures on M_{p}(i).
13 Oct 2014
Toward the Formal Theory of Higher Homotopical Categories
One framework for stating and proving theorems in abstract homotopy theory uses quasi-categories (aka ∞-categories): for instance, the result of
Francis that homology theories for topological
n-manifolds are equivalent to
n-disk algebras is formalized in this language. The foundational category theory of quasi-categories is developed in thousands of pages of dense mathematics by
Joyal,
Lurie, and others. Our project is to redevelop these foundations using techniques from formal category theory. We show that the accepted definitions (e.g., of equivalence, limits, adjunctions, cartesian fibrations) can be formulated inside the “homotopy 2-category” of quasi-categories. From this new perspective the proofs that they satisfy the expected relationships (e.g., that right adjoints preserve limits) mirror the classical categorical ones. Importantly, this 2-categorical work can also be applied to other homotopy 2-categories, e.g., for
n-fold complete
Segal spaces, which were used by Lurie to prove the
Baez-
Dolan cobordism hypothesis. This is joint work with
Dominic Verity.
20 Oct 2014
Topological Analogs of the Radon Transform
We define topological analogs to the Radon transform using persistent homology and Euler characteristic curves. From these we construct metrics on the space of all embedded finite simplicial complexes in R^{3} or R^{2}. This can be applied to shape recognition and morphology.
27 Oct 2014
The Algebraic K-Theory of the Sphere Spectrum, the Geometry of High-Dimensional Manifolds, and Arithmetic
Waldhausen showed that the algebraic K-theory of the “spherical group ring” on the based loops of a manifold captures the stable concordance space of the manifold. In the simplest case, this result says that for high-dimensional disks, information about BDiff is encoded in K(S), the algebraic K-theory of the sphere spectrum. This talk explains recent work with
Mike Mandell that provides a complete calculation of the homotopy groups of K(S) in terms of the homotopy groups of K(
Z), the sphere spectrum, and a certain Thom spectrum.
3 Nov 2014
Chromatic Unstable Homotopy Theory
I will show how the
Bousfield-
Kuhn functor enables us to study monochromatic behavior of unstable homotopy theory using stable techniques. As an example, I will compute the K(2)-local unstable homotopy groups of the three sphere.
10 Nov 2014
The Homotopy Calculus of Categories and Graphs
There are several known examples of categories in which the
Goodwillie derivatives of the identity functor have the structure of an operad, including based spaces, bounded-below differential graded Lie algebras over
Q, and algebras over a symmetric spectral operad. I will show that this is also the case in the category of small categories. Additionally, I will discuss my efforts to compute the derivatives of the identity functor in two categories of graphs.
1 Dec 2014
Homological Algebra of Complete and Torsion Modules
Let
R be a finite-dimensional regular local ring with maximal ideal
m. The category of
m-complete
R-modules is not abelian, but it can be enlarged to an abelian category of so-called
L-complete modules. This category is an abelian subcategory of the full category of
R-modules, but it is not usually a
Grothendieck category. It is well known that a Grothendieck category always has a derived category, however, this is much more delicate for arbitrary abelian categories. In this talk, we will show that the derived category of the
L-complete modules exists, and that it is in fact equivalent to a certain
Bousfield localization of the full derived category of
R.
L-complete modules should be dual to
m-torsion modules, which do form a Grothendieck category. We will make this precise by showing that although these two abelian categories are clearly not equivalent, they are derived equivalent.
— Spring 2015 —
26 Jan 2015
Localization Sequences in the Algebraic K-Theory of Ring Spectra
The algebraic K-theory of the sphere spectrum encodes significant information in both homotopy theory and differential topology.
Waldhausen’s chromatic convergence conjecture attempts to approximate K(S) by localizations K(L
_{n}S). The L
_{n}S are in turn approximated by the
Johnson-
Wilson spectra E(
n)=BP<
n>[
v_{n}^{–1}], and K(BP<
n>) is in principle computable. This would lead inductively to information about K(E(
n)), and hence K(S), via the conjectural fiber sequence K(BP<
n–1>) → K(BP<
n>) → K(E(
n)). In this talk, I will define the ring spectra of interest and construct some actual localization sequences in their K-theory. I will then use trace methods to show that the actual fiber of K(BP<
n>) → K(E(n)) differs from K(BP<
n–1>), meaning that the situation is more complicated than was originally hoped. This is joint work with
Ben Antieau and
Tobias Barthel.
2 Feb 2015
Coassembly in Algebraic K-Theory
The coassembly map allows us to approximate any contravariant homotopy-invariant functor by an excisive functor, i.e. one that behaves like a cohomology theory. We’ll apply this construction to
Waldhausen’s algebraic K-theory of spaces, and its corresponding THH functor. The results are somewhat surprising: a certain dual form of the A-theory
Novikov conjecture is false, but when the space in question is the classifying space B
G of a finite
p-group, coassembly on THH is split surjective after
p-completion. Even better, we can show that the coassembly map links up with the more familiar assembly map to produce the equivariant norm. As a result, we get some splitting theorems after K(
n)-localization, and a surprising connection between the Whitehead group and
Tate cohomology. If there is time, we will also discuss related work on the equivariant structure of THH.
9 Feb 2015
Norms, Transfers, and Operads
I’ll discuss joint work with
Blumberg in which we introduce a class of equivariant operads that parameterize infinitely homotopy commutative multiplications. This gives a language for describing what structure is preserved when we
Bousfield localize commutative ring spectra while also allowing us to see how the transfer is encoded operadically.
16 Feb 2015
Schematic Homotopy Types of Operads
The rational homotopy type
X_{Q} of an arbitrary space
X has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space
X are not accessible through the space
X_{Q}. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces
Toën introduced the notion of a pointed schematic homotopy type over a field
k, (
X×
k)
^{sch}.
In his recent study of the pro-nilpotent
Grothendieck-Teichmüller group via operads,
Fresse makes use of the rational homotopy type of the little 2-disks operad E
_{2}. As a first step in the extension of Fresse’s program to the pro-algebraic case we discuss the existence of a schematization of the little 2-disks operad.
23 Feb 2015
Combinatorial Models of Moduli Spaces CANCELLED
Ribbon graphs provide a powerful combinatorial tool in the study of the moduli space of Riemann surfaces. The theory of quadratic differentials in complex analysis gives a cellular decomposition of the moduli space indexed by ribbon graphs, and this allowed the computation of the Euler characteristic and
Kontsevich’s proof of
Witten’s intersection number conjecture.
Costello found a different ribbon graph model in his work constructing the B-model counterpart to
Gromov-Witten theory in terms of topological field theories. In this talk I will review these ideas and describe how to produce Costello-type combinatorial models of moduli spaces of many related classes of objects, such as unoriented, spin and
r-spin surfaces, surfaces with
G-bundles, and 3-dimensional handlebodies.
2 Mar 2015
T-Duality and the Chiral de Rham Complex
T-dual pairs are distinct manifolds equipped with closed 3-forms that admit isomorphism of a number of classical structures including twisted de Rham cohomology, twisted K-theory, and twisted Courant algebroids. An ongoing program is to study T-duality from a loop space perspective; that is, to identify structures attached to the loop spaces that are isomorphic under T-duality. In this talk, I’ll explain how the chiral de Rham complex of
Malikov,
Schechtman, and
Vaintrob, gives rise to such structures. This is a joint work with
Varghese Mathai.
