Topology Seminar
The Topology Seminar has talks on a variety of topics in topology, including algebraic geometry, chromatic homotopy theory, configuration spaces, continuum theory, functor calculus, graph cohomology, homotopical algebra, Nielsen fixedpoint theory, operads, simplicial sets, spaces of knots and links, span theory, and topological groups.
Fall 2024
For the Fall 2024 semester we will meet 1:00  2:00 on Fridays in 208 Maxim Doucet Hall or on Zoom.
For more information or connection details contact Robin Koytcheff.

25 October 2024, 1:00  2:00
Edge subdivision and L^{2}homology of certain rightangled Coxeter groups
Kevin Schreve (LSU)
Abstract: Any flag simplicial complex determines a rightangled Coxeter group and a corresponding CAT(0) cubical complex on which the group acts geometrically. If the simplicial complex is a triangulation of a sphere, the corresponding cube complex is a manifold. I will discuss some recent computations of L^{2}homology groups of these cubical complexes in certain cases. Joint work with Grigori Avramidi and Boris Okun. 
22 November 2024, 1:00  2:00
A geometric approach to cohomology
Greg Friedman (Texas Christian University)
Abstract: Cohomology is often more useful than homology due to its ring structure, but the cost is a definition of cochains and cup products that is in many ways more algebraic than geometric. Additionally, the standard definition provides a cup product that is not (graded) commutative at the cochain level, but only after passing to cohomology. By contrast, Lipyanksiy's geometric cohomology, defined on smooth manifolds, takes a different approach, representing cochains geometrically by maps from manifolds with corners and providing a geometricallydefined graded commutative product via fiber products. In this case, the penalty is that the product is only defined when the representing maps are transverse. We will discuss recent work to fully flesh out the theory of geometric chains and cochains and their products, as well as recent and anticipated applications. Joint work with Anibal MedinaMardones and Dev Sinha.
Spring 2024
For the Spring 2024 semester we will meet 1:00  2:00 on Fridays in 208 Maxim Doucet Hall or on Zoom.
For more information or connection details contact Robin Koytcheff.

19 January 2024, 1:00  2:00 (on Zoom: https://ullafayette.zoom.us/j/95338274163)
2holonomy representations via the Kontsevich integration map
Matthew Jackson (University of Lille)
Abstract: The KnizhnikZamolodchikov connection is a very powerful tool in constructing representations of braid groups. We investigate a higher order version of this connection due to Komendarczyk, Koytcheff and Volic, and construct a representation of the path 2groupoid of the configuration space of m points in R³ via the Kontsevich integration map. We show that this representation behaves nicely with respect to compositions of homotopies. This work was done during my internship at the University of Tokyo and Meiji University under the supervision of Toshitake Kohno. 
9 February 2024, 1:00  2:00 (on Zoom: https://ullafayette.zoom.us/j/95338274163)
A formalism for operadic structures with applications to Hopf algebras and B_{+} operators
Michael Monaco (Purdue University)
In the 1970s and 1990s, Gromov pointed out the nonexplicit nature of constructions in algebraic topology, leading him to propose a program of quantitative topology: asking about the "size" or "complexity" of the objects (a homotopy between two maps; a filling of a nullcobordant manifold) whose existence is implied by the results of algebraic and geometric topology. I will discuss some sample questions and the spectacularly varied answers we know (or don't know) to them. 
16 February 2024, 1:00  2:00 (on Zoom: https://ullafayette.zoom.us/j/95338274163)
Introduction to quantitative topology
Fedor Manin (UC Santa Barbara)
Abstract: There has been a good deal of investigation into ropelength of knots and links: what is the length of 1unitthick rope needed to realize them? In a recent preprint, Michael Freedman, motivated by questions in topological topology, introduced the idea of forcing only some components of a link to be far apart. For example, one may take a link made up of n separated copies of a link and try to find a realization such that the two components of each Hopf link are 1 unit apart. Freedman showed that for the Hopf link, such a realization must take up at least log n space, but hypothesized that the true answer should be linear in n. In joint work with Elia Portnoy, we showed that log n is in fact sufficient for any link type. We also improved and generalized Freedman's lower bounds based on Milnor invariants. 
23 February 2024, 1:00  2:00 (on Zoom: https://ullafayette.zoom.us/j/95338274163)
FIcalculus, representation stability, and generalizations
Kaya Arro (UC Riverside)
Abstract: Representation stability FIcalculus, representation stability, and generalizationsis a quality enjoyed by many functors (FImodules) from the category of finite sets and injections (FI) to an category of modules that ensures that the symmetric group representations it determines eventually become "constant" in an appropriate sense.
In this talk, I'll discuss how the machinery of functor calculus allows one to extend this notion to the stable ∞categorical world and provides tools for describing the behavior of FImodules. Time permitting, I'll also discuss avenues for generalizing representation stability enabled by the functor calculus approach. 
1 March 2024, 1:00  2:00 (room 208)
A Serre spectral sequence for moduli spaces of tropical curves
Nir Gadish (University of Michigan)
Abstract: The moduli space of genus g tropical curves with n marked points is a fascinating topological space, with a combinatorial flavor and deep algebrogeometric meaning. In the algebraic world, forgetting the n marked points gives a fibration whose fibers are configuration spaces of a surface, and Serre's spectral sequence lets one compute the cohomology "in principle". In joint work with Bibby, Chan and Yun, we construct a surprising tropical analog of this spectral sequence, manifesting as a graph complex and featuring the cohomology of compactified configuration spaces on graphs. 
