Topology Seminar
Past Talks
Winter 2018

 Feb 272018
Kirsten Wickelgren (Georgia Institute of Technology)
Motivic Euler numbers and an arithmetic count of the lines on a cubic surface
A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1homotopy theory. The resulting count is valued in the GrothendieckWitt group of nondegenerate symmetric bilinear forms. This is joint work with Jesse Kass.

 Feb 202018
Mingcong Zeng (University of Rochester)
Mackey functors and equivariant EilenbergMac Lane spectra
In the HillHopkinsRavenel solution of Kervaire invariant one problem, they give an analysis of norms of M \mathbbR, the C_{2}equivariant version of MU. They show that these spectra's slices are suspensions of H \mathbbZ, the EilenbergMac Lane spcetrum of constant Mackey functor. Specially, their 0th homotopy and slice are the constant Mackey functor. In this talk, we will compute RO(G)graded homotopy of H\mathbbZ, and talk about computation in homological algebra of \mathbbZmodules. These are ingredients of computation of RO(G)graded slice spectral sequences.

 Feb 132018
Craig Westerland (University of Minnesota)
Structure theorems for braided Hopf algebras
Special time: 3pm
The PoincaréBirkhoffWitt and MilnorMoore theorems are fundamental tools for understanding the structure of Hopf algebras. Part of the classification of pointed Hopf algebras involves a notion of “braided Hopf algebras.” I will present work in progress which will establish analogues of the PoincaréBirkhoffWitt and MilnorMoore theorems in this setting. The main new tool is a notion of a braided Lie algebra defined in terms of braided operads. This can be used to establish forms of these results, and also presents an unexpected connection to profinite braid groups and related operads.
This talk will take place at 3:00pm in room 203. There will also be a pretalk from 1:302:30pm in the same room.

 Feb 062018
Michael Farber (Queen Mary)
Robot motion planning and equivariant Bredon cohomology
The motion planning problem of robotics leads to an interesting invariant of topological spaces, TC(X), depending on the homotopy type of X  the configuration space of the system. TC(X) is an integer reflecting the complexity of motion planning algorithms for all systems (robots) having X as their configuration space. Methods of algebraic topology help to compute or to estimate TC(X) in many examples of practical interest. In the case when the space X is aspherical the number TC(X) depends only on the fundamental group of X. In the talk I will describe some recent results expressing the topological complexity of aspherical spaces in the case when the fundamental group is hyperbolic in the sense of Gromov.

 Jan 232018
Marc Hoyois (University of Southern California)
Multiplicative transfers in the cohomology of algebraic varieties
Cohomology theories usually come with additive transfers, like the proper pushforward on Chow groups or the flat proper pushforward in Ktheory. Many cohomology theories also admit multiplicative transfers along finite covering maps, called norms. For example, Fulton and MacPherson constructed norms on Chow groups of smooth varieties in order to compute characteristic classes of vector bundles, and Joukhovitski constructed norms on the Grothendieck group K_0. In joint work with Tom Bachmann, we extend these norms to higher Chow groups (aka motivic cohomology) and higher Ktheory. Our approach is inspired by equivariant homotopy theory and in particular by the machinery of norm functors developed by Hill, Hopkins, and Ravenel. The structure so obtained on motivic cohomology also recovers Voevodsky’s power operations. (Joint work with Tom Bachmann)

 Jan 162018
Martina Rovelli (Johns Hopkins University)
A uniform approach to characteristic classes as obstructions
We propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group of a principal bundle, and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber. By plugging in the correct parameters, we recover several classical theorems. We then define a family of invariants for principal bundles that detect the number of group reductions associated to characteristic classes that a principal bundle admits. We show that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.

 Jan 092018
Agnes Beaudry (University of Colorado Boulder)
The Equivariant Dual of Morava ETheory
In this talk, I will discuss the equivariant functional dual of Morava Etheory D(E). Strickland has proved that D(E) is self dual up to a shift. However, Etheory comes equipped with an action of the Morava stabilizer group G, and Strickland only determines the action of stabilizer group on the homotopy groups of D(E). In this talk, I will explain the Linearization Conjecture, which identifies the representation sphere shifting D(E). I will discuss computational consequences for the duals of homotopy fixed points of finite subgroups.
Fall 2017

 Nov 282017
Elden Elmanto (Northwestern University)
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. An analogous theorem for MorelVoevodsky'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, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry.
Time permitting, I will also provide 1) an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and, 2) a model for all motivic EilenbergMaclane spaces as simplicial indsmooth schemes.

 Nov 212017
Nima Rasekh (UIUC)
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.

 Nov 142017
Jeremy Hahn (Harvard)
Multiplicative structures in the Stable Splitting of Affine Grassmannians
After suspension, the space of based loops in SL_n(C) is known to split as an infinite wedge sum. I will describe joint work with Allen Yuan that upgrades this result to a coherently multiplicative, or A_{∞} splitting. The proof involves use of the original factorizable cosheaf, namely the BeilinsonDrinfeld Grassmannian.

 Nov 072017
Eva Belmont (MIT)
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 computing the Adams E_2 page at p = 3 in an infinite region, by computing its localization by the nonnilpotent element b_10. 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.

 Nov 022017
Kathryn Hess (EPFL)
The Acyclicity Theorem for transfer of model structures
2:00 pm in Eckhart 203. Note the strange day and time.
A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically more challenging is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, a wellknown 'acyclicity' condition must hold. I'll explain that for a broad class of accessible model structures, which generalize the wellknown combinatorial model structures, this necessary condition is also sufficient in both the rightinduced and leftinduced contexts, and the resulting model category is again accessible. I'll outline a few techniques for proving the acyclicity condition and explain how to apply these these techniques to constructing new model structures in concrete cases of interest, for example, in developing model category frameworks for homotopical Galois theory and homotopical MoritaTakeuchi theory for coalgebras. I'll explain moreover how these techniques also provide a new perspective on (generalized) Reedy model category structures. (Joint work with Kedziorek, Riehl, and Shipley, with Beaudry, Kedziorek, Merling, and Stojanoska, and with Berglund.)

 Oct 312017
Lauren Bandklayder (Northwestern University)
The DoldThom theorem via factorization homology
The DoldThom 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.

 Oct 242017
No talk.

 Oct 172017
Peter Patzt (Purdue University)
Representation stability for filtrations of Torelli groups
We show, finitely generated rational VIC_Qmodules and SI_Qmodules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the Torelli subgroups of Aut(F_n) and Mod(Σ_g,1) are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups, respectively. Furthermore we prove an analogous statement for their Johnson filtrations.

 Oct 102017
Paul VanKoughnett (Northwestern University)
Notes on the margins of Etheory
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 Etheory. The K(t)localizations of Etheory, 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.

 Oct 052017
John Berman (University of Virginia)
Towards homotopical Tambara functors
3:30 pm in Eckhart 207. Note the strange day, time, and place!
When spaces are acted upon by groups, strange phenomena can occur. Homotopy and cohomology groups see more structure than just a group action. In recent years, we have begun to understand why: equivariant spaces and spectra arise from a Lawvere theory called the Burnside category, which gives rise to Mackey functor structures on the homotopy groups. By studying Lawvere theories and where they come from, we hope to give a similar description of highly structured equivariant ring spectra  and to explain a mystery in the foundations of equivariant homotopy theory.

 Oct 032017
Clover May (University of Oregon)
A structure theorem for RO(G)graded cohomology
Computations in RO(G)graded Bredon cohomology can be challenging and are not well understood, even using coefficients in a constant Mackey functor. In this talk I will present a structure theorem for RO(C_{2})graded cohomology with \mathbbZ/2 coefficients that substantially simplifies computations. The structure theorem says the cohomology of any finite C_{2}CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will sketch the proof, which depends on a Toda bracket calculation, and give some examples.
Spring 2017

 Jun 012017
Christopher SchommerPries (University of Notre Dame)
The structure of tensor categories via 3dimensional topology
This is a Thursday talk, and will take place 3:004:00 in Eckhart 203.
Fusion tensor categories arise in many areas of mathematics: as representation categories for finite quantum groups, certain Hopf algebras, and loop groups; as the "basic invariants" of subfactors of von Neumann algebras in the theory of operator algebras; and also in the study of conformal field theory. Fusion tensor categories have a rich and fascinating structure. The goal of this talk will be to describe how 3dimensional topology and topological field theory allow this structure to be understood and explained. This is joint work with Christopher Douglas and Noah Snyder.

 May 302017
Jeremy Miller (Purdue)
Localization and homological stability
Traditionally, homological stability concerns sequences of spaces with maps between them that induce isomorphisms on homology in a range tending to infinity. I will talk about homological stability phenomena in situations where there are no natural maps between the spaces. The prototypical example of this phenomenon is configuration spaces of particles in a closed manifold. In this and other situations, the homological stability patterns depend heavily on what coefficients one considers.

 May 152017
Doug Ravenel (Rochester)
Model categories and spectra
This is a Monday talk, and will be held at 1:30pm in Eckhart 206.
I will explain how the passage from unstable to stable homotopy is a form of Bousfield localization. This will feature explicit generating sets for the stable category as a cofibrantly generated model category. All of this these terms will be gently explained during the talk. No prior knowledge of model categories will be assumed.