9 Mar 2015
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.
23 Mar 2015
The Chromatic Splitting Conjecture at n=p=2
I will discuss why the strongest form of
Hopkins’s chromatic splitting conjecture, as stated by
Hovey, cannot hold at chromatic level
n=2 at the prime
p=2. More precisely, let V(0) be the mod 2 Moore spectrum. I will give a summary of how one uses the duality resolution of
Bobkova,
Goerss,
Henn, Mahowald and
Rezk to show that π
_{k}L
_{1}L
_{K(2)}V(0) is not zero when
k is congruent to 5 modulo 8. I explain how this contradicts the decomposition of L
_{1}L
_{K(2)}S predicted by the chromatic splitting conjecture.
30 Mar 2015
Analysing Grothendieck Ring of Varieties Using K-Theory
The Grothendieck ring of varieties K
_{0}[V
_{k}] is defined to be the free abelian group generated by varieties, modulo the relation that for a closed subvariety
Y of
X, [
X]=[
Y]+[
X\
Y]. This ring is used extensively in motivic integration. There are two important structural questions about the ring:
- If X and Y are two varieties such that [X]=[Y], are X and Y birationally isomorphic?
- Is the class of the affine line [A^{1}] a zero divisor?
In a recent paper,
Borisov constructed an example showing that the answer to question 2 is “yes”; in a beautiful coincidence this construction also produced varieties
X and
Y which show that the answer to question 1 is “no.” In this talk we will construct a spectrum K(V
_{k}) such that π
_{0}K(V
_{k})=K
_{0}[V
_{k}], and such that the higher homotopy groups contain further geometric information. We will then analyze the structure of this spectrum to show that Borisov’s coincidence is not a coincidence at all, and that in fact the kernel of multiplication by [
A^{1}] is generated by counterexamples to question 1.
6 Apr 2015
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”. I will explain how “
J-homogeneous” functors are classified by suitably equivariant spectra, and address some convergence issues of the Taylor tree.
3pm Mon 13 Apr 2015
Kitchloo’s Category of Symplectic Motives
The geometric quantization program attempts to define a functor from classical mechanics (AKA symplectic geometry) to quantum systems; but it is known to have problems, associated to failure of transversality.
Kontsevich has made progress in the related deformation quantization program, which is less intrinsically global.
Kitchloo [in
arXiv:1204.5720] defines a stabilized category of symplectic manifolds, with morphism objects enriched over spectra, analogous to the algebraic geometers’ motives. He identifies its associated ‘motivic group’ as the (Hopf-Galois, ring) spectrum defined by the topological Hochschild homology of a Thom spectrum associated to a certain Lagrangian Grassmannian. After tensoring with
Q, this is remarkably like the motivic group Kontsevich finds acting on his deformation quantization constructions.
20 Apr 2015
Noncommutative Bialgebras in Spectra and Hopf-Galois Extensions
We briefly review the notions of infinity operads and infinity categories before describing definitions of noncommutative bialgebras in spectra, allowing us to define Hopf-Galois extensions of noncommutative ring spectra. We give a number of interesting geometrically motivated examples of the latter objects.
27 Apr 2015
Topological Persistence via Category Theory
Topological persistence originated as a strategy for measuring the topology of a statistical data set. The naive approach is to build a simplicial complex from the data and measure its homological invariants; but this approach is extremely sensitive to noise and is therefore unusable. The correct approach (made effective by
Edelsbrunner,
Letscher and
Zomorodian in 2000) is to represent the data by a multiscale family of simplicial complexes, and to measure the homology as it varies across all scales. The resulting multiscale invariants, known as persistence diagrams, are provably robust to perturbations of the data (
Cohen-Steiner, Edelsbrunner,
Harer 2007).
In this talk, I will explain how these ideas may be expressed in the language of category theory. The basic concepts extend quite widely. In particular, I hope to explain how Reeb graphs and join-trees—well known constructions in data analysis—can be thought of as persistent invariants, enjoying some of the same properties as persistence diagrams. My collaborators in this work include
Peter Bubenik,
Jonathan Scott,
Elizabeth Munch,
Amit Patel.
— Fall 2015 —
31 Aug 2015
Real Johnson-Wilson Theories and Computations
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 η-fibration, and discuss recent computations of the ER(n)-cohomology of some well-known spaces, including CP^{∞}.
21 Sept 2015
Ilya Grigoriev, University of Chicago
Characteristic classes of
manifold bundles
For every smooth fiber bundle $f: E\to B$ with fiber a closed,
oriented manifold $M^d$ of dimension $d$ and any characteristic class
of vector bundles $p \in H^* \left( BSO(d) \right)$, one can define a
``generalized Miller-Morita-Mumford class" or ``kappa-class"
$\kappa_p \in H^*(B)$. We are interested in the ideal $I_M$ of all
the polynomials in the kappa classes which vanish for \textit{every}
bundle with fiber diffeomorphic to $M$, as well as the algebraic
structure of the quotient $R_M = \mathbb{Q}\left[\kappa_p\right]/I_M$
of the free polynomial algebra by this ideal. I will talk mainly about
the case where the manifold is a connected sums of $g$ copies of $S^n
\times S^n$, with $n$ odd. In this case, we can compute the ring $R_M$
modulo nilpotents, and show that the Krull dimension of $R_M$ is $n-1$
for all $g>1$. This is joint work with Søren Galatius and Oscar
Randal-Williams.
28 Sept 2015
Rational homology of configuration spaces via factorization homology
The study of configuration spaces is particularly tractable over a field of characteristic zero, and there has been great success over the years in producing complexes simple enough for explicit computations, formulas for Betti numbers, and descriptive results. I will discuss recent work identifying the rational homology of the configuration spaces of an arbitrary manifold with the homology of a Lie algebra constructed from its cohomology. The aforementioned results follow immediately from this identification, albeit with hypotheses removed; in particular, one obtains a new, elementary proof of homological stability for configuration spaces.
5 Oct 2015
$E_n$ cells and homological stability
When studying objects with additional algebraic structure, e.g. algebras over an operad, it can be helpful to consider cell decompositions adapted to these algebraic structures. I will talk about joint work with Jeremy Miller on the relationship between $E_n$-cells and homological stability. Using this theory, we prove a local-to-global principle for homological stability, as well as give a new perspective on homological stability for various spaces including symmetric products and spaces of holomorphic maps.
12 Oct 2015
Topological Complexity of Spaces of Polygons
The topological complexity of a topological space X is the number of rules required to specify how to move between any two points of X.
If X is the space of all configurations of a robot, this can be interpreted as the number of rules required to program the robot to move from any configuration to any other.
A polygon in the plane or in 3-space can be thought of as linked arms of a robot. We compute the topological complexity of the space of polygons of fixed side lengths.
Our result is complete for polygons in 3-space, and partial for polygons in the plane.
19 Oct 2015
On equivariant infinite loop space machines
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 describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Bert Guillou, Peter May and Mona Merling.
26 Oct 2015
The Unbearable Lightness of 2-Being
(Title with apologies to Milan Kundera)
Joint work with Dr. Emily Riehl, Johns Hopkins University.
