Notre Dame Topology Seminar

Felix Klein Seminar (on geometric things)
Notre Dame Department of Mathematics

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Questions? Contact Mark Behrens

Fall Semester 2015 Schedule

10:00–11:00 (Eastern Time) Wednesdays @ 258 Hurley Hall (unless otherwise noted)

2 September: Zhouli Xu (University of Chicago); Uniqueness of smooth structures on spheres.
9 September: Prasit Bhattacharya (University of Notre Dame); Higher associativity of Moore spectra.
30 September: David Gepner (Purdue University); Cohomology theories associated to infinity-topoi, globally equivariant spectra, and elliptic cohomology
7 October: John Harper (The Ohio State University); Derived Koszul duality of spaces and structured ring spectra
Thursday, 8 October, 12:50-1:50, Hurley 258: Sarah Yeakel (UIUC); A chain rule for Goodwillie calculus
14 October: Ben Knudsen (Northwestern); Rational homology of configuration spaces via factorization homology
19 October: Fall Break (no seminar)
Tuesday, 27 October, 4:30-5:30, Hayes-Healy 229: Don Larson (Penn State Altoona); Modular forms and the beta family
18 November: Pedram Hekmati (University of Adelaide); Topological T-duality and Hodgkin’s theorem
2 December: Dylan Wilson (Northwestern); Orienting tmf with level structure
9 December: Brian Williams (Northwestern); Universal holomorphic factorization algebras

Spring Semester 2016 Schedule

3:00–4:00 (Eastern Time) Tuesdays @ 258 Hurley Hall

19 January: Yifei Zhu (Northwestern); Local moduli for elliptic spectra
26 January: COLLOQUIUM: Christopher Schommer-Pries (MPI); 4:00; TBA
MONDAY, 1 February: COLLOQUIUM: Agnes Beaudry (Chicago); 4:00; TBA
16 February: Scott Slinker (Virginia); Naming 2-variable modular forms at 3
23 February: Dev Sinha (U. Oregon); Goodwillie-Weiss calculus and knot theory
1 March: Andrew Salch (Wayne State); Complex multiplication in homotopy theory
8 March: Spring Break (no seminar)
15 March: Augusto Stoffel (Notre Dame); Holonomy of gerbes around super loops and twisted field theories
22 March: Fabian Hebestreit (Bonn); Stable diffeomorphism groups of odd-dimensional manifolds
29 March: Vasily Dolgushev (Temple); The Intricate Maze of Graph Complexes
5 April: Ryan Grady (Perimeter Institute); The BV formalism and some applications to index theory
19 April, 3:00-3:50, Hayes-Healy 229: John Francis (Northwestern); The cobordism hypothesis
26 April: Hiro Tanaka (Harvard); Lagrangian cobordisms and the Fukaya category
7 June: Eric Peterson; Determinantal K-theory and a few applications

Abstracts of Invited Talks

2 September 2015: Zhouli Xu

Uniqueness of smooth structures on spheres

In this talk, I will report recent progress that the 61-sphere has a unique smooth structure. Following results of Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, we show that the only odd dimensional spheres with a unique smooth structure are in dimension 1, 3, 5 and 61. Following recent work of Isaksen, we also show that in dimensions from 5 through 61, the only spheres with a unique smooth structure are in dimension 5, 6, 12, 56 and 61. Recent work of Behrens-Hill-Hopkins-Mahowald shows that the next sphere with a unique smooth structure, if exists, is in dimension at least 126. The computation of the stable homotopy groups of spheres at the prime 2 is essential to this result. I will review classical techniques and explain our new technique. This work is joint with Guozhen Wang.

9 September 2015: Prasit Bhattacharya

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 self-map on the sphere spectrum S^0), especially when i > 1. Stasheff developed a hierarchy of coherence for homotopy associative multiplications called A_n structures. The only known results are that M_p(1) is A_{p-1} and not A_p and that M_2(i) are at least A_3 for i > 1. In this talk, techniques will be developed to get estimates of `higher associativity' structures on M_p(i).

30 Septenber 2015: David Gepner

Cohomology theories associated to infinity-topoi, globally equivariant spectra, and elliptic cohomology

In this talk I will explain how an infinity-topos equipped with a suitable ring or module object gives rise to a (co)homology theory, locally defined on the infinity-topos, and how this specializes to various versions of (co)homology, such as sheaf cohomology, motivic cohomology, etc. in algebraic geometry, twisted cohomology, equivariant cohomology, etc. in algebraic topology, as well as similar sorts of things in other contexts. As an application, we will recover Schwede's global spectra as well as Lurie's equivariant elliptic cohomology. Finally, in the presence of a ring structure, we will see how to find invertible and dualizable objects and therefore maps which admit Thom isomorphisms or transfers with respect to these theories. This is joint work in progress with Thomas Nikolaus.

