Notre Dame Topology Seminar

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

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Unless otherwise noted - all talks have two parts - Part I (2:30–3:30 Eastern Time) is a general talk, intended for a general topology audience, Part II (4:00–5:00) can be aimed at specialists. The default room is 258 Hurley Hall.

Fall Semester 2016 Schedule

Sept 6: Andrew Putman (Notre Dame); The high dimensional cohomology of the moduli space of curves with level structures
Sept 13: Anibal Medina (Notre Dame); A Non-linear Dold-Kan Theorem and Algebraic Surgery
Sept 27: Chris Schommer-Pries (Notre Dame); Tori detect invertible topological field theories
Oct 11: Ivan Conteras (UIUC); Poisson geometry and BV-BFV theories
Oct 18: Fall Break (no seminar)
Oct 25: Nick Rozenblyum (Chicago); Topological applications of quantization in derived geometry
Nov 1: Bogdan Gheorge (Wayne State); Motivic fields and w_n-periodicity
WED, Nov 9, 3:00-4:00, DeBartolo 138: Leanne Merrill (Oregon); Algebraic v_n self maps at the prime 2
Nov 15: Irina Bobkova (Rochester); The K(2)-local Picard group at the prime 2
Nov 22: Luis Alexandre Pereira (Virginia); Genuine equivariant operads
Nov 29: Clark Barwick (MIT); Talk 1: 2:30-3:30 - The Hill–Hopkins Program; Talk 2: 4:00-5:00 - oo-Categories for End-Users
THURS, Dec 1, 11:00-12:00, Hurley 258: Nathaniel Stapleon (Regensburg); The character of the total power operation
Dec 6: Katrin Wehrheim (Berkeley); Talk 1: 2:30-3:30 - Quilts and Floer field theory I; Talk 2: 4:00-5:00 - Quilts and Floer field theory II
THURS, Dec 8, 11:15-12:15, Hurley 258: Jesse Wolfson (Chicago); Coincidences of homological densities, predicted by arithmetic
Dec 13: Gabriel Angelini-Knoll (Wayne State); Talk 1: 2:30-3:30 - Approximating algebraic K-theory of ring spectra; Talk 2: 4:00-5:00 - Periodic phenomena in iterated algebraic K-theory of finite fields

Spring Semester 2017 Schedule

Jan 24: Pelle Steffens (Amsterdam); Rozansky-Witten Theory: Towards an Atiyah-Segal TQFT from BV-BFV quantization
Jan 31: Nicolas Ricka (Wayne State); Motivic Modular Forms
Feb 14: 2:30-3:30: Donald Youmans (Geneva); Geometric Quantization and Hitchin's Connection; 4:00-5:00: Riccardo Iraso (SISSA); 2-dimensional Yang-Mills theory and perturbative BV-BFV formalism
Feb 21: Aaron Mazel-Gee (The Ohio State University); The geometry of the cyclotomic trace
Feb 28: Noah Snyder (Indiana University); The SO(3) action on the space of finite tensor categories
March 7: Corey Bregman (Rice); Kaehler manifolds and abelian-by-surface extensions
Mar 14: Spring Break (no seminar)
March 21: Mona Merling (JHU); Equivariant A-theory
THURSDAY, March 30 - 2:00-3:00 Hurley 258: Tom Church; The topological category of graphs
April 4: Arunima Ray (Brandeis); 4-dimensional analogues of Dehn’s lemma
April 11: Henry Horton (Indiana University); Symplectic instanton homology: SO(3)-bundles, functoriality, and Dehn surgery
April 23: Chris Schommer-Pries; The algebraic K-theory of manifolds via manifolds
May 2: Bena Tshishiku (Harvard); Mapping class group of a K3 surface

Abstracts of Invited Talks

Sept 6: Andrew Putman

The high dimensional cohomology of the moduli space of curves with level structures

I will explain how to find a vast amount of unstable rational cohomology in the moduli space of curves with level structures. This is joint work with Neil Fullarton.

Sept 13: Anibal Medina

A Non-linear Dold-Kan Theorem and Algebraic Surgery

The normalized chain functor from simplicial sets to chain complexes of abelian groups, admits a factorization by the free functor to simplicial abelian groups followed by the Dold-Kan functor to chain complexes. The Dold-Kan Theorem states that this second functor is full and faithful. The first part of this talk will focus on the construction of a new factorization using a full and faithful functor and a forgetful one. The proposed answer uses the category of chain complexes enriched with an E-infinity coalgebra structure. The second part of this talk will use the category of comodule over such coalgebras in the study of topological manifold structures on a given homotopy type.