For a few years now, Emily and I have been engaged in a project to study the extent to which the category theory of $(\infty,n)$-categories can be developed using some elementary tools from 2-category theory. This idea, which dates back to Andre Joyal's earliest work on quasi-categories, turns out to be remarkably and somewhat unbearably fruitful. Much to our own surprise, a very great deal of the category theory of such stuctures can be rendered directly from a study of {\em strict\/} 2-categories and bicategories of $(\infty,n)$-categories, using techniques largely developed by the Australian Category Theory School in the 1970s.
To date we have developed a largely 2-categorical theory of adjunctions, monadicity, cartesian fibrations, modules, limits and colimits, (pointwise) Kan extensions, final and initial functors, Yoneda's lemma, presheaf categories and many other important categorical structures besides. All of this work follows a mild re-working of a well trodden path of 2-being, which not only provides for a foundational redevelopment of the category theory of quasi-categories but is also couched in terms that is both model independent and amenable to application to $(\infty,n)$-categories and $(\infty,\infty)$-categories of various kinds.
One of the enduring themes in this work is the inspiration it draws from the {\em derivator\/} approach to abstract homotopy theory, as pioneered independently by Grothendieck and Heller. Such gadgets light up the theory of homotopy limits and colimits by observing that homotopy categories themselves can provide us with enough information so long as they come parameterised by {\em internal diagram types}. In this context a homotopy theory simply consists of a category fibred (or indexed) over the category of all small categories. The fibre of this structure over some small category $\mathbb{C}$ is thought of as the homotopy category of diagrams on $\mathbb{C}$ and adjunctions between fibres track the existance of homotopy Kan extensions. It is then the interplay between weak notions in the {\em external\/} world of each fibre and strong coherent notions in the internal world of abstract diagrams that allows one to develop an abstract homotopy theory to rival that available in the model category theoretic setting.
In this talk I plan to illustrate this relationship between weak external notions in 2-categories of $\infty$-categories and internal notions expressed in simplicially enriched form. The importance of this interplay will be illustrated with reference to specific examples of this phenomenon, such as the theory of homotopy coherent adjunctions, monads and the Beck monadicity theorem. While I will not define precisely what a 2-dimensional derivator might be, since as yet we have no completely compelling axiomatisation, rather I hope to motivate the importance of drawing inspiration from this 2-derivator point of view. This also seems like a apposite moment to celebrate the recent publication of our paper on homotopy coherent monads and Barr monadicty, which has very recently appeared in Advances in Mathematics.
2 Nov 2015
The Integration Pairing and Extended Topological Field Theories
Given a hermitian line bundle with a hermitian connection on a manifold there is a straightforward way of producing 1-dimensional topological field theory over the manifold. One can generalize this procedure by replacing complex line bundles with $n$-gerbes bound by $U(1)$, and replacing connections with appropriate connective structures. The resulting topological field theories are n-dimensional and fully extended. In this talk, we will describe a procedure of extracting these topological field theories using the higher categorical machinery of Jacob Lurie. If time permits we will discuss how the Wess-Zumino-Witten model, Chern-Simons theory and Dijkgraaf-Witten theory fit into this context.
9 Nov 2015
A chain rule for Goodwillie calculus
In the homotopy calculus of functors, Goodwillie defines a way of assigning a Taylor tower of polynomial functors to a homotopy functor and identifies the homogeneous pieces as being classified by certain spectra, called the derivatives of the functor. Michael Ching showed that the derivatives of the identity functor of spaces form an operad, and Arone and Ching developed a chain rule for composable functors. We will review these results and show that through a slight modification to the definition of derivative, we have found a more straight forward chain rule for endofunctors of spaces.
16 Nov 2015
Ernie Fontes, University of Texas
Weight structures and the algebraic K-theory of stable $\infty$-categories
Algebraic K-theory is a spectral invariant of module categories with applications to number theory and manifold geometry. Recently, various people have used the technology of $\infty$-categories to establish universal characterizations for K-theory. Many of the basic structural results about K-theory have been elevated to apply in the $\infty$-categorical context. I will describe Waldhausen's sphere theorem, a new analogous result for the algebraic K-theory of stable $\infty$-categories, and some applications of the new theorem.
23 Nov 2015
THANKSGIVING BREAK
30 Nov 2015
K-theory computations for topological insulators
Over the past decade a new class materials called topological insulators, often with counter-intuitive electric properties, have been discovered. The mathematics involved in classifying such materials is twisted equivariant Real K-theory. I will briefly describe the setup and survey new K-theory computations coming from these considerations. This is joint work with Dan Freed.
— Spring 2016 —
25 Jan 2016
Philip Egger, Northwestern
Computations in $v_2$-periodic homotopy theory
The subalgebra $A(1)$ of the mod 2 Steenrod algebra $A$ can be
seen as an $A$-module in four different ways. Davis and Mahowald showed
that all four of these $A$-modules are the mod 2 cohomologies of type 2
finite CW-complexes $A_1[ij]$ for $i,j\in\{0,1\}$. We prove that all four
of these complexes admit a 32-periodic $v_2$-self-map. This work is joint
with Bhattacharya and Mahowald.
We have also found a complex $Z$ which will admit a 1-periodic
$v_2$-self-map, and possibly disprove the telescope conjecture at $n=p=2$.
This work is joint with Bhattacharya.
1 Feb 2016
Complex multiplication in homotopy theory
If A is a ring and F is a one-dimensional formal group law over an A-algebra R, we say that F "admits complex multiplication by A" if there is a ring homomorphism from A to the endomorphism ring of F whose induced action on the tangent space of F coincides with the given action of A on R. A formal group with a choice of complex multiplication by A is also called a "formal A-module."
Formal A-modules of height 1 were used by Lubin and Tate in their solution to the p-adic version of Kronecker's Jugendtraum, that is, the computation of the abelian closure of any p-adic number field. Formal A-modules play a central role in both Drinfeld's and Carayol's approaches to local Langlands correspondences: in each case, the correspondence is realized by an action of a Galois group, a group of Hecke operators, and the automorphism group of a formal A-module, all acting on the cohomology of an appropriate deformation space of a formal A-module.
In this talk I will describe new results in this area and, depending on how much coffee I drink before the talk, some number of topological and algebraic applications:
1. The computation of the classifying ring of formal A-modules. (This computation was done for A = Z by M. Lazard, for A a field or the ring of integers of a nonarchimedean local field by V. Drinfeld, and for A a Dedekind domain of class number 1 by M. Hazewinkel; in each case the classifying ring is a polynomial algebra. In this talk I give the computation the classifying ring of formal A-modules for a larger class of Dedekind domains A, including all number rings regardless of class number; these include the first full computations of classifying rings of formal A-modules which are not polynomial algebras.)
2. The solution to Ravenel's problem on topological realization of formal A-modules. (In 1983, D. Ravenel posed the problem of constructing a spectrum X whose BP-homology recovers, as a BP_*-module, the classifying ring V^A of A-typical formal A-modules, where BP_* acts on V^A by the ring map BP_* -> V^A classifying the underlying formal group law of the universal A-typical formal A-module. In this talk, I give a topological nonrealizability result, that topological properties of the chromatic tower imply that certain BP_*-modules cannot occur as the BP-homology of a spectrum, and then I use algebraic computation in V^A to demonstrate that V^A satisfies these nonrealizability criteria, when A is the ring of integers in a finite extension of the p-adic rationals of degree > 1.)