 May 112017
Xiaolin (Danny) Shi (Harvard)
Hurewicz Images of Real Bordism Theory and Real JohnsonWilson Theories
This is a Thursday talk, and will take place 1:302:30 in Eckhart 203.
We show that the Hopf elements, the Kervaire classes, and the [ˉ(κ)]family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C_{2}fixed points of the Real bordism spectrum and the Real BrownPeterson spectrum. A subset of these families is detected by the C_{2}fixed points of Real JohnsonWilson theories E\mathbbR(n), depending on n. The proof relies on a computation of the map from the classical Adams spectral sequence of the sphere spectrum to the C_{2}equivariant Adams spectral sequence of BP\mathbbR. We also prove that the C_{2}equivariant May spectral sequence of BP\mathbbR is isomorphic to the associated graded C_{2}slice spectral sequence of BP\mathbbR. This is joint work with Guchuan Li, Guozhen Wang, and Zhouli Xu.

 Apr 112017
Magdalena Kedziorek (EPFL)
An algebraic model for rational SO(3) spectra
In algebraic topology we consider invariants of spaces with a G action, such as Gcohomology theories, which are represented by Gspectra. Since it is quite difficult to make systematic computations in the category of Gspectra, it is useful to have an algebraic model of it, i.e. a simpler, algebraic category Quillen equivalent to Gspectra. The work towards providing an algebraic model has been started by Greenlees with several cases established by Greenlees, Shipley and Barnes, including results for finite groups, tori and O(2).
In this talk I will present an overview of an algebraic model for rational SO(3)spectra. I will describe the general idea of a proof and concentrate on the main new ingredient, namely the interaction of the left Bousfield localisation and two adjunctions: induction – restriction and restriction – coinduction. I will justify why this approach will be useful in considering algebraic models for further groups.

 Apr 042017
Inbar Klang (Stanford)
Factorization homology and topological Hochschild cohomology of Thom spectra
By a theorem of Lewis, the Thom spectrum of an nfold loop map to BO is an E_{n}ring spectrum. I will discuss a project studying the factorization homology and the higher topological Hochschild cohomology of such Thom spectra, and talk about some applications, such as computations, and a duality between higher topological Hochschild homology and cohomology of such Thom spectra.
Winter 2017

 Mar 282017
Achim Krause (Max Planck Institute)
Nilpotence and Periodicity in Stable Motivic Homotopy Theory over \mathbbC
The world of stable motivic homotopy theory over \mathbbC looks very similar to classical stable homotopy theory. In particular, one can compare the Adams and AdamsNovikov spectral sequences to learn new things about classical homotopy groups as well.
However, one quickly notices a striking difference: The analogue of the celebrated nilpotence theorems by Nishida and DevinatzHopkinsSmith fail in the motivic setting, and in addition to the chromatic v_{i}, there are more types of selfmaps.
In this talk I want to explain how to detect and construct examples of additional, previously unexpected families of nonnilpotent selfmaps in stable motivic homotopy theory.

 Mar 072017
Prasit Bhattacharya (University of Notre Dame)
A very nice type 2 spectrum
Previously to this work, finite 2local complex with 32 periodic v_{2}selfmap were known to exist, e.g. M(1,4) and A_{1}. Which leads to the question whether there exist 2local finite complex with v_{2}periodicity less than 32. In a joint work with P.Egger we answer this question by producing a finite 2local spectrum Z which admits 1periodic v_{2}selfmap. The spectrum Z has some special properties, among which the most notable one is, tmf ∧Z ≅ k(2). We also give a complete calculation of the homotopy groups of its K(2)localization. Moreover, because of the property mentioned above the v_{2}periodic part of E_{2}page of tmfbased Adams spectral sequence can be computed as well, thereby providing a new gadget to attack the Telescope Conjecture at height 2 prime 2. Time permits, we will discuss wide array of possible future applications that the spectrum Z may have.

 Feb 212017
Saul Glasman (University of Minnesota)
Mackey functors, calculus and stable power operations
If F is an nexcisive functor from spectra to spectra, then the cross effects of F form a Mackey functor indexed on a certain category which shares many properties with the orbit category of a finite group; F can be recovered from this Mackey functor, and this correspondence forms an equivalence of categories. I'll give a brief description of this equivalence, and then discuss joint work in progress with Tyler Lawson in which we apply these ideas to the study of stable power operations in stable homotopy theory.

 Feb 142017
Gabe AngeliniKnoll (Wayne State Univeristy)
Approximating algebraic Ktheory of the image of J
The pcomplete connective image of J spectrum is a commutative ring spectrum that contains a periodic family of height one, called the alpha family, in the image of the unit map from the sphere spectrum. The AusoniRognes program suggests that algebraic Ktheory of this ring spectrum should contain chromatic height two information. To study this question, we first compute the linear approximation to algebraic Ktheory of the image of J, known as topological Hochschild homology, modulo p and v_{1} where p is an odd prime. This computation involves a spectral sequence, similar in style to the May spectral sequence, developed in joint work with Andrew Salch. We then show that S^{1}homotopy fixed points of THH of the image of J detects periodic classes of chromatic height 2 providing some evidence for chromatic redshift.

 Feb 072017
Dan Petersen (Copenhagen)
The gravity operad and Francis Brown's partial compactification of M_{0,n}
This talk is joint with the GeometryTopology seminar. Note that it will be held at 3pm in Eckhart 308.
The gravity operad is a certain operad built out of the cohomology of the moduli space M_{0,n} of npointed smooth genus zero curves. We show that as a nonsymmetric operad, it can be described combinatorially in terms of gluing together certain polygons with marked chords. In particular, this description implies that the nonsymmetric gravity operad is free. The result can be interpreted geometrically in terms of a partial compactification of M_{0,n} denoted M_{0,n}^{δ}, which was introduced by Francis Brown and whose cohomology plays a role in the study of MZVs. We see that the cohomology of M_{0,n}^{δ} gets identified with the generators of the gravity operad. The calculation of the cohomology of M_{0,n}^{δ} is new. (joint with Johan Alm)

 Jan 312017
Xing Gu (UIC)
On the Cohomology of BPU_n and its Application
In this talk I will introduce my work on the integral cohomology ring of BPU_{n} in degree less than or equal to 10, where PU_{n} is the projective unitary group of degree n and BPU_{n} is its classifying space. I will outline the proof of this result as well as its application on the topological periodindex problem.

 Jan 262017
Sam Nariman (Northwestern)
FriedlanderMilnor's problem for diffeomorphism groups
This talk is joint with the GeometryTopology seminar. Note that it will be held at 3pm in Eckhart 308.

 Jan 242017
Lukas Brantner (Harvard)

 Jan 172017
Alexandra Yarosh (Pittsburgh)

 Jan 102017
Akhil Mathew (Harvard)
Polynomial functors and algebraic Ktheory
The Grothendieck group K_{0} of a commutative ring is wellknown to be a λring: although the exterior powers are nonadditive, they induce maps on K_{0} satisfying various universal identities. The λoperations yield homomorphisms on higher Kgroups.
In joint work in progress with Barwick, Glasman, and Nikolaus, we give a general framework for such operations. Namely, we show that the Ktheory space is naturally functorial for polynomial functors, and describe a universal property of the extended Ktheory functor. This extends an earlier algebraic result of Dold for K_{0}. In this picture, the λoperations come from the ‘strict polynomial functors’ of FriedlanderSuslin.
Fall 2016

 Nov 292016
Haldun Bayindir (UIC)
Topological Equivalences of E_{∞} DGA
Using the Quillen equivalence between differential graded algebras (DGAs) and EilenbergMac Lane ring spectra, Dugger and Shipley defined a new equivalence relation between DGAs called topological equivalences. Topological equivalences play a central role in Morita equivalences of DGAs and topologically equivalent DGAs are also derived equivalent. Quasiisomorphic DGAs are topologically equivalent but the converse to this statement is not true; Dugger and Shipley constructed examples of DGAs that are topologically equivalent but not quasiisomorphic.
In this talk, I will talk about new results on topological equivalences of E_{∞} DGAs. I will discuss the cases where E_{∞} topological equivalences and quasiisomorphisms of E_{∞} DGAs agree as well as examples of E_{∞} differential graded algebras that are E_{∞} topologically equivalent but not quasiisomorphic. These results rely on the obstruction theories of Goerss, Hopkins and Miller and I will start by discussing these obstruction theories and their various applications.

 Nov 222016
Dominic Culver (Notre Dame)
On BP〈2〉cooperations
Inspired by Mark Mahowald’s work on boresolutions, Mark Behrens has initiated a program to use the Adams spectral sequence based on the connective spectrum of topological modular forms. In order to pursue this, Behrens, Ormsby, Stapleton, and Stojanoska developed techniques to compute the cooperations algebra of tmf. A closely related spectrum to tmf is the spectrum tmf_{1}(3), which is a form of BP〈2〉. I will describe techniques to calculate the BP〈2〉cooperations.

