26 Nov 2018
3 Dec 2018
The Linearization Conjecture
For G a nice profinite group (such as the Morava stabilizer groups), I will discuss the construction of a p-adic sphere which comes equipped with a natural action of G. The linearization conjecture predicts that this sphere is a kind of one point compactification of the p-adic Lie algebra of G. I will explain how to show that this holds when the action is restricted to certain finite subgroups of G and discuss an application to chromatic homotopy theory.
Spring 2014
16 Sept 2013
A Classifying Space for Commutative in Lie Groups
The goal of this talk is to study the classifying space for commutative in a topological group
G, B
comG. Such spaces are assembled from the different spaces of commuting elements in
G. In particular we show the geometric role that these spaces play in terms of transitionally commutative bundles. Some cohomological computations will be provided for the classical Lie groups SU(
n), U(
n) and Sp(
n). A decomposition as homotopy colimits will be provided for these spaces for the case of compact Lie groups and commutative K-theory is also introduced. This is a joint work with
Alejandro Adem.
30 Sept 2013
Generalizations of Milnor’s Invariants
In the 50s
Milnor defined invariants of multi-component linked circles in three-space as a generalization of the linking number in order to capture higher-order linking phenomena not detected by the linking number, such as the linking of the
Borromean rings. In the early 90s
Koschorke generalized Milnor’s invariants to linking of higher-dimensional spheres in higher-dimensional Euclidean spaces. The invariants are no longer numbers, but rather elements of the stable homotopy groups of spheres. I will present a generalization of Koschorke’s work to linking of arbitrary manifolds in Euclidean space. Here the invariants are points in the zero space of a Thom spectrum, are induced by a map of spaces rather than defined on the group level, and they are a part of an EHP-like sequence which naturally arises in the study of the Taylor tower of the identity functor in
Goodwillie’s homotopy calculus. A key feature of our generalization is that the invariants are “multirelative”. In particular, the classical linking number is really a relative invariant. Time permitting, geometric interpretations of the invariants will be discussed.
21 Oct 2013
Persistent Homology of Time-Delay Embeddings
Persistent homology is a topological method for measuring the shapes of spaces and features of functions. One of its most important applications is to point clouds, where shape is usually interpreted as the geometry of some implicit underlying object near which the point cloud is sampled. Time-delay embeddings, on the other hand, have been used mostly in the study of time series and dynamical systems to understand the nature of their attractors. In this talk we analyze the geometry and topology of time-delay embeddings through the lens of persistent homology. In particular, we study maximum persistence as a measure of periodicity at the signal level, present structural theorems for the resulting diagrams, and derive estimates for their dependency on the window size and embedding dimension. Time permitting, some biological applications will be presented.
28 Oct 2013
The Real K-Theory of Compact Lie Groups
In this talk I will first briefly review the previous work on the complex K-theory and
Atiyah’s Real K-theory of compact Lie groups. Then I will present a complete description of the ring structure of the equivariant Real K-theory of any compact, connected and simply-connected Lie group equipped with a Lie group involution. Along the way I will introduce the notion of Real equivariant formality, which is an important ingredient in obtaining the ring structure. Some applications and examples will also be discussed.
4 Nov 2013
Exotic Elements in the K(n)-Local Picard Group
Given a symmetric monoidal category we can study the group of invertible objects, known as the Picard group. For example the Picard group of the stable homotopy category is just the integers, generated by S
1. The situation is more interesting when we consider the K(
n)-local Picard group, where K(
n) is the
n-th
Morava K-theory. I will review the basic theory, as well as outline work in progress in constructing so called ‘exotic’ elements of the Picard group at height
n=
p-1.
11 Nov 2013
Determinantal K-Theory and a Few Applications
Chromatic homotopy theory is an attempt to divide and conquer algebraic topology by studying a sequence of what we’d first assumed to be “easier” categories. These categories turn out to be very strangely behaved—and furthermore appear to be equipped with intriguing and exciting connections to number theory. I’ll describe the most basic of these strange behaviors, then I’ll describe an ongoing project which addresses a small part of the “chromatic splitting conjecture”.