3. The solution to Ravenel's Global Conjecture. (In 1983, D. Ravenel made a conjecture about number-theoretic properties of the orders of the flat cohomology groups of the moduli stack of formal A-modules, generalizing J. F. Adams' "J(X) IV" description of the orders of the groups in the image of the J-homomorphism. I give a proof of the Global Conjecture, as well as some new number-theoretic results: given two Galois extensions K/Q, L/Q with rings of integers A,B respectively, the fields K,L are arithmetically equivalent (i.e., they have the same Dedekind zeta-function) if and only if the flat cohomology group H^1 of the moduli stack of A-modules agrees with the flat cohomology group H^1 of the moduli stack of B-modules.)
4. The computation of the cohomology of the automorphism group of a height 4 formal group law over a field of characteristic p > 5. (This is a group of cohomological dimension 16, whose full cohomology turns out to be have total dimension 3440 as an F_p-vector space.) This uses new methods, namely "height-shifting" techniques that build up this cohomology from the cohomology of the automorphism groups of the formal A-modules appearing in the local Langlands correspondence for representations of GL_2(A), with A the ring of integers in a ramified quadratic extension of Q_p. This also results in a computation of some new periodic families in the stable homotopy groups of spheres; when p=7, for example, this is a family of 7-torsion elements which repeats every 4800 dimensions. These are the first substantial computations in the divided delta-family in the stable homotopy groups of spheres. As a consequence of the computation, I give the homotopy groups of the K(4)-local Smith-Toda complex V(3). It is an open conjecture of Mahowald that V(3) is the last (largest height) Smith-Toda complex which exists; if Mahowald's conjecture is correct, then this computation completes the computation of the K(n)-local homotopy groups of all the Smith-Toda complexes V(n-1). This computation also leads to a natural conjecture about the F_p-vector space dimension of the cohomology of the automorphism group of a height n formal group law over F_p for all n, for p >> n.
8 Feb 2016
Comparing continuous and discrete homotopy
fixed points, with an application to the Ausoni-Rognes
conjecture
Let G be a profinite group and X a naive
G-spectrum. In certain cases, X can be realized as a
discrete G-spectrum in a concrete way. Thus, there
are two different ways to form a homotopy fixed point
spectrum for X: (1) by viewing G as a discrete
group; and (2) by letting G have its profinite topology
and respecting the discrete action. We will show that
in some cases these two homotopy fixed point spectra
are equivalent. We give an application of this result
to the Ausoni-Rognes conjecture in the n=1, p > 3 case.
15 Feb 2016
Zachary Stone, University of Maryland (Linguistics)
RESCHEDULED Tuesday Fen 23 1:30-2:30pm in Krieger 413
Special Lecture by a Linguist:
Spatial Methods in Minimalist Syntax
I will be sharing some ongoing research about mathematical models of bare phrase structure (BPS), including a new proposal for representing phrase structure as a topological site.
I will demonstrate that the category of BPS phrase markers, viewed as membership-ordered sets, is not rich enough to keep track of the syntactically significant “spatial” data in the category of phrase markers. It can be shown that the data needed to enrich this category are roughly all of the idiosyncratic properties of MERGE. I then define a category of spaces which can model bare phrase structure derivations that is an abstraction of the order-relations in a tree. I will show how this category can be set up to naturally represent many “derivational” generalizations of the important spatial properties of phrase markers, such as equidistance, comp/spec/adjunct positions, projection, terms, and dominance. This is made possible by the ability to represent feature configurations in the derivation itself. Taken together, this will naturally force an encoding of the properties missing from the category of membership-ordered sets needed to represent phrase markers.
22 Feb 2016
Lagrangian Cobordisms and Fukaya Categories
Unlike other cobordism categories, the category of Lagrangian
cobordisms inside a fixed symplectic manifold is stable.
This means it has, for instance, the structure of a triangulated category.
Moreover, we construct a functor to the Fukaya category
of that manifold respecting exact triangles. As a corollary, we will
see that Floer theory behaves similar to characteristic classes for
Lagrangian cobordisms. We'll also talk about a program for how
to show that the homotopy theory of Lagrangian cobordisms may
completely recover the Floer theory of the symplectic manifold.
29 Feb 2016
T-duality and iterated algebraic K-theory
T-duality arose in string theory as an equivalence between
the physics of two different but suitable related spacetimes. By
considering only the underlying topological quantities, T-duality can
be distilled into a mathematical theorem which states that the twisted
K-theories of certain pairs of circle bundles equipped with
U(1)-gerbes are isomorphic via a Fourier-Mukai transform. In this
talk, I will describe a generalization of T-duality to higher rank
sphere bundles. I will construct twists of the iterated algebraic
K-theory of connective complex K-theory by higher gerbes and describe
a T-duality isomorphism between the twisted iterated K-theories of a
pair of suitably related sphere bundles. (Joint with H. Sati and C.
Westerland)
7 Mar 2016
Duality for Real spectra
We will consider in this talk Real spectra, i.e. Z/2-spectra that can be built from the Real bordism spectrum MR by quotiening by a regular sequence and inverting elements. This includes both connective and periodic KR-theory and more generally the spectra BPR and ER(n). We will compute their Anderson duals, in particular providing new universal coefficient theorems.
21 Mar 2016
$N_{\infty}$-operads, equivariant trees and equivariant $G-\infty$-operads
One striking feature of equivariant operad theory is that equipping the classical $E_{\infty}$ operad with a trivial $G$-action does not provide
the correct "up to homotopy" replacement of the commutative operad (which necessarily has a trivial $G$-action).
Indeed, the correct notion of $G-E_{\infty}$ operad has n-th spaces which,
while $G$-contractible and $\Sigma_n$-free just as for the aforementioned $E_{\infty}$ operad), have nonetheless different $G \times \Sigma_n$ homotopy types.
More specifically, the $n$-th spaces in the $G-E_{\infty}$ operad have many more non empty fixed point sets, and these are essential to capture the Hill-Hopkins-Ravenel norms.
Following these ideas, Blumberg and Hill defined $N_{\infty}$ operads, a family of operads that interpolates between the $E_{\infty}$ and $G-E_{\infty}$ operads and which possess only some norms.
Further, they identified necessary conditions that the family $\mathcal{F}_n$ of non-empty fixed points of the $n$-th space of an $N_{\infty}$ must satisfy, though the sufficiency of those conditions was far from obvious.
In recent work, Gutierrez and White showed those conditions to be indeed sufficient, but their construction of an $N_{\infty}$ operad is via a small object argument, hence far from explicit.
In this talk, I will recall the theory of trees of Weiss, Moerdijk et al, and show how a suitable generalized notion of equivariant trees leads to,
via the grafting operation, an explicit model for $N_{\infty}$ operads as equivariant operads internal to simplicial sets.
Time permitting, I will also discuss partial work in the direction of connecting this model to regular operads via the world of equivariant dendroidal sets.
Namely, I will define $G-\infty$ operads, equivariant dendroidal sets satisfying new equivariant inner horn conditions which are closely related to the notion of norms,
and explain how these expand the "homotopy operad" of such a $G-\infty$ operad.