7 October 2015: John Harper

Derived Koszul duality of spaces and structured ring spectra

Consider a flavor of structured ring spectra that can be described as algebras over an operad O in spectra. A natural question to ask is when the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated Koszul dual comonad K can be modified to turn it into an equivalence of homotopy theories. In their 2012 Selecta Math. paper, Francis and Gaitsgory conjecture that replacing O-algebras with the full subcategory of homotopy pro-nilpotent O-algebras will do the trick. In joint work with Kathryn Hess we show that every 0-connected O-algebra is homotopy pro-nilpotent. This talk will describe recent work, joint with Michael Ching, that resolves in the affirmative the 0-connected case of the Francis-Gaitsgory conjecture. If time permits, we will also outline recent work, joint with Jake Blomquist, on derived Koszul duality for spaces.

Thursday, 8 October 2015, Hurley 258: Sarah Yeakel

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, we have found a more straight forward chain rule for endofunctors of spaces.

14 October 2015: Ben Knudsen

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.

Tuesday, 27 October 2015, 4:30-5:30, Hayes-Healy 229: Don Larson

Modular forms and the beta family

Let p be a prime greater than 3. In 2008, M. Behrens proved the existence of a 1-1 correspondence between beta elements in the p-primary Adams-Novikov spectral sequence and modular forms over Z up to certain congruence conditions depending on p. The proof used homotopical properties of a spectrum denoted Q. In this talk, I will briefly highlight some previous work on the homotopy of Q at the prime 3 (where Behrens' correspondence is not known to exist), and then I will describe work in progress at higher primes that attempts to make the correspondence explicit.

18 November 2015: Pedram Hekmati

Topological T-duality and Hodgkin’s theorem

A famous problem in topology, first solved by Hodgkin in 1967, is to determine the K-theory of compact simply-connected Lie groups. Hodgkin's original proof was extremely technical, motivating the discovery of a number of simpler proofs. In this talk I will present a new, surprisingly simple proof of Hodgkin's theorem using topological T-duality, an idea that originated in physics. This is based on joint work with David Baraglia.

2 December 2015: Dylan Wilson

Orienting tmf with level structure

Following Ando-Hopkins-Rezk, we will build various highly structured genera for Spin and String manifolds valued in modular forms with level structure. We will indicate which pieces of this argument are formal, and what must be done if we wish to give orientations by other ring spectra such as TAF or conjectural cohomology theories attached to the moduli of K3 surfaces.

9 December 2015: Brian Williams

Universal holomorphic factorization algebras

Factorization algebras are mathematical objects that conceptualize the notion of observables in physics. Costello and Gwilliam ([CG]) have made the explicit connection to physics by constructing a factorization algebra from the data of a perturbative quantum field theory. In this talk we focus on a particular type of factorization algebra that are present in 2-dimensional (chiral) conformal field theories. We show how to extract from certain 2d factorization algebras the structure of a vertex algebra, a central player of observables for 2d field theories. We highlight a construction of the Virasoro factorization algebra ([W]) and show how other important vertex algebras appear in this context. We then show that we can generalize the notion of a factorization algebra, which exists on a fixed manifold (space-time), to so-called "universal factorization algebras" which exist on all manifolds of a certain dimension and geometric type. In this setting we show that the objects mentioned in the first part of the talk exist as universal objects on the category of complex manifolds. If time permits we will mention progress on higher dimensional analogues of the above constructions.
[CG] Costello, Kevin; Gwilliam, Owen. "Factorization algebras in quantum field theory, Volume I"
[W] Williams, Brian. "The Virasoro vertex algebra and factorization algebras on Riemann surfaces"

19 January 2016: Yifei Zhu

Local moduli for elliptic spectra

A Morava E-theory 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 E-theory, in terms of explicit generators and quadratic relations.

26 January 2016: COLLOQUIUM - Christopher Schommer-Pries

TBA

1 February 2016: COLLOQUIUM - Agnes Beaudry

TBA

16 February 2016: Scott Slinker

Naming 2-variable modular forms at 3

The spectrum tmf is built from a sheaf on the moduli stack of elliptic curves, and so its homotopy is related to modular forms. If 2 and 3 are inverted, this gives the ring of classical modular forms generated by c_4, c_6, and the discriminant. The smash product tmf^tmf is related to “two-variable modular forms,” modular forms on pairs of elliptic curves together with an isogeny. Behrens, Ormsby, Stapleton, and Stojanovska have recently worked at the prime 2 to identify rational classes in terms of these classical forms. I am interested in working at 3. A curious phenomenon that arises here is that there is a subring isomorphic to a degree-shifted ring of one-variable modular forms with level structure. Rationally, this turns out to be a wedge summand of tmf^tmf and contains almost all of the alpha family.

23 February 2016: Dev Sinha

Goodwillie-Weiss calculus and knot theory

We present a model for the Goodwillie-Weiss tower for classical knots which is motivated by a homotopy theoretic view of linking number. To do so will require development of compactifications of configuration spaces. Then we will discuss the connections - both realized and conjectured - between this tower and the theory of Vassiliev finite-type invariants.