12 April 2024, 1:00  2:00 (on Zoom: https://ullafayette.zoom.us/j/95338274163)
Axioms for the category of finitedimensional Hilbert spaces and linear contractions
Matthew Di Meglio (University of Edinburgh)
Abstract: I will explain the motivation and main ideas behind recent joint work with Chris Heunen (arXiv:2401.06584) that characterises the category of finitedimensional Hilbert spaces and linear contractions. The axioms are about simple categorytheoretic structures and properties. In particular, they do not refer to norms, continuity, dimension, or real numbers. The proof is noteworthy for the new way that the scalars are identified as the real or complex numbers. Instead of resorting to Solèr’s theorem, which is an opaque result underpinning similar characterisations of other categories of Hilbert spaces, suprema of bounded increasing sequences of scalars are explicitly constructed using directed colimits of contractions. To keep the talk accessible, I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed. 
26 April 2024, 1:00  2:00 (on Zoom: https://ullafayette.zoom.us/j/95338274163)
A product operation on disk fiber bundles, and a configuration space with mouse diagrams
Xujia Chen (Harvard University)
Abstract: In this talk we will be concerned with smooth, framed fiber bundles whose fibers are the standard ddimensional disk, trivialized along the boundary. "Kontsevich's characteristic classes" are invariants defined for these bundles: given such a bundle \pi:E \to B, we can associate to it a collection of cohomology classes in H^*(B). On the other hand, there is a "bracket operation" for these bundles defined by Sander Kupers: namely, given two such bundles \pi_1 and \pi_2 as input, we can output a "bracket bundle" [\pi_1,\pi_2]. I will talk about this bracket bundle construction and a formula relating the Kontsevich's classes of [\pi_1,\pi_2] with those of \pi_1 and \pi_2. The main input of the proof is a new but very natural configuration space generalizing the FultonMacPherson configuration spaces. This is a work in progress joint with Robin Koytcheff and Sander Kupers.
Fall 2023
For the Fall 2023 semester we will meet 1:15  2:15 on Fridays in 208 Maxim Doucet Hall or on Zoom.
For more information or connection details contact Daniel Davis.

22 September 2023
Using the resolution of the Telescope Conjecture as a dropoff point for an introduction to Jannsen's continuous group cohomology, I
Daniel Davis (UL Lafayette)
Abstract: About 45 years ago, Doug Ravenel formulated the Telescope Conjecture, which consists of a claim for each pair (n, p), where n is a natural number and p is a prime. Around 1980, the conjecture was proved whenever n = 1. This summer, 4 researchers announced and gave talks related to a proof that all the other claims are false. When n = 2, a key object in the disproof is (L_{K(2)}K(E_1))^{hG_1}, a rich object that will partly be treated in the talk as a black box. When p \geq 5, there are 3 models for this object that can be extracted from the arXived and published literature and 2 of them are due to the speaker. One of those 2 is also valid for p = 2, 3, and we believe its construction can play a role in building the partially nonarXived and unpublished 3rd model, which is valid for all p and is the model that was used in the disproof. To fully understand the AusoniRognes Conjectures, one wants to know explicitly the homotopy groups of the key object. By using the speaker's first model and Jannsen's continuous cohomology instead of Tate's continuous cohomology, there is a 3column spectral sequence (hfpss) for computing these groups. Unlike the other two models, the first model is lowtech and we will explain that we believe there is also a lowtech way to show that this hfpss is actually 2column, reducing the desired computation to a short exact sequence that we think may be split exact. We have reduced our belief to verifying that a certain tower of abelian groups is MittagLeffler. We hope to verify this without using notyetpublished hightech work of Hahn, Raksit, and Wilson. 
29 September 2023
Using the resolution of the Telescope Conjecture as a dropoff point for an introduction to Jannsen's continuous group cohomology, II
Daniel Davis (UL Lafayette)
Abstract: This talk, a continuation of last week's seminar, will begin by reviewing the progression from Gset to topological Gmodule and their analogues in the world of spectra. We also say a little about the three models referred to last time. The focus will be on building the 3column hfpss for computing the stable homotopy groups of last week's beautiful black box. The key player here is Jannsen's continuous cohomology and a short exact sequence for it. 
6 October 2023
Using the resolution of the Telescope Conjecture as a dropoff point for an introduction to Jannsen's continuous group cohomology, III
Daniel Davis (UL Lafayette)
Abstract: This talk is a continuation of last week's seminar. Our focus is on building the 3column hfpss that previous abstracts in this series have referred to. We will discuss the definition of Jannsen's continuous cohomology and a short exact sequence that involves it, and then, as time permits, we will discuss work in progress on trying to push our results further. In particular, we will discuss the possibility of showing that the aforementioned hfpss is actually just a 2column one. 
3 November 2023 (on zoom)
Algebraic Ktheory of the twoperiodic first Morava Ktheory
Haldun Ozgur Bayindir (University of London)
Abstract: Using a root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of the algebraic Ktheory of the complex Ktheory spectrum. Furthermore, our computational methods also provide the algebraic Ktheory of the twoperiodic Morava Ktheory spectrum of height 1. 