28 Mar 2016
Galois descent in algebraic K-theory
Let $A \to B$ be a $G$-Galois extension of rings, or more generally of $E_\infty$-ring spectra in the sense of Rognes. A basic question in algebraic $K$-theory asks how close the map $K(A) \to K(B)^{hG}$ is to being an equivalence, i.e., how close $K$ is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after "periodic'' localization. We formulate and prove a general lemma that enables one to translate rational descent statements as above into descent statements after telescopic localization. As a result, we prove various descent results in the telescopically localized $K$-theory, $TC$, etc. of ring spectra, and verify several cases of a conjecture of Ausoni-Rognes. This is joint work with Dustin Clausen, Niko Naumann, and Justin Noel.
4 Apr 2016
The eta-local motivic sphere
The Hopf map eta is nilpotent in the stable homotopy groups of spheres. This is not so for the motivic Hopf map, considered as an element of the motivic stable homotopy groups of spheres. This suggests that the eta-local part of motivic stable homotopy theory is an interesting object of study. We will discuss joint work with Dan Isaksen which describes this for the base fields C and R.
11 Apr 2016
Constructing equivariant spectra
Equivariant spectra determine cohomology theories that
incorporate a group action on spaces. Such spectra are increasingly
important in algebraic topology but can be difficult to understand or
construct. In recent work, Angelica Osorno and I have developed a construction
for building such spectra out of purely algebraic data based on symmetric
monoidal categories. Our method is philosophically similar to classical
work of Segal on building nonequivariant spectra. In this talk I will
discuss an extension of our work to the more general world of Waldhausen
categories. Our new construction is more flexible and is designed to be
suitable for equivariant algebraic K-theory constructions.
18 Apr 2016
Marc Stephan, University of Chicago
Stable 2-types and categorical suspension
Joint work with Nick Gurski, Niles Johnson and Angélica Osorno. Recently, these three have established the 2-dimensional Stable Homotopy Hypothesis: every stable 2-type is modeled by a Picard 2-category. A Picard category is a symmetric monoidal category in which each object and each morphism is invertible, and a Picard 2-category is a 2-categorical analogue. A stable 2-type is a spectrum whose nth stable homotopy groups vanish for n different from 0, 1 and 2; in other words a connective spectrum with a 3-stage Postnikov tower. In practice, one would like to model the Postnikov data categorically as well, and in particular one would like to construct a nice model of the 2-type of the sphere spectrum.
In the 1-dimensional situation, every Picard category is equivalent to one which is strict and skeletal, and there is a nice model for the stable 1-type of the sphere spectrum. Contrary to this situation, we prove that there is no strict, skeletal model for the stable 2-type of the sphere spectrum by analyzing the Postnikov data. In the course of the proof, we establish a result of independent interest: considering a symmetric monoidal category as a one-object symmetric monoidal bicategory models the suspension on K-theory spectra.
— Fall 2016 —
12 Sept 2016
Alexander Campbell, Macquarie University
A coherent approach to 2-stacks
The category of 2-categories and 2-functors admits a monoidal model structure, due to Gray and Lack, in which the weak equivalences are the biequivalences. Categories enriched over this monoidal category, known as Gray-categories, were shown by Gordon, Power, and Street to model all tricategories (i.e. weak 3-categories). Indeed, one often finds that a tricategory of interest is triequivalent to the full sub-Gray-category of a Gray-enriched model category on the cofibrant-fibrant objects. Moreover, the tricategorical analogues of, inter alia, limits, colimits, and image factorisation systems, can be modelled by strict constructions of Gray-enriched category theory, in a fashion consonant with the definitions of enriched model category theory.
In this talk I will survey the results of the coherence theory of tricategories which I used and developed in my recent PhD studies on the higher categorical approach to non-abelian cohomology, which takes higher stacks as the coefficient objects. In particular, I will show how these results can be applied to prove that the tricategorical analogue of Grothendieck's plus construction for associated sheaves preserves finite tricategorical limits.
There will be no seminar for the rest of September. Mona will be away, but Emily will be organizing several category theory activities during that period -- check them out on her webpage.
3 Oct 2016
Quantum Field Theory As Positive Geometry:
Scattering Amplitudes And The Amplituhedron
Scattering amplitudes are a central observable in
fundamental physics. They are the backbone of experiments at giant
particle accelerators like the Large Hadron Collider. They have also
played a central role in the development of quantum field theory,
going back to the work of Schwinger, Feynman and Dyson nearly 70 years
ago. Recent years have revealed extremely surprising and deep hidden
simplicity and infinite-dimensional symmetries in gauge theory
scatteirng ampltiudes. These structures are completely obscured in the
conventional formulation of field theory
using Feynman diagrams. This suggests the existence of a new
understanding for scattering amplitudes where the usual rules of
space-time and quantum mechanics--locality and unitarity--
do not play a central role, but are derived consequences from a
different starting point. In this talk I will present such an
understanding for maximally supersymmetric scattering amplitudes,
which are identified as ``the volume" of a new
mathematical object--the Amplituhedron--generalizing the familiar
notion of convex polygons into the (positive) Grassmannian. I will
describe how locality and unitarity emerge hand-in-hand from positive
geometry
10 Oct 2016
No seminar. FALL COLLOQUIM.
17 Oct 2016
Computing Polynomial Approximations of Atomic Functors
A functor from finite sets to chain complexes is called atomic if it is completely determined by its value on a particular set. In this talk, we present a new resolution for these atomic functors, which allows us to easily compute their Goodwillie polynomial approximations. By a rank filtration, any functor from finite sets to chain complexes is built from atomic functors. Computing the linear approximation of an atomic functor is a classic result involving partition complexes. Robinson constructed a bicomplex, which can be used to compute the linear approximation of any functor. We hope to use our new resolution to similarly construct bicomplexes that allow us to compute polynomial approximations for any functor from finite sets to chain complexes.
24 Oct 2016
Asymptotic representation theory over Z
Representation theory over Z, i.e. the study of abelian groups with an action of a finite group G, is famously intractable. But "representation stability" provides a way to productively study asymptotic representation theory over Z, i.e. the study of sequences of abelian groups $V_n$ with an action of $G_n$, and specifically those aspects that persist for large n.
This has many applications in topology, but has also led to breakthroughs in other fields such as generic representation theory. I'll give a high-level overview of these techniques, including how important structural theorems in topology are reduced to "commutative algebra"-style questions in representation theory.
31 Oct 2016
Chromatic homotopy theory and certain perfectoid fields
At each prime p there is a complete algebraically closed topological
field $\mathbb{C}_p$, isomorphic to the classical complex field $\mathbb{C}$. Recent work
of Bhatt, Scholze, Hesselholt and others has made geometry over $\mathbb{C}_p$
accessible, and their methods seem applicable to interesting subfields $L_\infty$ of $\mathbb{C}_p$, defined by completing maximal totally ramified abelian
extensions of local number fields L.
In particular, work of Hesselholt suggests the existence of analogs
$k(L) \to THH(\mathcal{O}_{L_\infty},\mathbb{Z}_p)$
of the classical K-theory Chern character, defined on variants (indexed
by local fields) of the quotient of complex cobordism with homotopy
groups $\mathbb{Z}_p[v_n]$, with values in the topological Hochschild homology
of the valuation ring of $L_\infty$. This in turn suggests connections
between chromatic homotopy theory and geometry over the perfectoid
fields $L_\infty$.