1 March 2016: Andrew Salch

Complex multiplication in homotopy theory

If A is a ring and F is a one-dimensional formal group law over an A-algebra R, we say that F "admits complex multiplication by A" if there is a ring homomorphism from A to the endomorphism ring of F whose induced action on the tangent space of F coincides with the given action of A on R. A formal group with a choice of complex multiplication by A is also called a "formal A-module." Formal A-modules of height 1 were used by Lubin and Tate in their solution to the p-adic version of Kronecker's Jugendtraum, that is, the computation of the abelian closure of any p-adic number field. Formal A-modules play a central role in both Drinfeld's and Carayol's approaches to local Langlands correspondences: in each case, the correspondence is realized by an action of a Galois group, a group of Hecke operators, and the automorphism group of a formal A-module, all acting on the cohomology of an appropriate deformation space of a formal A-module. In this talk I will describe some new results in this area and some number of topological and algebraic applications: 1. The computation of the classifying ring of formal A-modules. 2. The solution to Ravenel's problem on topological realization of formal A-modules. 3. The solution to Ravenel's Global Conjecture. 4. The computation of the cohomology of the automorphism group of a height 4 formal group law over a field of characteristic p > 5.

15 March 2016: Augusto Stoffel

Holonomy of gerbes around super loops and twisted field theories

A gerbe with band C^x is analogous to a line bundle, but one categorical level higher - a bundle with fiber the category of complex lines. Just like a complex line bundle with connection determines a C^x-valued function on the loop space of the base (the holonomy), a gerbe with connection determines a line bundle on the loop space. Motivated by field theories, we will consider a super version of that construction, where loops are 1|1-dimensional, and its relation with complexified twisted K-theory. The base space will actually be an orbifold, so the above concerns equivariant cohomology with respect to finite group actions.

22 March 2016: Fabian Hebestreit

Stable diffeomorphism groups of odd-dimensional manifolds

Understanding characteristic classes of manifolds bundles is one of the key topics in the study of manifolds and their automorphisms. The advent of cobordism categories through the work of Madsen, Tillmann and Weiss introduced a new method for studying such groups. Its application in high dimensions was spearheaded by Galatius and Randal-Williams in the case of highly connected even dimensional manifolds by identifying groups of stable characteristic classes with the cohomology of certain computationally accessible infinite loopspaces. Results of Ebert, however, sharply limited the efficacy of usual cobordism categories in odd dimensions. I will report on joint work with Nathan Perlmutter on an enhancement of odd dimensional cobordism categories, that avoid these difficulties. In particular our results imply that stable characteristic classes also form the cohomology of a certain infinite loopspace in this case. In contrast to the even dimensional situation, however, the homotopy type of this space remains largely unidentified and its exploration is work in progress.

29 March 2016: Vasily Dolgushev

The Intricate Maze of Graph Complexes

In the paper "Formal noncommutative symplectic geometry", Maxim Kontsevich introduced three versions of cochain complexes GC_Com, GC_Lie and GC_As "assembled from" graphs with some additional structures. The graph complex GC_Com (resp. GC_Lie, GC_\As) is related to the operad Com (resp. Lie, As) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes GC_Com, GC_Lie and GC_\As (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces. In my talk, I will describe the links between these graph complexes (and their modifications) to the cohomology of the moduli spaces of curves, the group of outer automorphisms Out(F_r) of the free group F_r on r generators, the absolute Galois group Gal(Qbar/Q)$ of rationals, finite type invariants of tangles, and the homotopy groups of embedding spaces.

5 April 2016: Ryan Grady

The BV formalism and some applications to index theory

I will recall the Batalin-Vilkovisky formalism as interpreted by Costello and discuss several examples. The examples will be described using one take on (smooth) derived geometry. In computing the observable theory (after quantization) of these examples we will recover sheaves of twisted differential operators and also the algebraic index theorem of Fedosov and Nest-Tsygan. If time permits, I will discuss other examples coming from Lie algebroids and higher dimensional sigma models.

19 April 2016: John Francis (3:00-3:50, Hayes-Healy 229)

The cobordism hypothesis

The cobordism hypothesis—after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie—asserts that for a suitable target C, there is an equivalence TQFT(C) = obj(C) between C-valued framed topological field theories and objects of C. I'll describe a proof of the cobordism hypothesis based on factorization homology. This construction of factorization homology "integrates" a higher category over a "moduli space of stratifications" of a smooth manifold. This is joint work with David Ayala.

26 April 2016: Hiro Tanaka

Lagrangian cobordisms and the Fukaya category

I'll introduce Fukaya categories (a central object in symplectic geometry and mirror symmetry) and talk about how the homotopy theory of cobordisms may recover the whole Fukaya category in a large class of examples.

7 June 2016: Eric Peterson

Determinantal K-theory and a few applications

Chromatic homotopy theory is an attempt to divide and conquer algebraic topology by studying a sequence of what we’d first assumed to be “easier” categories. These categories turn out to be very strangely behaved — and furthermore appear to be equipped with intriguing and exciting connections to number theory. 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”.