Sept 27: Chris Schommer-Pries

Tori detect invertible topological field theories

Topological field theories assign invariants to manifolds of various dimension. These invariants are morphisms or objects in a possibly higher category. The simplest field theories are those which are invertible, those which assign invertible values at all manifolds. Such theories can be classified and computed using stable homotopy theory. We will discuss recent work which shows that if your field theory is at least one extended (so assigns values to d, d-1, and d-2 manifolds), then the field theory is invertible if and only if the value assigned to the (d-1)-torus is invertible.

Oct 11: Ivan Contreras

Poisson geometry and BV-BFV theories

The BV-BFV formalism for field theories, developed by Cattaneo, Mnev and Reshetikhin, has been successful in unifying different classes of field theories such as gauge, topological, supersymmetric theories, and describing their quantization, as well as the presence of boundary. The purpose of this talk is to describe two applications of this formalism arising from two problems Lie theory and deformation theory, respectively: the integration of Poisson brackets and the deformation quantization problem. It turns out that the symplectic formulation of the Poisson sigma model, a special BV-BFV theory, gives a procedure to integrate arbitrary Poisson brackets, whereas its integration produces Kontsevich's star products. If time allows, we will describe natural generalization of these two problems, namely, the integration of Poisson brackets in higher genus, and the quantization of relational symplectic groupoids. This is joint work with A. Cattaneo.

Oct 25: Nick Rozenblyum

Topological applications of quantization in derived geometry

Derived geometry, and particularly the theory of shifted symplectic structures, has recently become a central tool in mathematical physics. Roughly, the theory of shifted symplectic structures is a homotopical version of symplectic geometry. In addition to being a natural setting for the BV approach to Feynman integration, this theory provides a robust framework for various counting problems in geometry and topology, such as the Casson invariant and its generalizations. I will give a brief overview of the general theory and describe some of the topological applications.

Nov 1: Bogdan Gheorghe

Motivic fields and w_n-periodicity

The setting of this talk is stable motivic homotopy theory over Spec C, at p=2. Morel showed that eta is not nilpotent in the motivic stable stems over Spec C, by seeing it in Milnor-Witt K-theory. Since none of the motivic Morava K-theories detect eta, 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(eta) 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 perioficity operators w_0, w_1, ... , conjectured by Michael Andrews. We then show how to construct the fields K(w_n) as well as a Brown-Peterson spectrum wBP. If time permits, we indicate what is known about the w_n-family and formulate some conjectures about these periodic operators

WED, Nov 9: Leanne Merrill (3:00-4:00), DeBartolo 138

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 Hopkins-Devinatz-Smith together with the Periodicity Theorem of Hopkins-Smith describes non-nilpotent self maps of finite spectra. The Morava K- theories 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 Palmieri- Sadofsky 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 value of i.

Nov 15: Irina Bobkova

The K(2)-local Picard group at the prime 2

The group of invertible objects in a symmetric monoidal category is a basic invariant of the category. The Picard group of the category of spectra is known to consist only of the sphere spectrum and its suspensions. The problem becomes significantly richer when the category of spectra is localized with respect to Morava K-theories. The Picard group of the category of K(2)-local spectra is known for odd primes. I will talk about finite resolutions in the K(2)-local category and how we use them to compute the Picard group of the K(2)-local category in the last open case, p=2. This is joint work with Beaudry, Goerss and Henn.

Nov 22: Luis Alexandre Pereira

Genuine equivariant operads

A fundamental result in equivariant homotopy theory due to Elmendorf states that the homotopy theory of G-spaces, with w.e.s measured on all fixed points, is Quillen equivalent to the homotopy theory of G-coefficient systems in spaces, with w.e.s measured at each level of the system. Furthermore, Elmendorf's result is rather robust: suitable analogue results can be shown to hold for, among others, the categories of (topological) categories and operads. However, it has been known for some time that in the G-operad case such a result does not capture the "correct" notion of weak equivalence, a fact made particularly clear in recent work of Blumberg and Hill discussing a whole lattice of "commutative operads with only some norms" that are not distinguished at all by the notion of w.e. suggested above. In this talk I will talk about one piece of a current joint project with Peter Bonventre which aims at providing a more diagrammatic understanding of Blumberg and Hill's work using a notion of G-trees, which are a somewhat subtle generalization of the trees of Cisinski-Moerdijk-Weiss. More specifically, I will describe a new algebraic structure, which we dub a "genuine equivariant operad", which naturally arises from the study of G-trees and which we conjecture to be the analogue of coefficient systems in the "correct" analogue of Elmendorf's theorem for G-operads.