10 November 2023 (on zoom)
A normalizer decomposition for compact Lie groups
Eva Belmont (Case Western Reserve University)
Abstract: If G is a finite group, the normalizer decomposition is a way of expressing BG, up to pcompletion, as a homotopy colimit of smaller groups. Building on a construction due to Libman for plocal finite groups, we obtain a normalizer decomposition for compact Lie groups. We recover decompositions due to Dwyer, Miller, and Wilkerson of 2completed BSU(2) and BSO(3), and compute new decompositions of BSU(p) completed at p. The talk will also contain an introduction to fusion systems, which are related to a generalization of pcompleted classifying spaces, as that is the setting for our work. This is joint work with Natalia Castellana, Jelena Grbic, Kathryn Lesh, and Michelle Strumila.
Spring 2023
For the Spring 2023 semester we will meet 12:50  1:50 on Fridays in 208 Maxim Doucet Hall or on Zoom.
For more information or connection details contact Robin Koytcheff.

10 February 2023
Monoidal envelopes for dendroidal spaces
David Kern (Université de Montpellier)
Abstract: Segal dendroidal spaces are an arboreal model for ∞operads, allowing one to reason with them from an operadic point of view — in contrast with the more combinatorial viewpoint of Lurie's ∞categories of operators. However, certain constructions are more difficult to obtain in this model, chief among which the relation to monoidal ∞categories, and more generally the notion of cocartesian fibrations of operads. We present a definition of a monad on dendroidal spaces that constructs their monoidal envelopes, and which can be put to use to define cocartesian fibrations and thus monoidal ∞categories, as well as interpret certain of Lurie's constructions as an (un)straightening correspondence for ∞operads. The construction relies on a bridge between the dendroidal and combinatorial points of view provided by a plus construction of Barwick and of Berger. 
3 March 2023
Properads of Riemann surfaces in symplectic topology
Yash Deshmukh (Columbia University)
Abstract: I will introduce properads of framed and stable Riemann surfaces relevant to symplectic topology. I will then discuss a homotopical relationship between these properads motivated (partially) by their actions on symplectic invariants. Time permitting, I will discuss some ongoing work on using formal constructions from the theory of properads to give a systematic construction of a new family of operations on these symplectic invariants. (No prior knowledge of symplectic topology will be assumed.) 
17 March 2023
Haefliger's approach for spherical embeddings modulo immersions
Neeti Gauniyal (Kansas State University)
Abstract: I will discuss that one can deduce from Haefliger's work (1966) that the set of connected components for the space $\overline{Emb}(S^n,S^{n+q})$ of spherical embeddings modulo immersions is isomorphic to a certain homotopy group for $q\geq 3$. As a consequence, we see that all the terms of the homotopy long exact sequence of the pair $(SG/SG_q,SO/SO_q)$ have a geometric meaning relating to spherical embeddings and immersions. Also, I will briefly discuss the case when $q=2$. 
24 March 2023 (on zoom)
Embedding calculus of singular manifolds
Connor Malin (Notre Dame University)
Abstract: We extend embedding calculus to the setting of manifolds with a single singularity. In this setting, we produce a PontryaginThom collapse map which for a compact, connected, framed manifold $M$ yields an interesting automorphism of $T_\infty \Sigma^\infty_+ \mathrm{Emb}^{fr}(M,)(M)$. Our approach is to extend the Koszul selfduality of the $E_n$ operad to a compactly supported Koszul selfduality of the right module $E_M$. We also discuss implications for Goodwillie calculus and factorization homol 
31 March 2023
Limits in higher categories
Martina Rovelli
University Massachusetts Amherst
Amherst, Massachusetts
Abstract: The universal properties of many objects in mathematics are encoded as those of limits and colimits of certain diagrams valued in an ordinary category. With the rising growth of fields that rely on the language of higher categories – such as derived algebraic geometry, higher topos theory, categorification of knot invariants, homotopy type theory – it becomes necessary to develop a useable and consistent theory of limits and colimits for diagrams valued in an ncategory or an (∞, n)category. We’ll describe work in progress with Moser and Rasekh towards addressing the case of n = 2 or higher.
Fall 2022
For the Fall 2022 semester we will meet 1:15  2:15 on Fridays in 208 Maxim Doucet Hall.
For more information or connection details contact Daniel Davis.

27 October 2022 (4:50 pm Thursday Maxim Doucet room 208)
Frobenius algebras, spans, and symplectic categories
Walker Stern (University of Virginia)
Abstract: In this talk, I will discuss ongoing work seeking to classify variants of Frobenius objects in categories of spans, as well as motivation coming from Topological Field Theories and symplectic geometry. This talk is in part based on joint work with Ivan Contreras and Rajan Mehta. 
18 November 2022 (on Zoom)
How the Grothendieck construction treats categories equipped with extra structure
Joe Moeller
Cyber Infrastructure Group
National Institute of Standards and Technology
Abstract: The Grothendieck construction exhibits one of the most fundamental relations in category theory, namely the equivalence between fibrations and indexed categories. We give two monoidal variations of the construction, one where monoidal structure is present in the fibers, and one where it is present on the total category. As monoidal categories are principal players in applied category theory, it has proven to be a useful tool for building and studying categories that arise in applications. We extend these results from monoidal categories to linearly distributive categories (LDCs). LDCs generalize monoidal categories by providing two different ways of composing in parallel. They have applications in many areas including computer science and quantum mechanics. After reviewing the monoidal case and giving examples of how it has been used in applications, we give variants of the Grothendieck construction with linearly distributive structure on the fibers and on the total category. 
2 December 2022
We read a recent theorem by Mathew et al., then looked a term up in the dictionary and made progress for an AusoniRognes conjecture.