 Nov 152016
Dylan Wilson (Northwestern)
Equivariant power operations and the Real BrownPeterson spectrum
The speaker is interested in following a program due to Hill, Hopkins, and Ravenel for resolving the 3primary Kervaire invariant. As a warmup, we revisit the 2primary case and give a new construction of a spectrum that lies at the heart of the HillHopkinsRavenel theorem. Along the way, we build and investigate equivariant power operations, which are of independent interest and behave surprisingly well.

 Nov 082016
Leanne Merrill (University of Oregon)
Algebraic v_{n} self maps at the prime 2
A central question of algebraic topology is to understand homotopy classes of maps between finite cell complexes. The Nilpotence Theorem of HopkinsDevinatzSmith together with the Periodicity Theorem of HopkinsSmith describes nonnilpotent self maps of finite spectra. The Morava Ktheories K(n)_{*} are extraordinary cohomology theories which detect whether a finite spectrum X supports a v_{n}self map. Such maps are known to exist for each finite spectrum X for an appropriate n but few explicit examples are known. Working at the prime 2, we use a technique of PalmieriSadofsky to produce algebraic analogs of v_{n} maps that are easier to detect and compute. We reproduce the existence proof of Adams’s v_{1}^{4} map on the Mod 2 Moore spectrum, and work towards a v_{2}^{i} map for a small values of i.

 Nov 012016
Jay Shah (MIT)
Parameterized higher category theory
In this talk, I will first give an overview of a program, joint with Barwick, Dotto, Glasman, and Nardin, to extend the theory of ∞categories so as to incorporate and provide common footing for certain constructions arising in a host of examples, ranging from equivariant stable homotopy theory to cyclotomic spectra to the calculus of functors. As I will explain, to do this one must in the first instance develop the ∞category theory of the (∞,2)category of Cat_{∞}valued presheaves on an ätomic orbital" ∞category. Delving into the details, I will then explain how Lurie's model categories of marked simplicial sets provide an effective formalism for accomplishing this task.

 Oct 252016
Bogdan Gheorghe (Wayne State University)
Motivic fields and w_{n}periodicity
The setting of this talk is stable motivic homotopy theory over \operatornameSpec \mathbbC, at p=2. Morel showed that η is not nilpotent in the motivic stable stems over \operatornameSpec \mathbbC, by seeing it in MilnorWitt Ktheory. Since none of the motivic Morava Ktheories detect η, this implies that the obvious analogue of the Nilpotence Theorem of Devinatz, Hopkins and Smith is wrong motivically. The first step towards a motivic Nilpotence Theorem is thus to construct a motivic field K(η) that detects eta. In this talk, we will show how to construct such a motivic field. In the process, an obvious pattern appears, leading to the periodicity operators w_{0}, w_{1}, … conjectured by Michael Andrews. We then show how to construct the fields K(w_{n}) as well as a BrownPeterson spectrum wBP. If time permits, we indicate what is known about the w_{n}family and formulate some conjectures about these periodic operators.

 Oct 182016
Bert Guillou (University of Kentucky)
From motivic to equivariant homotopy groups  a worked example
The realization of a motivic space defined over the reals inherits an action of \mathbbZ/2\mathbbZ, the Galois group. This realization functor allows for information to pass back and forth between the motivic and equivariant worlds. I will discuss one example: an equivariant Adams spectral sequence computation for ko, taking the simpler motivic computation as input. This is joint work with M. Hill, D. Isaksen, and D. Ravenel.

 Oct 112016
Angélica M. Osorno (Reed College)
2monads in homotopy theory
The classifying functor from categories to topological spaces provides a way of constructing spaces with certain properties or structure from categories with similar properties of structure. An important example of this is the construction of infinite loop spaces from symmetric monoidal categories. The particular kinds of extra structure can typically be encoded by monads on the category of small categories. In order to provide more flexibility in the kinds of morphisms allowed, one can work with the associated 2monad in the 2category of categories, functors, and natural transformations. In this talk I will give the categorical setup required, and I will give examples of interest to homotopy theorists. I will also outline how this method of working can give general statements about strictifications and comparisons of homotopy theories. This is partially based on work with two different sets of collaborators: Nick Gurski, Niles Johnson, and Marc Stephan; Bert Guillou, Peter May, and Mona Merling.

 Oct 042016
Krishanu Sankar (Harvard University)
The equivariant symmetric power filtration
The EilenbergMaclane spectrum H\mathbbZ has a filtration by the symmetric powers of the sphere spectrum. The plocal behavior of this filtration has figured prominently in major questions of homotopy theory, starting with Nakaoka in the 1950s who showed that it respects the length filtration of the Steenrod algebra, to work on the Whitehead Conjecture by Kuhn and Priddy in the 1980s, and a duality with the Goodwillie tower of S^{1} studied by Arone and Mahowald in the 1990s.
We'll describe current efforts to study a Gequivariant version of this filtration when G=\mathbbZ/p, which is now a filtration for H\mathbbZ, where \mathbbZ is the constant Mackey functor. Our main results are about the plocal structure of the geometric fixed points, underlying points, and how they relate.
Our computations have many potential applications, such as computing an equivariant version of the Steenrod algebra (via Bredon cohomology), an equivariant analogue of the Whitehead conjecture, and a potentially simpler proof of the Reduction theorem of Hill, Hopkins, and Ravenel.
Spring 2016

 May 312016
Anna Marie Bohmann (Vanderbilt)
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 Ktheory constructions.

 May 252016
Doug Ravenel (University of Rochester)
What is a Gspectrum?
This is a department colloquium; it will be from 34pm in Eckhart 206.
Spectra in the sense of stable homotopy theory have been a major object of study in algebraic topology for half a century. During that time the basic definition has undergone some major revisions, including a major breakthrough in 1993 due to Peter May and three coauthors. Remarkably, these shifting foundations have not affected any of the computations made using earlier definitions. In the talk I will describe how the use of category theory has led to major simplifications.

 May 242016
Dondi Ellis (University of Michigan)
Motivic Analogues of the Cobordism Theories MO and MSO
I will begin by reviewing the foundations of equivariant and nonequivariant stable motivic homotopy theory, as well as the construction of unoriented cobordism MO and oriented cobordism MSO. In the nonequivariant stable motivic homotopy category, I will construct a kspectrum MGLO whose topological realization over the field k=\mathbbC is MO. I will give a complete description of the coefficient ring of MGLO up to knowledge of the coefficients of motivic H\mathbbZ/2. Next I will discuss the relation of MGLO to the \mathbbZ/2equivariant kspectrum MGLR. MGLR is a motivic analogue of Landweber's real oriented cobordism MR. Just as taking fixed points of MR at the prespectrum level gives MO, taking fixed points MGLR at the prespectrum level gives MGLO. Restricting attention to the field k=\mathbbC, I will briefly discuss new research relating to MGLR. Finally I will construct a kspectrum MGLSO whose topological realization over the field k=\mathbbC is MSO. I will describe MGLSO_{(2)} as a wedge sums of EilenbergMacLane spectra H\mathbbZ_{(2)} and H\mathbbZ/2.

 May 172016
Namboodiri Lectures

 May 102016
Carolyn Yarnall (University of Kentucky)
Slices and Suspensions
The equivariant slice filtration is an analogue of the Postnikov tower for Gspectra. However, unlike the Postnikov tower, the slice tower does not commute with taking ordinary suspensions and often the result of suspension is quite mysterious. In this talk, after recalling the construction and basic properties of the slice tower, we will look at the slice towers for integergraded suspensions of HZ and compare them to complementary results of Hill, Hopkins, and Ravenel concerning lambdasuspensions. We will then discuss some future directions regarding the classification of more general slices.

 May 032016
No seminar

 Apr 262016
No seminar

 Apr 212016
Paul Goerss (Northwestern)
GrossHopkins Duality
The pretalk for this talk will be 34pm in Eckhart 203. The talk will take place from 4:30 to 5:30 in Ryerson 277.
There is always plenty of confusion on this topic, because when we talk about GrossHopkins duality we often mean two (or four, if you throw in the algebra) different dualities and the connections between them. The combined effect is that in the K(n)local category, BrownComentz duality behaves very much like SerreGrothendeick duality. I will do my best to explain all this, then I will then explain how calculations in the K(2)local setting, usually regarded as an impenetrable mess, can be called to order and organized into a comprehensible, even inevitable, pattern.

 Apr 192016
Simona Paoli (University of Leicester)
Segaltype models of weak ncategories
This talk will be from 34:30pm.
In the third talk I discuss how to associate to a weakly globular Tamsamani ncategory a Segalic pseudofunctor. I illustrate first the case n=2, which is fairly straightforward and an instance of a general categorical technique known as transport of structure along an adjunction. I will then illustrate the case n > 2, which is much more complex and requires approximating every weakly globular Tamsamani ncategory with objects of a subcategory to which transport of structure can be applied. Together with the results of talk 2 this gives the rigidification functor from weakly globular Tamsamani ncategories to weakly globular nfold categories. I will then illustrate the ‘discretization functor’ from weakly globular nfold categories to Tamsamani ncategories, completing the comparison between the two models. Finally, I will discuss the groupoidal case of weakly globular n fold categories and how it satisfies the homotopy hypothesis.
I will end this talk series with a discussion of further directions and open questions.