Nov 29: Clark Barwick

Talk 1: 2:30-3:30 “The Hill–Hopkins Program”

Abstract: With motivation from various algebraic examples, we describe a far-flung project to realize a program suggested Mike Hill and (in part) Mike Hopkins.

Talk 2: 4:00-5:00 “oo-Categories for End-Users”

Abstract: I’ll explain an under-appreciated feature of quasicategories which makes them extremely well-adapted for giving very concrete constructions; we’ll explain how to make many of the constructions in our main talk precise.

THURS, Dec 1, 11:00-12:00, Hurley 258: Nathaniel Stapleton

The character of the total power operation

In the 90's Goerss, Hopkins, and Miller proved that the Morava E-theories are E_oo-ring spectra in a unique way. Since then several people including Ando, Hopkins, Strickland, and Rezk have worked on explaining the affect of this structure on the homotopy groups of the spectrum. In this talk we will show how a form of character theory due to Hopkins, Kuhn, and Ravenel can be used to reduce this problem to a combination of combinatorics and the GL_n(Q_p)-action on the Drinfeld ring of full level structures which shows up in the local Langlands correspondence.

Dec 6: Katrin Wehrheim

Quilts and Floer field theory

Part I: 2:30-3:30

Part II: 4:00-5:00

Floer field theory is a construction principle for e.g. 3-manifold invariants (from my joint work with Chris Woodward and motivated by Atiyah-Floer type conjectures) which proceeds by decomposition in a bordism category and a partial functor to the symplectic category. This talk will provide an introduction to the 2-categorical structures in topology (Bor), symplectic geometry (Symp), and algebra (Cat), with which the Floer field construction principle can be formalized and generalized to a Lurie-type extension principle "Any Floer field theory Bor_2+1 --> Symp --> Cat which satisfies a quilted naturality axiom has a natural extension to a 2-functor Bor_2+1+1 --> Symp --> Cat." The core of this theory is a generalization of string diagrams to so-called quilt diagrams, which in the symplectic category are realized in terms of a PDE by "pseudoholomorphic quilts".

THURS, Dec 8, 11:15-12:15, Hurley 258: Jesse Wolfson

Coincidences of homological densities, predicted by arithmetic

Basic questions in analytic number theory concern the density of one set in another (e.g. square-free integers in all integers). Motivated by Weil's number field/function field dictionary, we introduce a topological analogue measuring the “homological density” of one space in another. In arithmetic, Euler products can be used to show that many seemingly different densities coincide in the limit. By combining methods from manifold topology and algebraic combinatorics, we discover analogous coincidences for limiting homological densities arising from spaces of 0-cycles (e.g. configuration spaces of points) on smooth manifolds and complex varieties. We do not yet understand why these topological coincidences occur. This is joint work with Benson Farb and Melanie Wood.

Dec 13: Gabriel Angelini-Knoll

Talk 1: 2:30-3:30 - Approximating algebraic K-theory of ring spectra

In order to understand arithmetic properties of ring spectra, which generalize rings, we need to compute invariants like algebraic K-theory. The best approach to computing algebraic K-theory in recent years is to approximate algebraic K-theory by a simpler invariant called topological Hochschild homology. Topological Hochschild homology has a rich equivariant structure that can be used to give closer and closer successive approximations to algebraic K-theory. In my talk I will survey the general theory, called trace methods, and I will describe a new tool for computing (higher order) topological Hochschild homology that is joint work with A. Salch.

Talk 2: 4:00-5:00 - Periodic phenomena in iterated algebraic K-theory of finite fields

In the early 2000’s, Ausoni and Rognes conjectured that a “higher chromatic height” version of the Quillen-Lichtenbaum conjecture holds for algebraic K-theory of certain ring spectra. Loosely, the so called red-shift conjecture suggests that algebraic K-theory should take spectra of chromatic height n to spectra of chromatic height n+1. I will discuss formulations of this conjecture that apply to the iterated algebraic K-theory of certain finite fields. In particular, one may conjecture that if a spectrum detects the nth Greek letter family then algebraic K-theory of that spectrum detects the (n+1)st Greek letter family. Due to work of D. Quillen and J. F. Adams, algebraic K-theory of certain finite fields detects the alpha family. In my talk, I will show that iterated algebraic K-theory detects the beta family confirming a case of this conjecture. The proof involves successive approximations to algebraic K-theory using the fixed points of topological Hochschild homology with respect to cyclic group actions. I will also give some evidence that suggests that the “telescopic complexity” formulation of the red-shift conjecture does not hold in the case of finite fields.