Relevant preprint
7 Nov 2016
The homology of the Higman-Thompson groups
The symmetric groups, the general linear groups, and the automorphism groups of free groups are examples of families of groups that arise as symmetry groups of algebraic structures but that are also dear to topologists. There are many other less obvious examples of interest. For instance, in joint work with Nathalie Wahl, this point of view has led to a computation of the homology of the Higman-Thompson groups. It is also key to progress on groups related to mapping class groups and braid groups that I would like to present in this talk.
14 Nov 2016
More stable stems
I will describe a joint project with Guozhen Wang and Zhouli Xu to compute stable homotopy groups at the prime 2.
The cofiber of tau is a curious motivic spectrum that allows us to combine information from the Adams spectral sequence and from the Adams-Novikov spectral sequence in ways that were previously not possible. The story begins with the surprising observation that the Adams spectral sequence for the cofiber of tau is the same as the algebraic Novikov spectral sequence. These spectral sequences have been computed by machine in a large range.
From this information about the Adams spectral sequence for the cofiber of tau, we can deduce information about the Adams spectral sequence for the sphere by passing to the top and bottom cells.
This method yields almost the entire structure of the Adams spectral sequence in a large range. It makes the computation of the first 60 stems essentially routine. So far, we have extended computations to the 70 stem, with ongoing progress into higher stems.
21 Nov 2016
THANKSGIVING BREAK
28 Nov 2016
Functor Precalculus
Functor calculi have been developed in a variety of forms and contexts. These include Goodwillie's calculus of homotopy functors, Weiss' orthogonal calculus, the manifold calculus of Goodwillie and Weiss, and the discrete calculi for abelian and simplicial model categories. Each of these calculi comes equipped with its own definition of polynomial or degree $n$ functor. Such definitions are often formulated in terms of the behavior of the functor on certain types of cubical diagrams. Using the discrete calculus as a starting point, we identify a category-theoretic framework, which we call a precalculus, that provides a means by which notions of degree for functors can be defined via cubical diagrams. We show how such precalculi can be used to produce functor calculi. As part of this talk we will include examples of precalculi associated to both known and perhaps new functor calculi.
This is work in progress with Kathryn Hess.
5 Dec 2016
Dg-manifolds as a model for derived manifolds:
Given two smooth maps of manifolds f:M \to L and g:N \to L, if they are not transverse, the fibered product M \times_L N may not exist, or may not have the correct cohomological properties. Thus lack of transverality obstructs many natural constructions in topology and differential geometry. Derived manifolds generalize the concept of smooth manifolds to allow arbitrary (iterative) intersections to exist as smooth objects, regardless of transversality. In this talk we will describe recent progress of ours with D. Roytenberg on giving an accessible geometric model for derived manifolds using differential graded manifolds.
— Spring 2017 —
6 Feb 2017
Real Johnson-Wilson theories: computations toward applications
Complex cobordism and its relatives, the Johnson-Wilson theories, E(n), carry an action of C_2 by complex conjugation. Taking fixed points of the latter yields Real Johnson-Wilson theories, ER(n). They are generalizations of real K-theory and are similarly amenable to computations. We will describe their properties, survey recent work on the ER(n)-cohomology of some well-known spaces, and describe how this brings new information to bear on the immersion problem for real projective spaces. This is joint work with Nitu Kitchloo and W. Stephen Wilson.
13 Feb 2017
Keith Pardue, NSA
Products in a Category With One Object
A category with one object is, of course, a monoid. For the one object to admit a product with itself, the monoid requires an extra binary operation with properties that are not difficult to write down. We call these monoids Categorical Product (CP) Monoids. The monoid of endomorphisms of an infinite dimensional vector space is an example of such a monoid. We construct a universal CP monoid U — an initial object in the category of CP monoids — that is a submonoid of every nontrivial CP monoid. We further explore the rich combinatorial structure of U and show that every finite monoid injects into U.
20 Feb 2017
The Lubin-Tate Theory of $K(n)$-local Lie Algebras
Quillen models the rational homotopy type of spaces by d.g. Lie algebras over $\mathbb{Q}$. Recent work of Behrens-Rezk and Heuts considers $K(n)$-local Lie algebras as a modular generalisation. The homotopy groups of these Lie algebras are out of computational reach.
We study a more accessible variant, namely Lie algebras in complete $O_D^x$-equivariant module spectra over Lubin-Tate space. These also appear as the basic building blocks in the chromatic Goodwillie spectral sequence.
We compute the operations which act on the homotopy groups of said Lie algebras. For this, we apply a new general technique for intertwining unstable power operations with operadic Koszul duality and use discrete Morse theory to study the equivariant topology of the partition poset. This also yields a short and purely combinatorial proof of an old theorem of Goerss on the cotangent complex in derived algebraic geometry.
27 Feb 2017
Detecting periodicity in iterated K-theory
Classically, due to Adams and Quillen, algebraic K-theory of finite fields is known to detect the alpha family, a periodic family of height one in the homotopy groups of spheres. The red-shift conjecture of Ausoni and Rognes suggests that iterated algebraic K-theory of finite fields should contain height two information. In my talk, I will show that the beta family, a periodic family of height 2, is detected in iterated algebraic K-theory of finite fields of order q, under conditions on p and q. I will also describe some evidence that suggests that iterated algebraic K-theory of finite fields may not have telescopic complexity two, so other versions of the red-shift conjecture may not hold.
6 Mar 2017
Motivic modular forms
Motivated by the study of chromatic phenomenon in the classical and motivic Adams spectral sequence, we set up a machinery to build a spectrum (over $Spec(R)$ or $Spec(C)$) of motivic modular forms (mmf), that is, a ring spectrum whose cohomology is $A//A(2)$. This answers positively a conjecture made by Dan Isaksen.
The approach we suggest makes a detour by $C_2$-equivariant stable homotopy theory, and uses the proximity between the equivariant and motivic Steenrod algebra, a relationship which is not shared by the classical Steenrod algebra.
If time permits, we will talk about uniqueness of such spectra, and the chromatic consequences of mmf.
13 Mar 2017
Nilpotence and Perdiodicity in Motivic Homotopy Theory
Classically, it is proven by Devinatz-Hopkins-Smith that the complex cobordism spectrum $MU$ detects nilpotence. Equivalently, one can use the whole family of Morava K-theories $K(n)$ as their wedge sum sits in the same Bousfield class as $MU$. In the motivic setting one has analogous spectra, the algebraic cobordism spectrum is denoted by MGL and the motivic Morava K-theories by $K(n)$. These spectra fail to detect nilpotence, for example because they do not see the periodic Hopf map eta. In this talk we will construct a motivic field spectrum $K(\eta)$ that does detect eta, as well as discover a whole new infinite family of motivic periodicities denotes by $w_n$. A central tool that we will explore is a motivic 2-cell complex known by the name of the cofiber of tau.