 Apr 142016
Simona Paoli (University of Leicester)
Segaltype models of weak ncategories
This talk will be from 34:30pm.
In the second talk I define the most general of the three Segaltype models, called weakly globular Tamsamani ncategories. This contains as special cases the TamsamaniSimpson model as well as the weakly globular nfold categories defined in the first talk. Our main goal is to build a ‘rigidification’ functor from weakly globular Tamsamani ncategories to weakly globular nfold categories. This functor factors though a certain class of Catvalued pseudofunctors called Segalic pseudofunctors. In this talk I introduce Segalic pseudofunctors and I prove that their strictification produces weakly globular nfold categories.

 Apr 122016
Simona Paoli (University of Leicester)
Segaltype models of weak ncategories
This talk will be from 34:30pm.
Higher categories are structures generalizing a category where there are higher arrows and compositions between them. They find applications in diverse areas such as homotopy theory, mathematical physics, algebraic geometry, logic and computer science. When compositions of higher arrows are associative and unital, we obtain a strict ncategory. For the applications that higher categories are called for, the wider class or weak ncategories is needed.
In this talk series I introduce a new model of weak ncategories called weakly globular nfold category, based on a new paradigm to weaken higher categorical structures. This is an instance of a more general class of Segaltype models which includes the TamsamaniSimpson model of weak n categories. In the first talk I give an overview of the main features of three Segaltype models. I then define weakly globular nfold categories, and illustrate as examples some lowdimensional cases.

 Apr 052016
David Gepner (Purdue)
On the stable homotopy theory of stacks and elliptic cohomology
In this talk, we'll discuss what it means to be a cohomology theory for topological stacks, using a notion of local symmetric monoidal inversion of objects in families. While the general setup is abstract, it specializes to many cases of interest, including Schwede's global spectra. We will then go on to discuss various examples with particular emphasis on elliptic cohomology. It turns out that TMF sees more objects as dualizable (or even invertible) than one might naively expect.
Winter 2016

 Mar 292016
No seminar

 Mar 082016
Jeremiah Heller (UIUC)
Equivariant motivic cohomology
Motivic cohomology is an important invariant of smooth varieties and a fundamental tool for understanding algebraic Ktheory. Joint with Mircea Voineagu and Paul Arne Ostvaer we construct a "Bredonstyle" equivariant motivic cohomology, for a finite group G. The motivating case is G=Z/2, where these are expected to be related to Hermitian Ktheory of rings with involution. In this talk I'll discuss these invariants, their properties, and how these compare to topological Bredon cohomology (with constant Mackey functor coefficients) for complex varieties with involution.

 Mar 012016
Inna Zakharevich (University of Chicago)
The annihilator of the Lefschetz motive
The Grothendieck ring of varieties is defined to be the free abelian group geneerated by varieties, modulo the relation that for a closed subvariety Y of X, [X] = [Y] + [X \Y]; the ring structure is defined via the Cartesian product. For example, if X and Y are piecewise isomorphic, in the sense that there exist stratifications on X and Y with isomorphic strata, then [X] = [Y] in the Grothendieck ring. There are two important questions about this ring:
 What does it mean when two varieties X and Y have equal classes in
the Grothendieck ring? Must X and Y be piecewise isomorphic?
 Is the class of the affine line a zero divisor?

 Feb 232016
John Lind ( Universität Regensburg)
Duality in bicategories and the THH transfer
Associated to a fibration E → B with homotopy finite fiber is a stable wrong way map LB → LE of free loop spaces coming from the transfer map in THH. This transfer is defined under the same hypotheses as the BeckerGottlieb transfer, but on different objects. I will use duality in bicategories to explain why the THH transfer contains the BeckerGottieb transfer as a direct summand. The corresponding result for the Atheory transfer may then be deduced as a corollary. When the fibration is a smooth fiber bundle, the same methods give a three step description of the THH transfer in terms of explicit geometry over the free loop space. (Joint work with C. Malkiewich)

 Feb 162016
No seminar

 Feb 092016
Lennart Meier (Bonn)
Equivariant TMF of a point
Over the years, different versions of equivariant elliptic cohomology have been proposed, the most recent being Lurie's version of equivariant TMF. This talk will present work in progress how to understand equivariant TMF of a point by decomposing it into better known spectra. This will include giving a (short) introduction to equivariant TMF.

 Feb 022016
Vesna Stojanoska (UIUC)
Comparing flavors of selfduality and their descent properties
There are many different ways to import algebraic notions of duality to homotopy theory, and depending on the intended applications, these notions can have quite different flavors. At the same time, spectra such as complex or real Ktheory have many versions themselves, but all seem to be selfdual with respect to some kind of duality. In this talk, I will discuss an explicit way to relate Gorenstein duality for connective ring spectra in the sense of DwyerGreenleesIyengar, and Anderson duality. The relation will be crucial in studying descent properties for either version of duality. This is based on joint work in progress with John Greenlees.

 Jan 262016
Yifei Zhu (Northwestern)
Modular equations and Hecke operators for local elliptic spectra
A Morava Etheory at height 2 can be modeled by an elliptic spectrum whose formal group is the universal deformation of the formal group of a supersingular elliptic curve. By studying the moduli of elliptic curves at a supersingular point and near the cusps, we determine the algebra of power operations on such an Etheory, in terms of generators and quadratic relations. Building on this explicit structure, we study certain analogs of numbertheoretic constructs for elliptic spectra, with a specific application toward Eorientations of vector bundles via Rezk's logarithmic operations on units of ring spectra.

 Jan 192016
Aaron Royer (UT Austin)
Ktheory computations for topological insulators
This talk will take place at a special time: 1:30 for the pretalk and 3 for the actual talk.
Over the past decade a new class materials called topological insulators, often with counterintuitive electric properties, have been discovered. The mathematics involved in classifying such materials is twisted equivariant Real Ktheory. In the pretalk I will review the relevant equivariant stable homotopy theory, namely Bredon cohomology, equivariant Ktheory and the equivariant AtiyahHirzebruch spectral sequence. In the main talk I will briefly describe the setup and survey new Ktheory computations coming from these considerations. This is joint work with Dan Freed.

 Jan 142016
Elizabeth Vidaurre (CUNY)
Cohomology of Polyhedral Product Spaces and Real Moment Angle Complexes
This is a Thursday talk. The pretalk will be 1:302:30 in Eckhart 203 and the main talk will be 34 in Eckhart 308.
Certain subspaces of a product of pairs of spaces whose factors are labeled by the vertices of a simplicial complex are referred to as "polyhedral product spaces". Polyhedral products are given by taking the union of subproducts associated to each simplex. Such polyhedral products are realized by objects studied in combinatorics, commutative algebra and algebraic geometry. In algebraic geometry, the labeled pairs are 2disks and their boundaries; the associated polyhedral product is called a momentangle complex. The real versions of momentangle complexes, where the pairs are intervals and their boundaries, is also considered. We will study how the cohomology of polyhedral products can be given in terms of the underlying simplicial complex. We will illustrate this by considering different classes of simplicial complexes in different settings.

 Jan 122016
Peter Nelson (UIUC)
A Small Presentation for Morava Etheory Power Operations
Commutative ring spectra have a great deal of additional structure than in the classical case. Power operations give manifestations of this structure on homotopy groups, which have been helpful in many computational problems. I'll talk about some structural results regarding the theory of power operations for an interesting family of ring spectrathe Morava Etheoriesand how this relates to a story in algebra.

 Jan 052016
Robert Legg (Northwestern University)
An obstruction theory for producing exotic Picard elements
Chromatic homotopy theory posits that we should study certain localized homotopy categories, namely, the K(n) and E(n)local categories, as the building blocks of the stable homotopy category. The Picard group of a symmetric monoidal category is an essential datum of the category, and we would like to know the Picard groups of these chromatic categories. I will discuss an obstruction theory useful for producing socalled exotic Picard elements of these categories and, as an example, will show how theory provides an alternate proof for the existence of a certain elusive exotic Picard element of the 3primary K(2)local category.
Fall 2015

 Dec 012015
Sune Precht Reeh (MIT)
HopkinsKuhnRavenel character theory for fusion systems
A saturated fusion system associated to a finite group G encodes the pstructure of the group as the Sylow psubgroup enriched with additional conjugation. The fusion system contains just the right amount of algebraic information to for instance reconstruct the pcompletion of BG, but not BG itself. Abstract saturated fusion systems F without ambient groups exist, and these have (pcompleted) classifying spaces BF as well. The pretalk will be an introduction to fusion systems and how BF is a stable retract of BS, for the Sylow psubgroup S, as encoded by a characteristic idempotent in the double Burnside ring of S. The main talk will be on a joint project with Tomer Schlank and Nat Stapleton: We use characteristic idempotents to generalize the HKR character map from finite groups to all abstract fusion systems, and show that when tensored with a certain algebra we get an isomorphism. This also involves constructing and studying transfer maps for free loops on classifying spaces.