Daniel Davis
Abstract: Land, Mathew, Meier, and Tamme have a recent deep theorem about sifted colimits for T(n)local ring spectra commuting with the T(n+1)localized algebraic Ktheory functor. We will state this theorem, explain a little about what the concepts/objects in it are  though algebraic Ktheory will basically remain a black box, go into some detail about what sifted colimits (in 1category theory) are, and then describe our result that is related to an AusoniRognes conjecture about the LubinTate spectrum E_n and the extended Morava stabilizer group, a profinite group. This result, valid for all positive integers n and primes p, immediately yields a spectral sequence that should be closely related to one whose existence is a part of the conjecture (perhaps it is the conjectured spectral sequence).
Spring 2022
For the Spring 2022 semester we will usually meet 1:00  2:00 on Fridays. Some meetings will be on Zoom and others will be inperson.
For more information or connection details contact Philip Hackney.

20 January 2022 (on Zoom at 4:40 pm Thursday)
Operadic Unitarization
Olivia Borghi (University of Melbourne)
Abstract: In the 1990 paper "QuasiHopf Algebras" V.G. Drinfe'ld describes a notion of hopf algebra with a weakened coassociativity called a quasiHopf algebra. He builds upon this notion and defines several different iterations of quasiHopf algebras with weakened cocommutativity. These include triangular, quasitriangular and coboundary quasiHopf alegbras. Drinfe'ld further describes a process of "unitarization" whereby we can provide certain quasitiangular quasiHopf algebras (quantum groups) with coboundary qausiHopf algebra structures. Tanaka duality tells us there is a correspondence between these weakened Hopf algebras and monoidal categories. Using Donald Yau's theory of Gmonoidal categories I build an operadic counterpart to this "unitarization" whereby we can provide certain braided monoidal categories with coboundary monoidal structures. I then build a theory of infinite dimensional Gmonoidal categories to describe this operadic unitarization in that setting. 
11 February 2022 (on zoom)
Smooth embeddings and their families
Danica Kosanović (ETH Zürich)
Abstract: Configuration spaces of manifolds are examples of spaces of embeddings, which can be employed for studying all other embedding spaces, via GoodwillieWeissKlein calculus. We will discuss how certain classes in homotopy groups of configuration spaces give rise to nontrivial families of embeddings, that generalise lower central series of braid groups and VassilievGusarovHabiro constructions in knot theory. 
25 February 2022 (on zoom)
Open problems in graph models for operadic structures
Philip Hackney (UL Lafayette)
Abstract: Last Fall we gave a talk about a new way to express the category of undirected graphs that is used to model modular operads. We will briefly recall this, and then present several related open problems. 
4 March 2022 (on zoom)
Quasi2Segal sets
Matt Feller (University of Virginia)
Abstract: There are various models of "categories up to homotopy," such as Segal spaces which are simplicial spaces satisfying a condition that encodes unique (up to homotopy) composition of morphisms. In the last decade, the study of simplicial spaces which satisfy a particular weaker algebraic condition was independently begun by both Dyckerhoff–Kapranov, who call them 2Segal spaces, and Gálvez–Kock–Tonks, who call them decomposition spaces. Meanwhile, quasicategories are an alternative model of uptohomotopy categories whose theory has been massively developed in the past two decades, thanks largely due to Joyal and Lurie. With the ultimate hope of harnessing the technology for quasicategories in the 2Segal setting, we set out to define a 2Segal version of quasicategories as simplicial sets with a particular horn lifting condition. In particular, our goal is to describe an associated model structure on the category of simplicial sets. 
25 March 2022 (on zoom)
Graphing, homotopy groups of spheres, and spaces of links and knots
Robin Koytcheff (UL Lafayette)
Abstract: We determine the homotopy groups of spaces of 2component long links in a range that depends on the dimensions of the source manifolds and target manifold. Our result applies in roughly the triplepointfree range, if one incorporates the dimension of the homotopy group into the source manifold. Within this range, graphing maps identify each group with the direct sum of a homotopy group of a sphere and a group of isotopy classes of higherdimensional knots. We also show that the homotopy groups of a sphere generally form a direct summand of the homotopy groups of a space of links. Finally, by joining components, we obtain a nontrivial map from a homotopy group of a sphere into the first nontrivial homotopy group of a space of long knots with odd codimension. These maps send the class of each Hopf fibration to a generator. 
1 April 2022 (on zoom)
Grothendieck–Teichmüller theory and modular operads
Luciana Basualdo Bonatto (Oxford)
Abstract: The absolute Galois group of the rationals Gal(Q) is one of the most important concepts in number theory. Although we cannot explicitly describe more than two elements in this infinite group, we know it acts on wellknown algebraic and topological objects in compatible ways. Grothendieck–Teichmüller theory uses these representations to study this Galois group. One of the most important representations comes from the compatible actions of Gal(Q) on all the profinite mapping class groups of surfaces. In this talk, we introduce an algebraic tool called an infinity modular operad and use it to construct an infinity modular operad of surfaces capturing the compatibility structure above. We show this admits an action of Gal(Q), translating the Grothendieck–Teichmüller program into the theory of infinity modular operads, which provides new ideas and tools to approach this problem. This is joint work with Marcy Robertson. 