Jan 24: Pelle Steffens

Rozansky-Witten Theory: Towards an Atiyah-Segal TQFT from BV-BFV quantization

The derived/homological methods of Batalin and Vilkovisky are generally regarded as the most powerful approach to quantizing a gauge theory while respecting its symmetries; accordingly, the technique lies at the heart of most modern mathematical treatments on perturbative Quantum Field Theory (see Costello-Gwilliam for example). Recently, Cattaneo, Mnev and Reshetikhin extended these ideas to topological gauge theories on manifolds with boundary, whose quantization would lead to TQFT's (valued in topological chain complexes) satisfying the Atiyah-Segal axioms. The first hour will serve as in introduction to these concepts , while in the second hour we wil see this construction at work in the example of Rozansky-Witten theory, an AKSZ TQFT (think odd Chern-Simons) connecting holomorphic symplectic geometry with invariants of rational homology 3-spheres.

Jan 31: Nicolas Ricka

Motivic Modular Forms

Motivated by the study of chromatic phenomenon in the classical and motivic Adams spectral sequence, we set up a machinery to build a spectrum (over Spec(R) or Spec(C)) of motivic modular forms (mmf), that is, a ring spectrum whose cohomology is A//A(2). This answers a question raised by Dan Isaksen. The approach we suggest makes a detour by C_2-equivariant stable homotopy theory, and uses the proximity between the equivariant and motivic Steenrod algebra, a relationship which is not shared by the classical Steenrod algebra. If time permits, we will talk about uniqueness of such spectra, and the chromatic consequences of mmf.

Feb 14: Donald Youmans (2:30-3:30)

Geometric Quantization and Hitchin's Connection

Geometric quantization arose from the attempt to put canonical quantization, which is used in physics to pass from classical mechanics to quantum mecahnics, on rigorous mathematical grounds. Starting with a symplectic manifold and its Lie algebra (under the Poisson bracket) of smooth functions, one tries to construct irreducible representations thereof. The irreducibility condition forces one to choose a polarization. A priori, the constructed quantum Hilbert spaces, i.e. the representations, depend on this choice. To probe how much the representations rely on this choice, one can organize the prequantum Hilbert spaces into a fiber bundle over the space of all polarizations. It was shown by N. J. Hitchin and others that in the case of Kähler polarizations this bundle is endowed with a projectively flat connection allowing the comparison of fibers. While the above ideas were extensively studied for manifolds, the case of super-manifolds still remains an open problem. A first step was given by S. Wu in who considered a purely odd vector space. In this talk I want to give a brief and by no means complete survey of the construction of the Hitchin connection in the case of the geometric quantization of the moduli space of flat connections. Moreover, I want to discuss what happens if one trys to geometrically quantize a purely odd Lie algebra rather than a purely odd vector space. In particular, I will outline possible interactions of Hitchin's connection with the underlying Lie theory.

Feb 14: Riccardo Iraso (4:00-5:00)

2-dimensional Yang-Mills theory and perturbative BV-BFV formalism

In this talk I will survey the non-perturbative solution of the 2-dimensional version of Yang-Mills theory on surfaces with boundaries and outline the functorial approach to the quantization of gauge theories given by the BV-BFV formalism. I will then describe some explicit perturbative computations in this formalism and compare the results with the non-perturbative answer.

Feb 21: Aaron Mazel-Gee

The geometry of the cyclotomic trace

The cyclotomic trace is a natural map running from algebraic K-theory to topological cyclic homology (TC). This trace map is important both conceptually and computationally, as it is known to be "locally constant" by the celebrated Dundas--Goodwillie--McCarthy theorem: its fiber remains unchanged under nilpotent extensions of connective associative ring spectra. However, the original construction of TC is quite subtle, and in particular it does not permit a precise interpretation of TC or of the cyclotomic trace at the level of derived algebraic geometry. In this talk, I will describe a new construction of TC that affords just such a precise geometric interpretation, which is based on nothing but universal properties (coming from Goodwillie calculus) and the geometry of 1-manifolds (via factorization homology). This represents joint work with David Ayala and Nick Rozenblyum.

Feb 28: Noah Snyder

The SO(3) action on the space of finite tensor categories

The cobordism hypothesis gives a correspondence between the framed local topological field theories with values in C and a fully dualizable objects in C. Changing framing gives an O(n) action on the space of local TFTs, and hence by the cobordism hypothesis it gives a (homotopy coherent) action of O(n) on the space of fully dualizable objects in C. One example of this phenomenon is that O(3) acts on the space of fusion categories. In fact, O(3) acts on the larger space of finite tensor categories. I'll describe this action explicitly and discuss its relationship to the double dual, Radford's theorem, pivotal structures, and spherical structures. In the first hour I'll talk about topological field theories, local TFTs, the cobordism hypothesis, and tensor categories. In the second hour I'll talk about O(n) actions. This is part of work in progress joint with Chris Douglas and Chris Schommer-Pries.