 Nov 242015
Vitaly Lorman (Johns Hopkins)
Real JohnsonWilson Theories and Computations
Complex cobordism and its relatives, the JohnsonWilson theories, E(n), carry an action of C_{2} by complex conjugation. Taking homotopy fixed points of the latter yields Real JohnsonWilson theories, ER(n). These can be seen as generalizations of real Ktheory and are similarly amenable to computations. We will outline their properties, describe a generalization of the ηfibration, and discuss recent computations of the ER(n)cohomology of some wellknown spaces, including CP^{∞}.

 Nov 172015
Tomer Schlank (MIT)

 Nov 102015
Nerses Aramian (UIUC)
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 1dimensional topological field theory over the manifold. One can generalize this procedure by replacing complex line bundles with ngerbes bound by U(1), and replacing connections with appropriate connective structures. The resulting topological field theories are ndimensional 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 WessZuminoWitten model, ChernSimons theory and DijkgraafWitten theory fit into this context.

 Nov 032015
Ben Knudsen (Northwestern)
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.

 Oct 272015
Jonathan Beardsley (Johns Hopkins)
A Construction of MU Without Manifolds
In his book "Complex Cobordism and Stable Homotopy Groups of Spheres," Ravenel suggested that one might be able to construct MU from formal group law data, without the aid of manifolds or vector bundles. We review HopfGalois extensions for E_{n}ring spectra and describe how this technology provides for a spectral construction of MU with distinct similarities to Lazard's construction of the Lazard ring.

 Oct 202015
Alexander Kupers (Stanford)
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 localtoglobal 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.

 Oct 132015
Glen Wilson (Rutgers University)
Motivic stable stems over finite fields
In the MorelVoevodsky motivic stable homotopy category over the complex numbers, Marc Levine proved that the motivic stable stems π_{n,0} are isomorphic to the topological stable stems π_{n}^{s}. What can we say about the motivic stable stems over fields of positive characteristic? In this talk, we will discuss calculations of the twocomplete motivic stable stems over finite fields of odd characteristic using the motivic Adams spectral sequence. For n < 19, we find that after twocompletion, π_{n,0} = π_{n}^{s} + π_{n+1}^{s}.

 Oct 062015
Sarah Yeakel (UIUC)
A chain rule for Goodwillie calculus
In the homotopy calculus of functors, Goodwillie defined a way of assigning a Taylor tower of polynomial functors to a homotopy functor and identified 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 using the category of finite sets and injective maps, we have found a more straight forward chain rule for endofunctors of spaces.

 Sep 292015
Alexander Neshitov (University of Ottawa)
Framed Correspondences and the MilnorWitt Ktheory
The theory of framed motives developed by Garkusha and Panin based on ideas by Voevodsky, gives a tool to construct fibrant replacements of spectra in A^{1}homotopy category. In the talk we will discuss how this construction gives an explicit identification of the motivic homotopy groups of the base field with its MilnorWitt Ktheory. In fact, this identification can be done similar to the theorem of SuslinVoevodsky which identifies motivic cohomology of the base field with Milnor Ktheory.
Spring 2015

 Jun 022015
Daniel G. Davis (University of Louisiana at Lafayette)
For the AusoniRognes conjecture at n=1, p > 3: a strongly convergent descent spectral sequence
Let p be a prime such that p ≥ 5, let K be any closed subgroup of the units of the padics, and let V(1) be the type 2 SmithToda complex S^{0}/(p,v_{1}). Also, let KU_{p} denote the pcompleted complex Ktheory spectrum, with K(KU_{p}) the associated algebraic Ktheory spectrum. We discuss our proof that there is a strongly convergent descent spectral sequence
with E_{2}^{s,t} = 0, for all s ≥ 2 and any t ∈ Z. When K is the units of the padics, this spectral sequence can be viewed as giving a construction of a spectral sequence that is conjectured by Ausoni and Rognes. (The AusoniRognes conjectures include statements for every n ≥ 1, and our work addresses only part of the n=1 case.)E_{2}^{s,t} = H^{s}_{c}(K; π_{t}(K(KU_{p}) ∧V(1))[v_{2}^{−1}]) ⇒ π_{t−s}((K(KU_{p}) ∧v_{2}^{−1}V(1))^{hK}), 
 May 262015
Naboodiri lectures

 May 192015
Anibal Medina (SUNY Stony Brook)
E_{∞} comodules and topological manifolds
The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using later language, his construction could be said to endow the chains with an Einfinity coalgebra structure.
The second story also begins with a question: when is a space homotopy equivalent to a topological manifold? For dimensions greater than 4, an answer was provided by the work of Browder, Novikov, Sullivan and Wall in surgery theory, which in a later development was algebraically expressed by Ranicki as a single chain level invariant: the total surgery obstruction.
After presenting the necessary parts of these stories, the goal of this talk will be to express the total surgery obstruction associated to a triangulated space in terms of comodules over the Einfinity coalgebra structure build by Steenrod on its chains.

 May 122015
No seminar

 May 052015
No seminar

 Apr 282015
Andrew Salch (Wayne State University)
A proof of some 1983 conjectures of Ravenel
In this talk I will describe some of the relationships between longperiod families in the stable homotopy groups of spheres and the moduli theory of formal Amodules (i.e., formal group laws "with complex multiplication by A"), where A is the ring of integers in a number field.
Then I will explain some conjectures of Ravenel, posed in 1983, about the moduli theory of formal Amodules (specifically, the first flat cohomology group H^{1} of the moduli stack of onedimensional formal Amodules), and I will explain how to prove (most of the cases of) these conjectures. The proof uses a combination of methods from computational homotopy theory (the chromatic spectral sequence, May and Bockstein spectral sequences) and methods from number theory (a new Hasse principle for a certain class of ideals in number rings, and some properties of Dedekind zetafunctions).
Time allowing, I will also talk about some of the topological connections and consequences of the proof and the ideas involved in itthis gets into a kind of "padic Hodge theory of spectra" which is still in its early stages.

 Apr 212015
Tyler Lawson (University of Minnesota)
Construction and mapping for commutative ring spectra
Many structural properties of the category commutative ring spectra are welldeveloped in several sources. However, even in many important examples it is not clear if we can construct certain desired commutative ring spectra, and when we can construct them it is not clear whether we can construct maps between them. I'll discuss several approaches to attacking problems like this in the cases of the complex bordism spectrum, pcomplete connective spectra, and "chromatic" spectra which live in the K(n)local category.

 Apr 142015
Nathan Perlmutter (University of Oregon)
Homological stability for diffeomorphism groups of odd dimensional manifolds
I will present a new homological stability result for the diffeomorphism groups of manifolds of dimension 2n+1 ≥ 9, with respect to forming the connected sum with copies of an arbitrary (n−1)connected, (2n+1)dimensional manifold that is stably parallelizable. This result is an odd dimensional analogue of a recent theorem of Galatius and RandalWilliams regarding the homological stability of the diffeomorphism groups of manifolds of dimension 2n ≥ 6, with respect to forming connected sums with S^{n} ×S^{n}.

 Apr 072015
Philip Egger (Northwestern)
v_{2}periodicity of A_{1}
We will construct four 2local finite spectra of type 2 whose cohomologies as module over the Steenrod algebra A are the four Amodule structures of the subalgebra A(1). We will call these spectra A_{1} and show that all four of these spectra admit a 32periodic v_{2} selfmap. This project is joint work with Bhattacharya and Mahowald.
Winter 2015

 Mar 312015
Andrew Blumberg (UT Austin)
The algebraic Ktheory of the sphere spectrum
Waldhausen showed that the algebraic Ktheory 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 highdimensional disks, information about BDiff is encoded in K(\mathbbS), the algebraic Ktheory of the sphere spectrum. This talk explains recent work with Mike Mandell that provides a complete calculation of the homotopy groups of K(\mathbbS) in terms of the homotopy groups of K(Z), the sphere spectrum, and a certain Thom spectrum. I will also explain what we know about the homotopy type of K(\mathbbS) in terms of a kind of Ktheoretic TatePoitou duality.

 Mar 102015
Jonathan Campbell (UT Austin)
The KTheory of the Category of Varieties
The Grothendieck ring of varieties is a fundamental object of study for algebraic geometers. As with all Grothendieck rings, one may hope that it arises as π_{0} of a Ktheory spectrum, K(Var_{k}). Using her formalism of assemblers, Zakharevich showed that this is in fact that case. I'll present an alternate construction of the spectrum that allows us to quickly see the E_{∞}structure on K(Var_{k}) and produce various character maps out of K(Var_{k}). I'll end with a conjecture about K(Var_{k}) and iterated Ktheory.

 Mar 032015
Pieter Hofstra (University of Ottawa)
Isotropy Theory
Grothendieck Toposes are often considered as generalized spaces; indeed, every space gives rise to a topos of sheaves, and various invariants and constructions from (algebraic) topology can be generalized to the level of toposes. In this talk, I will introduce a newly discovered invariant called the isotropy group of a topos and illustrate by considering special cases such as continuous group actions and etale groupoids. On the one hand, this group plays a key role in the study of crossed toposes, a direct generalization of crossed modules. On the other hand, the isotropy group has connections to the theory of monoidal categories and certain aspects of lowdimensional topology. Based on joint work with Jonathon Funk.