8 April 2022 (on zoom)
The homotopytype of diffeomorphism groups of "small" manifolds, and their relation to spaces of string links
Ryan Budney (University of Victoria)
Abstract: I will describe a technique for probing the homotopytype of diffeomorphism groups of "small" manifolds. These techniques are largely dimensionagnostic, although they only give new results in dimension 4 and up. We generate our diffeomorphisms via an understanding of the homotopytype of spaces of string links, and we also detect our diffeomorphisms using embeddingcalculus (alternatively, finitetype invariants) for different spaces of string links. This allows us to say that the mapping class group of S^1 x D^3 is not finitely generated. It also allows us to detect the HatcherWagoner diffeomorphisms of S^1 x D^{n1} for n>5, and it gives us some interesting mapping classes of S^1 x D^4. 
29 April 2022 (on zoom)
Twisted arrow categories of operads and Segal conditions
Sergei Burkin
Abstract: Several homotopy coherent structures can be defined via Segal conditions on presheaves over categories such as the simplex category, Moerdijk–Weiss dendroidal category, Segal's category, and Hackney–Robertson–Yau categories. We introduce a uniform construction of such categories from operads. The construction at least partially explains how and why these categories appear in homotopy theory.
Fall 2021
For the Fall 2021 semester we will usually meet 11:00  12:00 on Fridays. Some meetings will be on Zoom and others will be inperson.
For more information or connection details contact Philip Hackney.

10 September 2021 (on Zoom)
Graph categories for operadic structures
Philip Hackney
In this talk, I'll give a brand new definition of several categories of graphs with loose ends. These categories have appeared previously in joint work with Robertson and Yau. Each is designed to control a particular operadic structure, like modular operads, cyclic operads, (wheeled) properads and so on, as well as their "higher" variants. This is analogous to the situation with the Moerdijk–Weiss dendroidal category Ω and the simplicial category Δ, which are useful for describing higher operads and higher categories.
We will explain the Segal condition and the relationship with operadic structures, and also how the new descriptions facilitate comparison of structures.
This talk will not require any background on generalized operads or operadic structures, nor will it require any knowledge of the graphical categories under discussion. Indeed, a key feature of the new definitions is that they can be understood from scratch, and nerve theorems allow one to define operadic structures as certain presheaves on these graph categories. 
17 September 2021 (on Zoom)
Integrals, trees, and spaces of pure braids and string links
Robin Koytcheff
Abstract: The based loop space of configurations in a Euclidean space R^n can be viewed as the space of pure braids in R^{n+1}. In joint work with Komendarczyk and Volic, we studied its real cohomology using an integration map from a certain graph complex and recovered a result of Cohen and Gitler. Specifically, the map we studied is a composition of Kontsevich’s formality integrals and Chen’s iterated integrals. We showed that it is compatible with BottTaubes integrals for spaces of 1dimensional string links in R^{n+1}. As a corollary, the inclusion of pure braids into string links in R^{n+1} induces a surjection in cohomology for any n>2. More recently, we showed that the dual of the integration map embeds the homotopy groups of the space of pure braids into a space of trivalent trees. We also showed that a certain subspace of these homotopy groups injects into the homotopy groups of spaces of kdimensional string links in R^{n+k} for many values of n and k. 
24 September 2021 (on Zoom)
Khovanov's categorification of the Jones polynomial
Jake Sundberg
Abstract: This talk focuses primarily on the BarNatan paper of the same name, which gives a “picturerich glimpse” into Khovanov homology. In this construction, a 1+1dimensional TQFT is applied to the cube of resolutions of a knot to transition from a topological setting to an algebraic one. This construction gives the homological object at the core of this talk, which is strictly more powerful than the Jones polynomial. A long exact sequence used in computing Khovanov homology will be discussed if time permits. 
1 October 2021 (on Zoom at 11:15)
A GilletWaldhausen theorem for chain complexes of sets
Maru Sarazola (Johns Hopkins)
Abstract: In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGWcategory. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly wellsuited for algebraic Ktheory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to nonadditive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an exact category for the purpose of algebraic Ktheory.
In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to recent work with Brandon Shapiro where we further develop this program by defining chain complexes and quasiisomorphisms for finite sets. These satisfy an analogue of the classical Gillet–Waldhausen Theorem, providing an alternate model for the Ktheory of finite sets. 
15 October 2021 (on Zoom at 3:30)
A topological characterization of the Kashiwara–Vergne groups
Marcy Robertson (University of Melbourne)
Abstract: Solutions to the Kashiwara–Vergne equations in noncommutative geometry are a "higher dimensional" version of Drinfeld associators. In this talk we build on work of BarNatan and Dancso and identify solutions of the Kashiwara–Vergne equations with isomorphisms of (completed) wheeled props of "welded tangled foams" — a class of knotted surfaces in $\mathbb{R}^4$. As a consequence, we identify the symmetry groups of the Kashiwara–Vergne equations with automorphisms of our (completed) wheeled props.
The first part of this talk will be accessible to a general audience and I will not assume familiarity with the KashiwaraVergne equations, Drinfeld associators or wheeled props. Includes joint work with Z. Dancso and I. Halacheva. 
22 October 2021 (on Zoom at 3:30)
Torsion in Khovanov Homology, Part I
Jake Sundberg
Abstract: Przytycki discovered infinite families of links and knots whose Khovanov homology contains torsion different than $\mathbb{Z}_2$. We examine the methodology used to find these families and discuss some counterexamples to the Przytycki–Sazdanović braid conjecture that were found using these techniques. 