March 7: Corey Bregman

Kaehler manifolds and abelian-by-surface extensions

A question going back to Serre asks which finitely presented groups arise as the fundamental groups of compact Kaehler manifolds. In this talk we will survey some known results on this question and study extensions of abelian groups by hyperbolic surface groups. Topologically, these groups are realized as fundamental groups of surface bundles over tori. We show that if any such extension is Kaehler, then it is virtually a product.

March 21: Mona Merling

Equivariant A-theory

I will describe joint work with Cary Malkiewich on equivariant A-theory. I will describe an approach which is related the bivariant A-theory of Williams and another approach which we expect encodes information about equivariant pseudo-isotopies of G-manifolds.

THURSDAY, March 30, 2:00-3:00, Hurley 258: Tom Church

The topological category of graphs

We give a combinatorial model for the topological category of graphs. This globalizes the classic theorem of Culler-Vogtmann on the contractibility of Outer space. This model makes possible a description of generic representations of Out(F_n), analogous to FI-modules for S_n-representations or the VI-/V-modules studied by Putman-Sam for GL_n(R). Joint work with Jacob Lurie.

April 4: Arunima Ray

4-dimensional analogues of Dehn’s lemma

Dehn's lemma is a classical and fundamental result for 3-manifolds. We investigate certain 4- dimensional analogues, giving examples where they do or do not hold, in the smooth and topological categories. For instance, we show that an essential 2-sphere S in the boundary of a simply connected 4-manifold W such that S is null-homotopic in W need not extend to an embedding of a ball in W. However, if W is simply connected (or more generally, has abelian fundamental group) with boundary a homology sphere, then S bounds a topologically embedded ball in W. Moreover, we give examples where such an S does not bound any smoothly embedded ball in W. We give similar results for tori; in particular, we construct an incompressible torus T in the boundary of a contractible 4- manifold W such that T extends to a topological embedding of a solid torus in W but no smooth embedding. (This is joint work with Danny Ruberman.)

April 11: Henry Horton

Symplectic instanton homology: SO(3)-bundles, functoriality, and Dehn surgery

I will describe a new construction of a Floer theoretic invariant of a 3-manifold Y equipped with a principal SO(3)-bundle ω, as well as outline some of its properties. The invariant arises as the Lagrangian Floer homology of so-called traceless character varieties associated to a Heegaard splitting of the 3-manifold; we call it the "symplectic instanton homology" of (Y, ω) and denote it SI(Y, ω). If K is a (framed) knot in Y and Y₀, Y₁ denote 0- and 1-surgeries on K (with respect to the given framing), we indicate how the 2-handle cobordism maps induced by these surgeries fit into an exact triangle of symplectic instanton homologies, ··· → SI(Y, ω + ω_K) → SI(Y₀, ω) → SI(Y₁, ω) → ···, where ω_K is the SO(3)-bundle on Y dual to K. Time permitting, we will indicate how for link surgeries there is more generally a spectral sequence of symplectic instanton homologies, which gives a relationship to reduced Khovanov homology when applied to the branched double cover of a link.

April 23: Chris Schommer-Pries

The algebraic K-theory of manifolds via manifolds.

This talk will be a somewhat leisurely discussion of the speaker's attempt to understand Waldhausen's celebrated result that the algebraic K-theory of spaces, when applied to a smooth manifold, splits as a wedge A(X) = Q(X_+) x Wh^Diff(X). of the suspension spectrum of X_+ and a spectrum which is a certain double delooping of the stable concodrance group of X. Our aim will be to describe these spaces/spectra (and the splitting) in more or less elementary manifold-y terms as certain spaces of embedded manifolds or classifying spaces of bordism categories, similar to those arising in the work of Galatius, Madsen, Tillmann, Weiss, Randel-Williams, and others. Igusa functions (aka framed generalized Morse functions) and Atiyah duality will make an appearance when we connect these to Waldhausen's work.

May 2: Bena Tshishiku

Mapping class group of a K3 surface

The cohomology of the mapping class group Mod(M) of a manifold M is an important source of invariants for fiber bundles with fiber M. We use arithmetic groups to construct nontrivial classes in the cohomology of Mod(M) when M is a K3 surface, and discuss some geometric application.

6/22/2017