 Feb 242015
Birgit Richter (Universitat Hamburg)
Higher topological Hochschild homology of rings of integers in number fields
This is a report on joint work (in progress) with Bjørn Dundas and Ayelet Lindenstrauss. B ökstedt calculated topological Hochschild homology of the integers and LindenstraussMadsen covered the cases of number rings. We use these results to determine higher topological Hochschild homology for rings of integers in number fields with coefficients in the corresponding residue fields. Our approach is to use the decomposition of the nsphere into two hemispheres and to show that one can reduce the calculation to a one in homological algebra.

 Feb 192015
Justin Noel (University of Regensburg)
Derived induction and restriction theory
This is a Thursday talk, which will take place at 4:30pm in Eckhart 203. The pretalk will be at 3pm in Eckhart 203.
Let G be a finite group. Artin's theorem says that we can recover the complex representation ring of G from the representations of the cyclic subgroups of G up to torsion, or additive nilpotence. Quillen's Fisomorphism theorem says that we can recover the modp cohomology of G from the modp cohomology of the elementary abelian psubgroups of G up to multiplicative nilpotence.
These results naturally fit into Dress's theory of induction/restriction for Mackey and Green functors. These particular Mackey functors arise as the homotopy groups of Gspectra and it is relatively straightforward to lift Dress's framework to Gspectra. In this derived framework we associate to each family F of subgroups of G a subcategory Fnil, of Gspectra with the following properties: If E ∈ Fnil then:
 A generalized version of Artin's induction/restriction theorem holds for Eequivariant (co)homology.
 If E is a homotopy commutative ring spectrum, then a generalization of Quillen's Fisomorphism theorem holds for Eequivariant cohomology of Gspaces.
 If E is an E_{∞} ring spectrum then we can recover the category of Gequivariant Emodules from the categories of Hequivariant Emodules as H varies over F.
Moreover the category Fnil is closed under finite homotopy (co)limits, retracts, and (co)tensors with arbitrary Gspectra. Our theory applies to genuine equivariant complex and real Ktheory (extending Artin's theorem and a result of Fausk), and the Borel equivariant cohomology theories associated to modp cohomology (extending the result of Quillen), integral cohomology (extending a result of Carlson), complex oriented theories (extending a result of HopkinsKuhnRavenel), ko, the many variants of topological modular forms, L_{n}local spectra, and classical cobordism theories.
This is joint work with Akhil Mathew and Niko Naumann.

 Feb 172015
Deborah Vicinsky (University of Oregon)
Stabilizations of categories and graphs
I will construct the stabilization of the category of small categories with the canonical model structure, in which the weak equivalences are equivalences of categories and the cofibrations are injective on objects. In this case, the stabilization completely determines the Goodwillie derivatives of the identity functor. I will also discuss the homotopy theory of two categories of graphs.

 Feb 102015
Daniel Grayson (UIUC)
Homotopy Type Theory and Univalent Foundations
Homotopy type theory with the univalence axiom of Voevodsky provides both a new logical foundation for mathematics and a formal language usable with computers for checking the proofs mathematicians make daily. As a foundation, it replaces set theory with a framework where sets are defined in terms of a more primitive notion called "type". As a formal language, it encodes the axioms of mathematics and the rules of logic simultaneously, and promises to make the extraction of algorithms and values from constructive proofs easy. With a semantic interpretation in homotopy theory, it offers an alternative world where the proofs of basic theorems of mathematics can be formalized with minimal verbosity and verified by computer.
As a relative newcomer to the field, I will survey these recent developments and sketch the basic concepts for a general mathematical audience.

 Feb 032015
Bob Bruner (Wayne State University)
Characteristic Classes in Connective Ktheory
We will describe the connective Kcohomology of U(n), Sp(n), O(n), their oriented variants, the torus T(n) and the 'symplectic torus' Sp(1)^{n}, in so far as they are understood at present. Variants include the equivariant and completed versions: ku^{*}_{G} → ku^{*}BG. They specialize to the usual integral homology (at v=0) and to the representation theory (away from v=0).
The best results are available for ku^{*} and the more highly connected groups (e.g., Sp(n) rather than O(n)), but some results are available for ko^{*} as well.
We will start with some general facts about the relations between connective Ktheory, representation theory, and cohomology.

 Jan 272015
Benjamin Antieau (UIC)
Localization sequences in Ktheory and a question of Rognes
After giving background on the algebraic Ktheory of ring spectra, I will discuss recent joint work with Barthel and Gepner on the algebraic Ktheory of truncated BrownPeterson spectra and our answer to a question of Rognes about higher chromatic analogues of the famous fiber sequence K(\mathbbF_{p})→ K(\mathbbZ_{p})→ K(\mathbbQ_{p}) of Quillen.

 Jan 202015
Rick Jardine (University of Western Ontario)
Tspectra
The pretalk for this talk will be Monday, january 19th at 4pm in Eckhart 202.
Tspectra, or spectrum objects with generalized "suspension" parameters, first appeared in the construction of motivic stable categories. The motivic stable model structure lives within a localized model structure of simplicial presheaves in which the affine line is formally collapsed to a point, and the suspension object is the projective line. Because of these constraints and limitations of the tools then at hand, the original construction of the motivic stable category was technical, and made heavy use of the Nisnevich descent theorem.
This talk will begin with a general introduction to the concepts around Tspectra. I shall display a short list of axioms on the parameter object T and the ambient flocal model category which together lead to the construction of a well behaved flocal stable model structure of Tspectra. Examples include the motivic stable category, but the construction is much more general. The resulting stable category has many of the basic calculational features of the motivic stable category, including slice filtrations. The overall construction method is to suitably localize an easily defined strict model structure for Tspectra. The localization trick is an old idea of Jeff Smith, but assumptions (the axioms) are required for the recovery of normal features of stable homotopy theory from the localized structure.

 Jan 132015
Conflict with JMM

 Jan 062015
Tony Elmendorf (Purdue University Calumet)
A Presheaf Construction for Generalized Multicategories
This talk will present a construction of generalized multicategories that is designed to account for the classical nonsymmetric and symmetric cases, as well as the sort of equivariant permutative categories considered by Guillou and May as inputs for equivariant infinite loop space theory. It is a generalization of Leinster's monadic construction, which however does not account for (nonequivariant) permutative categories and symmetric multicategories. The main result is the construction of a left adjoint to the forgetful functor, generalizing the free permutative category on a symmetric multicategory. The construction of the left adjoint requires some machinery that may be of interest in its own right.
Fall 2014

 Dec 042014
Charmaine Sia (Harvard)
Structures on forms of Ktheory
This is a Thursday talk
In the early 1970's, Morava studied forms of Ktheory and observed that they have interesting number theoretic connections. Until very recently, forms of Ktheory have not been studied in greater depth and integrated into the theory of tmf. I establish some expected structured ring spectra results on forms of Ktheory. Forms of objects are usually classified by Galois cohomology. Based on the structured ring spectra results established, I give a criterion for distinguishing homotopy equivalence classes of forms of Ktheory via a computation in the second homotopy group of the spectrum.

 Dec 022014
Michael Donovan (MIT)
Operations in the Adams spectral sequence for simplicial F_{2}algebras
While in the homotopy theory of simplicial algebras, the homotopy of "spheres" is known, the unstable Adams spectral sequence is very far from degenerate. I’ll explain the various unstable operations that act on E_{2}, and discuss some composite functor spectral sequences that may be used to calculate E_{2}.

 Nov 252014
Zhen Huan (UIUC)
QuasiElliptic Cohomology
The elliptic cohomology associated to the Tate curve can be expressed as equivariant Ktheory. I will introduce a theory Ell constructed by my advisor Charles Rezk. Its form is very similar to that of Tate Ktheory but is not an elliptic cohomology, which brings its name "quasielliptic cohomology". Some constructions on the two theories have similar forms, such as the power operations, though some can be interpreted more easily on the new one. I will first set up the theory and show some properties of it. After that I will show my construction of its power operations following Nora Ganter's work, and then an application using its explicit form. I will also talk about the spectra representing it up to weak equivalence and show my construction of it. Moreover, the spectra can be "globalized" in Stefan Schwede's global homotopy theory.

 Nov 182014
Aaron MazelGee (UC Berkeley)
GoerssHopkins obstruction theory for ∞categories
GoerssHopkins 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 tractable 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.

 Nov 112014
Nick Gurski (University of Sheffield)
Homotopy coherent distributive laws
A distributive law is a categorical tool for combining two algebraic structures, each given via some monad, into a single algebraic structure in which the resulting monad is the composite of the original two. Moving into the world of homotopy theory, we might be interested in telling the same story with homotopy coherent monads on some quasicategory. I will give two approaches to this subject, one 2categorical and one using more combinatorial methods from simplicial homotopy theory. I will try to give some indication why both approaches have clear advantages, as well as why they are equivalent. This is joint work with James Cranch.

 Nov 042014
Eric Peterson (UC Berkeley)
Determinantal Ktheory and an application
Chromatic homotopy theory is an attempt to divide and conquer algebraic topology by studying a sequence of what were 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. To give an appreciation for the subject, 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."