29 October 2021 (on Zoom at 2:00)
Torsion in Khovanov Homology, Part II
Jake Sundberg
Abstract: A homologically thin knot/link is one whose Khovanov homology over a ring R is supported on two adjacent diagonals. As an introduction to the topic, several related concepts are defined and examples are given. We work toward proving a theorem which gives conditions for which all torsion of knots/links which are homologically thin over an interval is $\mathbb{Z}_2$torsion. If time permits, the Khovanov homology of closed 3braids will be discussed. 
5 November 2021 (on Zoom at 11:00 am)
Classifying model structures on finite total orders
Scott Balchin (MPIM Bonn)
Abstract: To do abstract homotopy theory in a category of interest, one may assign to it the data of a Quillen model structure. This naturally raises the question of the possibility of classifying all model structures on a fixed category. In this talk I will report on joint work with Ormsby, Osorno and Roitzheim where we classify all model structures on the category [n]. In the process we will see many well known combinatorial invariants, and tantalizing links to the theory of equivariant homotopy theory. The talk will involve plenty of examples, and no prior knowledge will be assumed! 
19 November 2021 (on Zoom at 11:00 am)
Torus knots and links
Jake Sundberg
Abstract: The primary goal of this talk is to classify knots and links which can be drawn on the surface of a torus with no points of intersection. To enrich the discussion, some necessary background on Seifert surfaces and matrices will be provided. The Khovanov homology of the torus links T_{2k, 2kn} and T'_{2k, 2kn}, where the latter is a variant of the former with exactly half of the components' orientations reversed.
Spring 2020
For the Spring 2020 semester we will meet 1:00  2:00 on Fridays in 208 Maxim Doucet Hall.

31 January 2020
PreTalbot Seminar I: an example of ambidexterity in chromatic homotopy theory
Daniel Davis
Abstract: In unpublished and transformative work, Hopkins and Lurie developed and studied a phenomenon known as ambidexterity (expressed in the framework of infinitycategories): when ambidexterity occurs, a certain limit of a diagram of a particular type of spectra is equivalent to a colimit of the same diagram. We will give an example of ambidexterity that was known before the work of HopkinsLurie, and time permitting, we will discuss a relationship of this kind of example to work in a paper in preparation by Thomas Credeur, Drew Heard, and Davis. 
7 February 2020
PreTalbot Seminar II: another (pre)example of ambidexterity and the theorem itself
Daniel Davis
Abstract: Last time, we gave a contemporary presentation of the main result of HoveySadofsky and an application of it in a paper of Behrens, which we generalized (just an exercise). In this talk, we give another application of it that is used in CredeurDavisHeard (in preparation) and we consider an interesting pattern that appears in a corollary of that work's main result. We present the ambidexterity theorem and possibly make some comments about BG and the simplicial replacement of a diagram of K(n)local spectra. 
14 February 2020
PreTalbot Seminar III: considering the finite group case of ambidexterity further
Daniel Davis
Abstract: Let K be a finite group and X a K(n)local spectrum upon which K acts. We review the "naive definition" of the morphism rho : BK \rightarrow Sp_{K(n)} that is associated to X. The speaker hasn't yet nailed down the exact content of the rest of the talk, but possible topics include using cosimplicial replacement to understand the homotopy fixed point spectrum, noting its relationship to BK, using simplicial replacement as a way to present the homotopy orbit spectrum, and noting relationships between these two replacements as a way of making this case of ambidexterity a little less mysterious.
Fall 2019
For the Fall 2019 semester we will meet 1:00  2:00 on Fridays in 208 Maxim Doucet Hall.
View the seminar scheduling calendar (ics calendar file).

20 September 2019
An introduction to factorization homology
Robin Koytcheff
Abstract: Factorization homology is a construction that takes a manifold and an E_nalgebra as an input and produces an algebraic object as output. The possible outputs include various structures in topology, algebra, and mathematical physics, such as the homology of a space, the Hochschild homology of an algebra, and certain observables in a TQFT. The construction essentially encodes global phenomena that can be built out of local ones. In this talk, I will explain what an E_nalgebra is and discuss some examples of factorization homology. 
27 September 2019
Group Actions and Cogroup Coactions in ∞topoi
Jonathan Beardsley
Georgia Institute of Technology
Abstract: In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞topoi. An ∞topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞topos, there is an equivalence of categories between the category of Gmodules in that topos and the category of BGcomodules, where BG is the classifying object for Gtorsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞topos with an ΩXaction, and objects with an Xcoaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between Gspaces and spaces over BG for G a topological group. 
11 October 2019
Learning Seminar: Factorization homology 
18 October 2019
Learning Seminar: Factorization homology 
25 October 2019 (ROOM CHANGE Maxim Doucet Hall room 311)
Broken techniques for disappearing things
Hiro Lee Tanaka
Texas State University
Abstract: I will discuss stacks classifying families of broken lines and broken cycles. These stacks give new ways to organize algebraic structures, and have enticing applications to symplectic geometry. For example, in joint work with Jacob Lurie, we've shown that families of broken lines classify nonunital A_∞ structures while giving a clear pathway to enrich Lagrangian Floer theory over spectra (which are more powerful invariants than chain complexes). 
1 November 2019
An introduction to left Kan extensions
Daniel Davis
Abstract: Factorization homology can be defined as the left adjoint to a particular functor between "structured functor categories." This left adjoint is obtained by forming the "operadic left Kan extension." This is a "structured" version of the higher categorical left Kan extension, which is an extension of classical left Kan extensions, which are the 1categorical gadgets that we study in this talk. 