20 Mar 2017
SPRING BREAK
27 Mar 2017
OPEN HOUSE FOR INCOMING GRAD STUDENTS
3 Apr 2017
Operads with Homological Stability
For a carefully constructed operad M of surfaces, Tillmann showed that algebras over M group complete to infinite loop spaces. This result relies, in part, on Harer's homological stability theorem for mapping class groups of surfaces. We will review this theorem and provide a more general framework which shows that operads satisfying a certain homological stability condition detect infinite loop spaces. This is joint work with Maria Basterra, Irina Bobkova, Kate Ponto, and Ulrike Tillmann.
10 Apr 2017
The Homotopy Theory of (Genuine) Equivariant Operads
The study of equivariant operads is enlivened by the existence of multiple distinct generalizations of $E_\infty$-operads, recording which norm maps are available to algebras. These are detected by fixed-point conditions over the graph subgroups of $G\times \Sigma_n$, or equivalently by a certain systems of finite $H$-sets. In this talk, we will encode this complexity homotopically. Specifically, we will define the new combinatorial gadget of $G$-trees, and show that they inspire a more general algebraic structure, called genuine equivariant operads. We will further show that, for any indexing system $\F$, the two categories of genuine and non-genuine $G$-operads can be equipped with appropriate $\F$-model structures, which are in fact Quillen equivalent.
17 Apr 2017
Lifting Motivic Measures
In this talk I'll define a spectral analogue of the cut-and-paste Grothendieck ring of varieties, $K_0 (Var)$. The construction of such a spectrum depends on a modification of Waldhausen's $S_\dot$ construction. With this in hand, we can construct spectral lifts of interesting ring maps $K_0 (Var) \rightarrow R$ (alias "motivic measures"). In particular, I'll discuss a lifting of the zeta function, which is joint work with Jesse Wolfson and Inna Zakharevich.
24 Apr 2017
The positive geometry of Wilson loop diagrams
In this talk, I introduce a class of diagrams (combinatorial objects) called Wilson Loop Diagrams. These diagrams are the analog of Feynman diagrams in the supersymmetric Yang Mills theory, SYM N=4. I then show that these diagrams correspond to cells of a CW complex of the positive Grassmannians, $Gr_+(k,n)$. I spend what time remains exploring the geometry of the complex labeled by these graphs.
1 May 2017
Binomial Coefficients, Koszul Homology, and Siegel Modular Forms
First, I will describe a new result on the arithmetic of binomial coefficients (in this month’s issue of the Amer. Math. Monthly
https://dx.doi.org/10.4169/amer.math.monthly.124.4.353), speculatively related to the homotopy of MO
for k>8. Second, I will outline a new proof that, unlike most familiar spectra, tmf is not a ring spectrum quotient – by relating the homotopy of such a quotient to Koszul homology. Third, I will describe ongoing work to identify cobordism invariants of Cayley plane bundles with cusps of certain genus-2 Siegel modular forms, equivalently degenerations in compactifications of certain moduli spaces of genus-2 surfaces.
— Fall 2017 —
4 Sept 2017
11 Sept 2017
2-Segal Spaces
Dyckerhoff-Kapranov and independently Galvez-Kock-Tonks observed that many simplicial spaces arising in different contexts fail to be Segal spaces, but still satisfy a related condition, now called "2-Segal condition". In particular, they observed that the result of a certain Waldhausen S_{\bullet}-construction for exact categories has this property. In a joint work with Bergner, Osorno, Rovelli, Scheimbauer, we slightly modified this construction to see that any unital 2-Segal set arises by this construction. In an ongoing project, we are promoting this result to a Quillen equivalence for unital 2-Segal spaces.
18 Sept 2017
Characteristics of Ring Spectra and a Canonical Construction of MU by Versal Algebras
I'll describe classes in the 2n-1th homotopy groups of the X(n) spectra and show that they are generators. I will describe some theorems that imply that X(n+1) is a versal X(n) algebra (which I will define!). If there is time, I will say what these classes have to do with the Greek letter classes (roughly) and how the above construction can be used (along with topological Hochschild cohomology) to obtain canonical E_1 complex orientations of ring spectra whose homotopy is concentrated in even degrees.
25 Sept 2017
Infinite Loop Spaces in Algebraic Geometry
A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. The analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception.
In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K-theoretic trivialization of its contangent complex. I will explain what this means and how it is not so different from finite pointed sets. In particular, I will explain all the requisite algebraic geometry.
Time permitting, I will also provide an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme.
2 Oct 2017
Localizing the $E_2$ page of the Adams spectral sequence
The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for localizing its $E_2$ page by the non-nilpotent element $b_{10}$ at the prime 3; this localization is isomorphic to the $E_2$ page itself in an infinite region. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.
9 Oct 2017
On negative K-theory, bounded t-structures, and the theorem of the heart
After recalling the definition of negative K-theory, we will show that the negative K-groups of a stable infinity category vanish whenever the stable infinity category supports a bounded t-structure with noetherian heart. We will then use this to extend the Barwick-Neeman "theorem of the heart" to negative K-theory, use this to compute the negative K-theory of certain ring spectra, and demonstrate how negative K-groups obstruct the existence of bounded t-structures. This is joint work with B. Antieau and J. Heller.
16 Oct 2017
Factorization homology and topological Hochschild cohomology of Thom spectra
By a theorem of Lewis, the Thom spectrum of an n-fold loop map to BO is an E_n-ring spectrum. I will discuss a project studying the factorization homology and the E_n topological Hochschild cohomology of such Thom spectra, and talk about some applications, such as computations, and a duality between topological Hochschild homology and cohomology of certain Thom spectra. Time permitting, I will discuss connections to topological field theories. This talk will include an introduction to factorization homology.
23 Oct 2017
Notes on the margins of E-theory
The deformation space of a height n formal group over a finite field has an exact interpretation into homotopy theory, in the form of height n Morava E-theory. The K(t)-localizations of E-theory, for t < n, force us to contend with the margins of the deformation space, where the formal group's height is allowed to change. We present a modular interpretation of these marginal spaces, and discuss applications to homotopy theory.
30 Oct 2017
A theory of elementary higher toposes
Topos theory was developed by Grothendieck in order to be able to study schemes in their proper categorical context, by applying the language of sheaves. Later people realized that it can be generalized to a theory of elementary toposes, which allow us to study sets from a categorical perspective and even give a categorical characterization of sets.
The theory of Grothendieck toposes was later generalized by Lurie and Rezk to the context of higher categories, giving rise to higher toposes. This theory has been successfully used in derived algebraic geometry. However, as of now, we still lack an analogous theory of elementary higher toposes.
The goal of this is to introduce such a definition of elementary higher toposes and show this definition satisfies some common sense conditions that we would expect of any such theory.
6 Nov 2017
Homology of infinite loop spaces
We compare two different methods to compute mod 2 homology of an infinite loop space. The first comes from studying the Adams resolution and that the homology of a space should be a coalgebra. The second comes from studying the Goodwillie tower for infinite loops and that the homology will form an algebra.
13 Nov 2017
The Dold--Thom theorem via factorization homology
The Dold--Thom theorem is a classical result in algebraic topology giving isomorphisms between the homology groups of a space and the homotopy groups of its infinite symmetric product. The goal of this talk is to outline a new proof of this theorem, which is direct and geometric in nature. The heart of this proof is a hypercover argument which identifies the infinite symmetric product as an instance of factorization homology.