 Oct 282014
Prasit Bhattacharya (Indiana University)
Higher Associativity of Moore spectra
Not much is known about homotopy coherent ring structures of the Moore spectrum M_{p}(i) (the cofiber of p^{i} selfmap 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).

 Oct 232014
Mark Behrens (University of Notre Dame)
On the tmfresolution
This is a Thursday talk. The pretalk will be from 34pm in E203, and the talk will be 4:305:30 in E207.
Mahowald used the bobased Adams spectral sequence to compute the 2primary v_{1}periodic stable homotopy groups of spheres, and from this he deduced the v_{1}periodic telescope conjecture for p = 2. I will discuss what I know about the tmfresolution at p = 2, incorporating work of many collaborators over the years, most significantly Tyler Lawson, Kyle Ormsby, Vesna Stojanoska, and Nat Stapleton.

 Oct 212014
Saul Glasman (MIT)
The Cyclotomic Hodge Filtration
In the 80s, Loday defined a filtration on the Hochschild homology of a commutative algebra A that recovers the Hodge filtration of the de Rham complex of A in the case where A is a smooth \mathbbQalgebra. The subject of this talk is a refinement of this to a filtration by spectra of the topological Hochschild homology of a commutative ring spectrum. I'll discuss why one would want such a thing; in particular, similar objects are implicated in the study of special values of Lfunctions. Then I'll talk about how to construct it, and if time permits, how to lift it to a filtration of topological cyclic homology using the techniques of equivariant stable homotopy theory.

 Oct 142014
Omar Camarena (Harvard)
The universal property of the multiplicative structure of Thom spectra
The theory of Thom spectra associated to stable spherical fibrations has been developed by many authors including Mahowald, Lewis, May, Quinn, Ray and Sigurdsson. A stable spherical fibration is classified by a map X → BGL_{1}(S) and Lewis showed that if this map is an infinite loop map or an nfold loop map then the Thom spectrum is an E_{∞} or E_{n}ring spectrum, respectively. Ando, Blumberg, Hopkins, Gepner and Rezk introduced a new approach to Thom spectra using the language of ∞categories. Using their approach, we will explain how to apply some simple (∞)category theory to study multiplicative structures on Thom spectra, proving a generalization of Lewis's theorem and moreover characterizing the ring structure by a universal property. We will also describe some applications of this characterization which are work in progress with Tobias Barthel.

 Oct 072014
Michael Andrews (MIT)
Nonnilpotent elements in motivic homotopy theory
Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that nonnilpotent self maps on finite plocal spectra induce nonzero homomorphisms on BPhomology. Motivically, this theorem fails to hold: we have a motivic analog of BP and whilst η:S^{1,1}→ S^{0,0} induces zero on BPhomology, it is nonnilpotent. Recent work with Haynes Miller has led to a computation of η^{−1}π_{*,*}(S^{0,0}); we found it to have a very simple description.
I'll introduce the motivic homotopy category, the motivic Adams SS and the motivic AdamsNovikov SS before describing this theorem. Then I'll talk about future directions and the possibility that there are more periodicity operators in motivic chromatic homotopy theory than in the classical story.
Pretalk: The music of spheres: the Adams spectral sequence and periodicity.
We’ll recall that the homotopy groups of spheres stabilize. The talk will be concerned with a tool for computing these stable groups. First, we’ll examine why Ext groups show up in topology and after discussing the purpose of a spectral sequence we’ll play with the Adams SS. We’ll get a feel for how algebraic relations are displayed by SS charts, how to read off homotopy groups from the charts, and we’ll make precise to what extent the algebra is reflecting the topology. Then we’ll take a step back to see repeating patterns in the charts and describe the topology underpinning these observations. If time permits, I'll display, in picture form, a theorem of mine that completely describes the Adams spectral sequence at an odd prime p above a line of slope 1/(p^{2}−p−1). 
 Sep 302014
Guozhen Wang (MIT)
Monochromatic unstable homotopy groups of spheres
The EHP sequence, Goodwillie tower, and the BousfieldKuhn functor are powerful tools in understanding unstable homotopy theory. I will show how we can combine these to study the v_{2} periodic phenomenon in unstable homotopy groups of spheres. As an example, I will compute the K(2)localized homotopy groups of the unstable 3sphere.
Spring 2014

 Jun 052014
Marc Stephan (EPFL)
This is a Thursday talk.

 Jun 032014
Kate Poirier (CUNY)

 May 292014
Alissa Crans (Loyola Marymount University)
This talk will be at 4pm.

 May 272014
Andrew Baker (University of Glasgow)
Characteristics for E_{∞} ring spectra
I will show how to extend the algebraic notion of characteristic at least for connective spectra, obtaining a homotopically meaningful concept. I will then discuss some examples of constructions that involve Hopf invariant 1 elements, leading to some conjectures about characteristics of HZ, ku, ko, tmf (at 2).
Pretalk: Power operations, coactions and cellular constructions for E_{∞} ring spectra
First I'll review the basic ideas involved in building power operations, then explain how to describe their interwining with coactions. Then I'll discuss cellular constructions, including the relationship with topological AndreQuillen theory.

 May 202014
Dan Isaksen (Wayne State Univeristy)
A tour of the Adams spectral sequence
I will present the results of a detailed computational analysis of the motivic and classical Adams spectral sequences at the prime 2. Some highlights include:
 corrections to previously published results about stable homotopy groups beyond the 47stem.
 a brute force approach to the existence of the 62dimensional Kervaire class.
 a conjectural description of the homotopy groups of the etalocal motivic sphere.
 an outline of a program to compute new stable stems by combining motivic Adams E2term data with classical AdamsNovikov E2term data.
Pretalk: Higher compositions in algebra and topology
I will give a general introduction to the subject of Massey products and Toda brackets, suitable for all graduate students. I will describe some of their many uses in algebra and topology, and I will present some new results about the general theory of fourfold Massey products.

 May 132014
Unni Namboodiri Lectures in Geometry and Topology

 Apr 292014
Tobias Barthel (Harvard University)

 Apr 242014
Daniel Schaeppi (University of Western Ontario)
This will be a Thursday talk. There will be a pretalk at 3pm, with the main talk at 4:30.

 Apr 222014
Irakli Patchkoria (University of Copenhagen)
Towards integral equivariant rigidity
Let p be an odd prime and G a finite group such that p does not divide the order of G. We show that the plocal Gequivariant stable homotopy category, indexed on a complete Guniverse, has a unique equivariant model in the sense of Quillen model categories. The proof is a generalization of Schwede's proof in the nonequivariant case and uses that the group algebra \mathbbF_{p}[G ×G] is semisimple. Combining this with our previous result about the 2local Gequivariant stable homotopy category, we get an integral equivariant rigidity theorem for any finite 2group.

 Apr 152014
Unni Namboodiri Lectures in Geometry and Topology

 Apr 012014
Emanuele Dotto (MIT)
Equivariant Diagrams and Equivariant Excision
In a nonequivariant setting, a functor is excisive if it takes homotopy pushout squares to homotopy pullback squares. Given a finite group G and a functor from Gspaces to Gspaces (or Gspectra), this definition of excision does not "capture enough equivariancy". For example the category of endofunctors of Gspaces with this property does not model Gspectra. One solution is to replace squares by "cubes with action", where the group is allowed to act on the whole diagram by permuting its vertices. I will talk about the homotopy theory of these equivariant diagrams and relate the resulting notion of equivariant excision to previous work of Blumberg. As an application of this theory, I will give a proof of the Wirthmuller isomorphism that uses only the equivariant suspension theorem and formal manipulations of limits and colimits.
Winter 2014

 Mar 252014
Spring break

 Mar 182014
Exam week

 Mar 112014
Marcy Robertson (University of Western Ontario)
Higher Prop(erad)s
Prop(read)s are an extension of the notion of operad which allow one to model structures with manytomany operations, such as various kinds of bialgebras. In this talk we will discuss uptohomotopy versions of properads, as well as applications.

 Mar 042014
No seminar

 Feb 252014
Akhil Mathew (Harvard University)
The Galois group of a stable homotopy theory
To a "stable homotopy theory" (a presentable, symmetric monoidal stable ∞category), we naturally associate a category of finite étale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. This construction builds on, and generalizes, ideas of many authors, and includes the étale fundamental group of algebraic geometry as a special case. We calculate the Galois groups in several examples, both in settings of rational and padic homotopy and in "chromatic" stable homotopy theories. For instance, we show that the Galois group of the periodic E_{∞}algebra of topological modular forms is trivial, and, extending work of Baker and Richter, that the Galois group of K(n)local stable homotopy theory is an extended version of the Morava stabilizer group.

 Feb 182014
Beren Sanders (UCLA)
Restriction to a subgroup as an étale localization
I will discuss some recent work (joint with Paul Balmer and Ivo Dell'Ambrogio) which demonstrates that in many equivariant settings (such as equivariant stable homotopy theory, modular representation theory, or equivariant Ktheory), restriction to a subgroup can be regarded as a generalized kind of localization which is conceptually analogous to localizing with respect to the étale topology rather than the Zariski topology. Aspects of the proof will be explained but my primary aim will be to advertise why this result is interesting and to mention potential applications.