18 Nov 2013
Unitary Embeddings of Finite Loop Spaces
In this talk I will discuss the existence of unitary embeddings for homotopical analogues of compact Lie groups, such as finite loop spaces and
p-compact groups. The fusion systems of these objects are used to build suitable faithful representations of their Sylow subgroups, and obstruction theory to study whether they lift to unitary embeddings. Some examples and consequences will be provided. This is joint work with
Natàlia Castellana.
25 Nov 2013
A New Proof of Strickland’s Theorem
Strickland proved that the scheme that classifies subgroups of order p
k of the formal group associated to
Morava E-theory is corepresented by the Morava E-theory of the symmetric group S
pk (modulo a transfer ideal). In this talk we will apply the transchromatic generalized character maps to provide a new proof of this theorem. This is joint work with
Tomer Schlank.
2 Dec 2013
Internal Languages for Higher Categories
Every category
C looks locally like a category of sets and further structure on
C determines what logic one can use to reason about these “sets”. For example, if
C is a topos, one can use full (higher order) intuitionistic logic. Similarly, one expects that every ∞-category looks locally like an ∞-category of spaces. A natural question then is: what sort of logic can we use to reason about these “spaces”? It has been conjectured that such logics are provided by variants of Homotopy Type Theory, a formal logical system, recently proposed as a foundation of mathematics by
Vladimir Voevodsky. After explaining the necessary background, I will report on the progress towards this conjecture.
4pm Thurs 5 Dec 2013
Three Hopf Algebras and Their Common Algebraic and Categorical Background
We discsuss the renormalization Hopf algebra of
Connes and
Kreimer,
Gontcharov’s Hopf algebra for multi-zeta values and the Hopf algebra appearing in
Baues’ double cobar construction. We show that these are a examples of a common algebraic framework. Moreover this framework is a manifestation of one of the properties of Feynman categories, which we briefly define and discuss at the end. These are a categorical universal frameworks for operad-like structures.
27 Jan 2014
Sheaves, Cosheaves and Topological Data Analysis
Sheaves and cosheaves, broadly interpreted, are data management tools with a local-to-global principle. A combinatorial description of (co)sheaves in terms of representations of a particular category (the entrance/exit path category) have enabled a streamlined development of applied (co)sheaf theory. These applications include, among many things, sensor networks, network coding/optimization problems, and topological data analysis (TDA). In this talk, I will focus on how sheaves and cosheaves offer insight into TDA, especially in level-set persistence. I will introduce an extended metric on the category of (co)sheaves, and explain how a derived perspective offers stability results.
10 Feb 2014
Bousfield Localization and Algebras over Operads
We give conditions on a monoidal model category
M and on a set of maps
S so that the
Bousfield localization of
M with respect to
S preserves the structure of algebras over various operads. 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 commutativity. As a special case of our general machinery we characterize which localizations preserve genuine equivariant commutativity. Our results are general enough to hold for non-cofibrant operads as well, and we will demonstrate this via a treatment of when localization preserve strict commutative monoids. En route we will introduce the commutative monoid axiom, which guarantees us that commutative monoids inherit a model structure. If there is time we will say a word about the generalizations of this axiom to other non-cofibrant operads, and about how these generalized axioms interact with Bousfield localization.
17 Feb 2014
GRT-Equivariance of Tamarkin’s Construction
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-Teichmüller group.
24 Feb 2014
Recent Developments in Representation Stability
I will give a gentle survey of the theory of representation stability, viewed through the lens of its applications. I will focus on the recent “second wave” of applications outside the domain of classical representation theory. These applications include: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; uniform generators for congruence subgroups and “congruence” subgroups; and distributional stability for random squarefree polynomials over finite fields.
3 Mar 2014
Positive Scalar Curvature and Twisted Spin Cobordism
The basic question for my talk is whether a closed manifold admits a metric of positive scalar curvature and I will explain how this question reduces to calculations in certain cobordism rings, due to a result of
Gromov and
Lawson. Following an overview of these calculations in the case of Spin-cobordism, where they were carried out by
Stolz and
Führing, I want to address their generalisation to the case of twisted Spin-cobordism, which is ongoing joint work of
Joachim and myself. In particular I will exhibit a generalisation of the
Anderson-
Brown-
Peterson splitting and compute the mod 2 cohomology of the twisted, connective, real K-theory spectrum.