8 November 2019
Examples of left Kan extensions, a model for them, and right Kan extensions
Daniel Davis
Abstract: We unpack the title a little further: under hypotheses that are often satisfied, we give a colimit that yields the left Kan extension. Also, we briefly note the adjunction that can be viewed as the definition of right Kan extensions and, time permitting, we will present a few other facts about these gadgets. 
15 November 2019
An isovariant Elmendorf's theorem
Sarah Yeakel
MSRI / UC Riverside
Abstract: An isovariant map is an equivariant map which preserves isotropy groups. Isovariant maps show up in equivariant surgery theory and in other settings when homotopy theory is applied to geometry. For a finite group G, we will discuss a cofibrantly generated model structure on the category of Gspaces with isovariant maps, along with an isovariant version of Elmendorf's theorem and possible applications to questions in fixed point theory. 
22 November 2019
Learning Seminar: Factorization homology 
6 December 2019
Learning Seminar: Factorization homology
Spring 2019
For the Spring 2019 semester we will meet 1:00  2:00 on Fridays in 208 Maxim Doucet Hall.

25 January 2019
Connectivity and growth in the homology of graph braid groups
Ben Knudsen
Harvard University
Abstract: I will discuss recent work with An and DrummondCole showing that the homology of configuration spaces of graphs exhibits eventual polynomial growth, an analogue of classical homological and representation stability results for manifolds. We compute the degree of this polynomial in terms of an elementary connectivity invariant, in particular verifying an upper bound conjectured by Ramos. Along the way, we uncover a new “edge stabilization” mechanism and a family of spectral sequences arising from a small chain model first introduced by Swiatkowski. 
1 February 2019 (TIME CHANGE 1:302:30)
The Taylor tower for the space of knots and additive finitetype knot invariants
Robin Koytcheff
UL Lafayette
Abstract: Any space of embeddings has a sequence of Taylor approximations which comes from Goodwillie–Weiss functor calculus. In joint work with Budney, Conant, and Sinha, we studied this Taylor tower for the space of long knots in 3space and showed that it is an additive finitetype (a.k.a. Vassiliev) invariant. Our result suggests that the tower could be a universal finitetype invariant over the integers. We will discuss a multiplicative structure on the tower which is compatible with stacking long knots and which was a key ingredient in our work. 
7 February 2019
Manifold calculus of functors
Paul Arnaud Songhafouo Tsopméné
University of Regina
Abstract: Manifold calculus, due to Goodwillie and Weiss, goes back to around 1999. It is concerned with the study of functors from the poset O(M) of open subsets of a manifold M to the category of topological spaces. Manifold calculus has been used by many authors to understand the topology of a variety of embedding spaces. This talk will go over three things. First I will explain the concept of manifold calculus by highlighting some analogies with the ordinary calculus. Next I will present a combinatorial model for manifold calculus. Lastly, if time permits, I will say few words about the classification of homogeneous functors from O(M) into any (simplicial) model category. 
15 February 2019
Configuration spaces as weak operads, II
Philip Hackney
UL Lafayette
Abstract: Last October, Robin Koytcheff showed us how one can compactify configuration spaces to yield an E_n operad. The goal of this talk (which is based on arXiv:1707.05027) is to explain how configuration spaces can be considered directly as (a part of) an "operad uptohomotopy." 
22 February 2019
Manifold calculus of functors for rimmersions
Ismar Volić
Wellesley College
Abstract: Manifold calculus of functors can be used for studying spaces of embeddings, including spaces of rimmersions, which are immersions where no more than r – 1 points are allowed to equal (embeddings are thus 2immersions). I will give some background on manifold calculus of functors and then present some recent work on understanding the manifold calculus Taylor tower that approximates the space of rimmersions. Manifold calculus in this context supplies interesting connections to combinatorial topology, such as the structure of certain subspace arrangements as well as Tverberglike problems, so some time will be devoted to these topics. This is joint work with Franjo Šarčević. 
8 March 2019
Configuration spaces as weak operads, III
Philip Hackney
UL Lafayette
Abstract: The goal of this talk is to explain how configuration spaces can be considered directly as (a part of) an "operad uptohomotopy." 
22 March 2019
Spaces of diffeomorphisms and spaces of knots
Robin Koytcheff
UL Lafayette
Abstract: I will discuss results on spaces of diffeomorphisms of lowdimensional manifolds and how these can be used to deduce results about spaces of knots. 
29 March 2019
Spaces of diffeomorphisms and spaces of knots, part II
Robin Koytcheff
UL Lafayette
Abstract: Last week, I surveyed results on diffeomorphisms of disks and spheres, and I explained how the (3dimensional) Smale Conjecture shows that the space of long unknots is contractible. This week, I will discuss how it can be used to calculate homotopy types of spaces of some more interesting knots. 
26 April 2019
A Model For S^{g}
Thomas Credeur
UL Lafayette
Abstract: For a fixed prime p, it is well known that the K(n)local SpanierWhitehead dual of the nth Morava Etheory, E_n, is equivalent to the n^2 suspension of E_n. This equivalence is equivariant with respect to the action of the Morava stabilizer group in the homotopy category, but is not equivariant at the pointset level. More recent work of Beaudry, Goerss, Hopkins, and Stojanoska has introduced a "dualizing spectrum" S^{g}, which has the property that the K(n)local dual of E_n is equivariantly equivalent to S^{g} \wedge E_n. In this talk we give a model for the spectrum S^{g}, and show that the smash product of E_n with this model is equivalent to the n^2 suspension of E_n.