 Feb 112014
Fabian Hebestreit (Münster University)
Twisted Spin cobordism and positive scalar curvature
In the first half I will describe the basic motivating question of the talk, namely when a closed manifold admits a metric of positive scalar curvature and what is known about the answer. In particular I will explain how this question reduces to calculations in certain cobordism rings, due to a result of Gromov and Lawson.
In the second half I will give an overview of these calculations in the case of Spincobordism, where they were carried out by Stolz and Fuehring, and then address their generalisation to the case of twisted Spincobordism, which is ongoing joint work of Joachim and myself. In particular I will exhibit a generalisation of the AndersonBrownPeterson splitting and compute the mod 2 cohomology of the twisted, connective, real Ktheory spectrum.

 Feb 042014
Kirsten Wickelgren (Georgia Tech)
Towards a motivic simlpicial EHP spectral sequence
This talk will start with a discussion of the classical EHP sequence of James and Toda, and then move towards a version in A^{1} homotopy theory using the simplicial suspension map. This is joint work in progress with Ben Williams.

 Jan 232014
Nora Ganter (University of Melbourne)
The Kac character formula as a pushforward in elliptic cohomology
The pretalk for this talk will be from 1:30 to 2:30, and the talk itself will run from 2:45 to 3:45.
I will explain how to obtain the Weyl character formula as a pushforward in equivariant Ktheory (this is due to Atiyah and Bott) and discuss what happens when Ktheory is replaced by elliptic cohomology.

 Jan 212014
Marc Hoyois (Northwestern University)
A fixedpoint theorem in motivic homotopy theory
I will explain how stable motivic homotopy gives an interesting arithmetic refinement of the GrothendieckLefschetz trace formula over nonalgebraically closed fields.

 Jan 072014
David White (Wesleyan University)
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 noncofibrant 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 noncofibrant operads, and about how these generalized axioms interact with Bousfield localization.
Fall 2013

 Dec 032013
Irina Bobkova (Northwestern University)
A resolution of the spectrum E^{hS21} at the prime 2
The homotopy of the plocal sphere spectrum S is determined by a family of localizations L_{K(n)}S with respect to Morava Ktheories K(n). We will discuss some computations when p, n=2. Considerable information here can be derived from the action of the Morava stabilizer group on the LubinTate theory. Goerss, Henn, Mahowald and Rezk have constructed a resolution of the K(2)local sphere at the prime 3 which allows to simplify computations of π_{*}L_{K(2)}S. We will discuss a generalization of their work to the prime 2 and construct a resolution of the spectrum E^{h\mathbbS21}, which is closely related to the K(2)local sphere at the prime 2.

 Nov 262013
Drew Heard (University of Melbourne)
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 nth Morava Ktheory. 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.
In the pretalk I'll discuss generalities of the Picard group of a symmetric monoidal category, giving examples from algebra, topology and homological algebra.

 Nov 212013
John Lind (Johns Hopkins University)
Equivariantly Twisted Cohomology Theories
This is a Thursday talk, which will take place at 1:30pm in E203.
Twisted Ktheory is a cohomology theory whose cocycles are like vector bundles but with locally twisted transition functions. If we instead consider twisted vector bundles with a symmetry encoded by the action of a compact Lie group, the resulting theory is equivariant twisted Ktheory. This subject has garnered much attention for its connections to conformal field theory and representations of loop groups. While twisted Ktheory can be defined entirely in terms of the geometry of vector bundles, there is a homotopytheoretic formulation using the language of parametrized spectra. In fact, from this point of view we can define twists of any multiplicative generalized cohomology theory, not just Ktheory. The aim of this talk is to explain how this works, and then to propose a definition of equivariant twisted cohomology theories using a similar framework. The main ingredient is a structured approach to multiplicative homotopy theory that allows for the notion of a Gtorsor where G is a grouplike A_{∞} space.

 Nov 192013
Chris Kapulkin (University of Pittsburgh)
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.

 Nov 122013
Robert Hank (University of Minnesota)
Higher multiplicative structure in homotopy theory
Given a multiplicative structure and some notion of homotopy, Toda brackets and Massey products are able to detect higher multiplicative structure. Being able to detect higher structure can be useful in spectral sequence computations or obstruction theory arguments. The common examples where this structure can be detected are chain complexes and spaces. In this talk we will revisit these known definitions, recast in a different light, and use that to understand more about the higher structure in more general settings such as A_{∞} categories and ∞categories.

 Nov 052013
Karol Szumilo (University of Bonn)
Cofibration categories and quasicategories
Classically, homotopy theories are described using homotopical algebra, e.g. as model categories or (co)fibration categories. Nowadays, they are often formalized as higher categories, e.g. as quasicategories or complete Segal spaces. These two types of approaches highlight different aspects of abstract homotopy theory and are useful for different purposes. Thus it is an interesting question whether homotopical algebra and higher category theory are in some precise sense equivalent.
In this talk I will concentrate on cofibration categories and quasicategories. I will discuss some basic features of both notions building up to a result that the homotopy theory of cofibration categories is indeed equivalent to the homotopy theory of cocomplete quasicategories.

 Oct 292013
Cary Malkiewich (Stanford University)
Coassembly maps, gauge groups, and Novikov conjectures
Calculus of functors is a powerful technique from homotopy theory, which studies computationally difficult constructions by means of linear approximations. We will describe a new variant of this calculus, based on the embedding calculus of Weiss and Goodwillie. This gives a sequence of approximations to the stable gauge group of a principal bundle, in which the linear approximation is the CohenJones string topology spectrum LM^{−TM}. We will finish with an application (in progress) to algebraic Ktheory, extending work of Bokstedt, Hsiang, and Madsen on the Atheory Novikov conjecture.

 Oct 222013
Angelica Osorno (Reed College)
A new equivariant infinite loop space machine
An equivariant infinite loop space machine turns categorical or algebraic data into genuine spectra. While infinite loop space machines have been a crucial part of homotopy theory for decades, equivariant versions have been less well understood until recently.
I will describe joint work with A.M. Bohmann in which we build an equivariant infinite loop space machine that starts with diagrams of categories on the Burnside category and produces a genuine Gspectrum via the work of GuillouMay. This machine readily applies to produce EilenbergMacLane spectra for Mackey functors and equivariant topological Ktheory.

 Oct 152013
Gregory Arone (University of Virginia)
The sphere operad
This talk has been canceled.
By a sphere operad we mean an operad in the category of pointed spaces, where the nth space is S^{n−1}, and which is stably equivalent to the coendomorphism operad of S^{1}. We describe a concrete model for such an operad, which has several properties that seem desirable.
As an application, we use it to define a tower of operads in spectra, which we call the sphere prooperad. It turns out that the derivatives of a homotopy functor from Spectra to Spectra have the structure of a module over the sphere prooperad, and one can recover the Taylor tower of a functor from this module structure.
The first part of the talk is joint work with M. Kankaanrinta and the second part is joint with M. Ching.

 Oct 082013
Aaron Royer (University of Texas in Austin)
Generalized String Topology and Derived Koszul Duality
The existence of intersectiontype operations on the homology of the free loop space of an oriented manifold was first brought to broad attention by Chas and Sullivan in the late 1990s. Building on the work of several previous authors, we place these structures in the modern context of highly structured spectra by constructing a lax symmetric monoidal functor from parametrized spaces (over a closed manifold) to symmetric spectra. Employing derived Koszul duality and the recentlydeveloped theory of ∞categories, we give an alternative characterization of this "generalized string topology" functor, proving a coherence theorem for certain generalized PontrjaginThom collapse maps in the process, and illuminating its role in parametrized stable homotopy theory.
Pretalk Title: ∞categories, parametrized homotopy and Thom spectra
Abstract: We will quickly introduce the theory of ∞categories, emphasizing their usefulness in making formal arguments about homotopical functors. As an application, we discuss the perspective on parametrized stable homotopy theory and Thom spectra from arxiv:0810.4535 and the resulting technical and conceptual advantages (and disadvantages.)

 Oct 012013
Emily Riehl (Harvard University)
Algebraic perspectives on (generalized) Reedy categories
Reedy categories are small categories equipped with a degree filtration on objects. The axioms allow for an inductive definition of diagrams and natural transformations, which is useful, in particular, for understanding the homotopical behavior of limits and colimits of diagrams with Reedy shape.
The pretalk will summarize a recent expository paper with Dominic Verity. We give a canonical presentation of the hom bifunctor as a cell complex built from pushoutproducts of "boundary inclusions", which translates to a canonical presentation of any diagram or natural transformation as a (relative) cell complex and as a (relative) Postnikov tower whose cells are built from the latching or matching maps. This makes the proof of the Reedy model structure essentially trivial and leads to a geometric criterion characterizing the Reedy categories which give formulae for homotopy (co)limits.
After tea I will report on work in progress to extend this theory to generalized Reedy categories: We propose that algebraic weak factorization systems are a natural tool to define the equivariant factorizations required to extend diagrams from one degree to the next.