10 Mar 2014
Counting Curves and the Euler Class
We shall discuss the enumerative problem of counting the number of complex curves (in complex projective space of dimension 2) which pass through the requisite number of generic points and has a prescribed singularity at one point. Our exposition will be from a topological point of view via the Euler class. This is joint work with
Ritwik Mukherjee.
Thurs 27 Mar 2014
in Remsen 101
A Manifestation of the Grothendieck-Teichmüller Group in Geometry
Inspired by Grothendieck’s lego-game,
Vladimir Drinfeld introduced, in 1990, the Grothendieck-Teichmüller group GRT. This group has interesting links to the absolute Galois group of rationals, moduli of algebraic curves, solutions of the
Kashiwara-
Vergne problem, and theory of motives. My talk will be devoted to the manifestation of GRT in the extended moduli of algebraic varieties, which was conjectured by
Maxim Kontsevich in 1999. My talk is based on
the joint paper with
Chris Rogers and
Thomas Willwacher.
7 Apr 2014
An Approach to the Homotopy Groups of Behrens’ Spectrum Q(2) at the Prime 3
In this talk I will describe an approach to computing the homotopy groups of a spectrum
Q(2), originally constructed by
Mark Behrens in an effort to clarify and extend previous work on the 3-primary
K(2)-local sphere by
Shimomura, Yabe,
Goerss,
Henn,
Mahowald, and
Rezk. The spectrum
Q(2) is built using degree 2 isogenies of elliptic curves (hence the “2” in its name) and spectra related to TMF, and as such, is conjectured to have interesting number-theoretic properties that we will discuss. Finally, we will point out potential connections between
Q(2) and the beta family in the 3-primary stable stems.
14 Apr 2014
Counting Electronic Excitations in Organic Systems Using Algebraic Topology
Excited electrons in organic semiconductors and insulators form metastable states, known as excitons. Excitons arise in many different physical and optical phenomena, such as photosynthesis. We formulate solutions to the Exciton Scattering equations (excitons) in terms of an intersection problem, and apply an index theorem to obtain a lower bound on the number of such excitons in a given system. This lower bound becomes exact when the molecule has long enough ‘arms’, or large translational symmetry.
21 Apr 2014
Towards a Cyclotomic Hodge Filtration
Loday’s Hodge filtration on the algebraic
Hochschild homology of a commutative ring, a relative of the classical Hodge filtration on the de Rham complex of a smooth algebra, is analogous to the weight filtration on algebraic K-theory: the Adams operations act by prescribed scalars on its graded pieces. We’ll describe how to generalize this to a filtration by cyclotomic spectra of the topological Hochschild homology spectrum of an E
∞-ring spectrum
R. We’ll explain how this might ease calculations with topological cyclic homology and help us understand the coherent algebraic structures on THH and K-theory.
28 Apr 2014
Gauged Sigma Models and Equivariant Elliptic Cohomology
The universal elliptic cohomology theory, TMF, is expected to have a rich equivariant refinement related to categorified groups called 2-groups. I will start by defining some flavors of 2-group representation theory in the language of field theories, drawing on the toolbox developed for Chern-
Simons theory of a finite group. This perspective leads to an intriguing relationship between characters of 2-group representations and a geometric model of equivariant TMF over the complex numbers built from the geometry of gauged sigma models.
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∗(v1-1S/p) explicitly. As a result of his work we understand the Adams spectral sequence for the Moore spectrum above a line of slope
1/(p2–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-1π∗(S0)=0 but motivically there is an interesting question. We will compute
η-1π∗,∗(S0,0) 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 E2-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 EhG-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 Mp(i) (the cofiber of the pi self-map on the sphere spectrum S0), especially when i>1. Stasheff developed a hierarchy of coherence for homotopy associative multiplications called An structures. The only known results are that Mp(1) is Ap-1 and not Ap and that M2(i) are at least A3 for i>1. In this talk, techniques will be developed to get estimates of ‘higher associativity’ structures on Mp(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 R3 or R2. 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
nS). The L
nS are in turn approximated by the
Johnson-
Wilson spectra E(
n)=BP<
n>[
vn–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
XQ 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
XQ. 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.
16 Mar 2015
—Spring vacation—
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
kL
1L
K(2)V(0) is not zero when
k is congruent to 5 modulo 8. I explain how this contradicts the decomposition of L
1L
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 [A1] 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
0K(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 [
A1] 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 C2 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 th