4 Dec 2017
Topological Equivalences of $E_\infty$ DGAs
In this talk, I present an idea for studying $E_\infty$ differential graded algebras ($E_\infty$ DGAs) using stable homotopy theory. Namely, I discuss new equivalences between $E_\infty$ DGAS that are defined using commutative ring spectra.
We say $E_\infty$ DGAs are $E_\infty$ topologically equivalent when the corresponding commutative
ring spectra are equivalent. Quasi-isomorphic $E_\infty$ DGAs are $E_\infty$ topologically equivalent. However, the examples I am going to present show that the opposite is not true; there are $E_\infty$ DGAs that are $E_\infty$ topologically equivalent but not quasi-isomorphic. This says that between $E_\infty$ DGAs, we have more equivalences than just the quasi-isomorphisms.
I also discuss interaction of $E_\infty$ topological equivalences with the Dyer-Lashof operations and cases where $E_\infty$ topological equivalences and quasi-isomorphisms agree.
— Spring 2018 —
29 Jan 2018
Real Orientations of Lubin-Tate Spectra
We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is an application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points.
5 Feb 2018
Milnor Operations in Equivariant Homotopy Theory
I will discuss work, joint with Dylan Wilson, defining equivariant analogues of the Milnor primitives in the odd-primary Steenrod algebra. Along the way, we develop an analogue of the Hopkins-Mahowald theorem that the free $E_2$ algebra with $p=0$ is $HF_p$
12 Feb 2018
Group Homology via Morita Theory
Morita theory was designed to address the question: When do two rings have equivalent categories of modules? Later, it became a jumping-off point to the theory of noncommutative motives, which provides a universal home for ring homology theories (including K-theory and Hochschild homology). Even more recently, noncommutative motives have been identified with stable infinity categories. We will survey this beautiful story, and then discuss a variant for group homology. The analogues of stable infinity categories are locally cartesian closed categories, and of the Morita category is the Burnside category. Morita theory in this context provides a duality between genuine and naive equivariant homotopy theory.
19 Feb 2018
Manifold Calculus and H-principle
In this talk, I'll explain the connection between Manifold Calculus (Goodwillie calculus for manifolds) and Gromov's h-principle. Manifold Calculus is a homotopy theoretic technique for studying embedding spaces of manifolds, and Gromov's h-principle is a classical tool for finding solutions to partial differential relations. I'll explain how h-principle can be used to extend Manifold Calculus to manifolds with tangential structures and present some of its applications.
26 Feb 2018
The character of the total power operation
In the 90's Goerss, Hopkins, and Miller proved that the Morava E-theories are $E_\infty$-ring spectra in a unique way. Since then several people including Ando, Hopkins, Strickland, and Rezk have worked on explaining the effect of this structure on the homotopy groups of the spectrum. In this talk, I will present joint work with Barthel that shows how a form of character theory due to Hopkins, Kuhn, and Ravenel can be used to reduce this problem to a combination of combinatorics and the $GL_n(Q_p)$-action on the Drinfeld ring of full level structures which shows up in the local Langlands correspondence.
5 Mar 2018
The Waldhausen S-construction as an equivalence of homotopy theories
The notion of unital 2-Segal space was defined independently by Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks as a generalization of a category up to homotopy. The notion of unital 2-Segal space was defined independently by Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks as a generalization of a category up to homotopy. A key example of both sets of authors is that the output of applying Waldhausen's S-construction to an exact category is a unital 2-Segal space. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we expand the input of this construction to augmented stable double Segal spaces and prove that it induces an equivalence on the level of homotopy theories. Furthermore, we prove that exact categories and their homotopical counterparts can be recovered as special cases of augmented stable double Segal spaces
(Friday!) 9 Mar 2018
The Johnson filtration is finitely generated
A recent breakthrough of Ershov-He shows that the Johnson
kernel subgroup of the mapping class group is finitely generated for g
at least 12. In joint work with Ershov and Church, I have extended
this to show that every term of the lower central series of the
Torelli group is finitely generated once the genus is sufficiently
large. A byproduct of our work is a proof that the Johnson kernel is
finitely generated for g at least 4 which is remarkably simple (so
simple, in fact, that I will be able to give it in nearly complete
detail in this talk).
26 Mar 2018
Stable module categories as categorified Tate cohomology
Tate cohomology is an important tool in group cohomology, and also features prominently in stable homotopy theory. It is closely related to the stable module category studied in modular representation theory. For example, Tate cohomology becomes corepresentable in the stable module category. I will explain this fact by exhibiting a more fundamental connection between the two concepts: the stable module category itself arises as a kind of Tate construction in stable $\infty$-categories. In particular, this provides a definition of the stable module category for ring spectra, suggesting a variety of interesting follow-up questions in homotopy theory. This is joint work with Aaron Royer and Saul Glasman.
2 Apr 2018
The Topological Hochschild Homology of Maximal Orders in Simple Q-Algebras
This is joint work with Ayelet Lindenstrauss. First, I will introduce topological Hochschild homology and do some survey calculations. And I'll go over the calculations of topological Hochschild homology pf maximal orders in simple Q-algebras.
9 Apr 2018
Spanier-Whitehead dual of TMF at p=2
I will introduce chromatic homotopy theory which uses Bousfield localization with respect to Morava K-theories K(n) to filter the category of spectra. This filtration by height allows us to simplify calculations of stable homotopy groups of spheres by working one prime and one chromatic height at a time. I will introduce the main tools from number theory that help with these computations.
Then I will talk specifically about current work at chromatic height 2 and describe how the sphere at height 2 can be decomposed in terms of spectra related to the spectrum of topological modular forms TMF. I will talk about computing the Spanier-Whitehead dual of TMF and describe how this is useful for understanding the K(2)-local sphere.
16 Apr 2018
On a $(\infty,2)$-category of homotopy coherent adjunctions in an $\infty$-cosmos.
Riehl and Verity initiated a program to study the category theory of $(\infty, 1)$-categories in a model-independent way, through the study of $\infty$-cosmoi.
Homotopy coherent monads in an $\infty$-cosmos are of particular interest since they determine an Eilenberg-Moore object of (homotopy coherent) algebras.
In this talk, I will generalize the graphical calculus of Riehl and Verity to provide a combinatorial description of the simplicial categories $\mathbf{Adj}_{hc}[n]$ and $\mathbf{Mnd}_{hc}[n]$.
They encode homotopy coherent diagrams of homotopy coherent adjunctions and monads of the shape of the simplex $\Delta[n]$, and in particular for $n=1$, provide appropriate definitions for homotopy coherent morphisms of adjunctions and monads.
I will briefly introduce weak complicial sets and show that the nerve induced by $\mathbf{Adj}_{hc}[- ]: \Delta \to \mathrm{sSet-Cat}$ gives rise to an $(\infty,2)$-category of homotopy coherent adjunctions. If time permits, I will discuss a related conjecture for homotopy coherent monads.
23 Apr 2018
About Bredon motivic cohomology of a field
We introduce Bredon motivic cohomology of an Z/2-equivariant smooth scheme and show that complexes of equivariant equidimensional cycles compute this cohomology. We use this and other methods to identify the Bredon motivic cohomology of a field in weight 0 and 1 as well as the Bredon motivic cohomology of the field of complex numbers. This is a joint work with J. Heller and P.A. Ostvaer.