Fall 2018
For the Fall 2018 semester we will meet 1:00  2:00 on Fridays in 208 Maxim Doucet Hall.

31 August 2018
A complete definition of E_\infty ring spaces, pt. I: E_\infty spaces
Daniel Davis 
14 September 2018
A complete definition of E_\infty ring space, pt. II
Daniel Davis 
21 September 2018
Spaces over operad pairs and the special case of E_\infty ring spaces
Daniel Davis 
28 September 2018
Spaces over operad pairs: the final leg in their definition and a few examples
Daniel Davis
Abstract: Building on the three previous talks, we give the definition of a space over an operad pair, which enables us to complete the definition of an E_\infty ring space. Then we give several examples of operad pairs and what algebraic structures on spaces these yield. Time permitting, the next goal is to understand how symmetric bimonoidal categories give E_\infty ring spaces in a canonical way. 
12 October 2018
The state of the art on a few problems in chromatic homotopy theory
Daniel Davis 
26 October 2018
Manifoldtheoretic E_n operads
Robin Koytcheff
Abstract: An E_n operad is an operad equivalent to the little ndisks operad. I will begin by recalling the definition of the little disks operad. The Fulton—Macpherson operad and Kontsevich operad are two E_n operads, both of which can be defined via a manifoldtheoretic compactification of the configuration space of points in ndimensional space. I will discuss this compactification, the operad structure maps, and some applications. 
2 November 2018
Homotopy theory of algebras over operads
Dmitri Pavlov
Texas Tech University
Abstract: Suppose O is an operad in simplicial sets, chain complexes, motivic spectra, etc. Consider the category Alg_O of algebras over this operad equipped with degreewise weak equivalences. We are interested in the following questions:
1) Under what conditions on O does Alg_O present the "homotopically correct" category of algebras over an operad (e.g., in the sense of \inftycategories)?
2) Under what conditions on O does Alg_O possess a model structure?
3) Under what conditions on O and O' does a weak equivalence $O \to O'$ induce a Quillen equivalence $Alg_O \to Alg_O'$?
We provide a complete answer to 1) and 3) in terms of an ifandonlyif criterion that is easy to verify in practice, and we also give a sufficient condition for 2) that is applicable to all known practical examples. Our criteria work in abstract monoidal model categories, such as simplicial sets, chain complexes, motivic spectra, topological spaces, and many others. Our work espouses the yoga of "synthetic" model category theory, which postulates that model structures and their properties can be most easily established inductively by tracing the construction of the category under consideration stepbystep, as opposed to constructing model structures directly in one step.
The above is joint work with Jakob Scholbach (Muenster). If time permits, I will also discuss the case of coalgebras over operads, as well as Leinsterstyle homotopy algebras over operads. 
9 November 2018
Configuration spaces as weak operads
Philip Hackney
Abstract: Two weeks ago, Robin Koytcheff told us one can compactify configuration spaces to yield an E_n operad. In this talk, I'll explain how configuration spaces can be considered directly as (a part of) an "operad uptohomotopy." The beginning of the talk will be a brief introduction to the dendroidal objects of Moerdijk & Weiss. This talk is based on arXiv:1707.05027. 
16 November 2018
Cellular E_2algebras and the unstable homology of mapping class groups
Alexander Kupers
Harvard University
Abstract: We discuss joint work with Soren Galatius and Oscar RandalWilliams on the application of higheralgebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability  in contrast to the "additive" one due to Quillen  which has the advantage of providing information outside of the stable range. 
30 November 2018
Transfer Maps and a Perfect Pairing of Spectra
Thomas Credeur
Abstract:Let G be a profinite group. Given any discrete Gspectrum X and a closed subgroup K of G, one can form the homotopy fixed point spectrum X^{hK}. Given an open subgroup K of G and an open subgroup H of K, there is a "wrong way" map in the stable homotopy category, the transfer map from X^{hH} to X^{hK}. Using these maps we can construct a transfer map from E_n^{hU} to E_n^{hG_n}, where G_n is the Morava stabilizer group and E_n is a LubinTate spectrum. The transfer maps can then be used to construct a perfect pairing in the homotopy category. As a consequence of this perfect pairing, we will see that E_n^{hU} is self dual. 
14 December 2018
Special meeting: 2:15  3:45 (talk proper: 2:453:45)
Dependent paths in homotopy type theory
Jonathan Steven Prieto Cubides
Informatics PhD student
University of Bergen (Norway)
Abstract: We introduce some basic notions about HoTT to talk about dependent paths. We show a joint work with Marc Bezem that gives us two proofs about the geometrical intuition in HoTT behind the dependent paths, also called pathovers, which has been only mentioned but not proved before. We typecheck in Agda the proofs and show some lemmas for the second one that makes a shorter proof.
Spring 2018
For the Spring 2018 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

26 January 2018
Graph complexes, formality, and configuration space integrals for spaces of braids, part I
Robin Koytcheff
Abstract: This series of talks will culminate in a report on recent work joint with Komendarczyk and Volic, where we connect two integrationbased approaches to the cohomology of spaces of braids. This first part will include an introduction to the configuration space of distinct points in Euclidean space, as well as the related braid groups. I will discuss the basic algebraic topology of these configuration spaces and some of the underlying geometric intuition. 
2 February 2018
Graph complexes, formality, and configuration space integrals for spaces of braids, part II
Robin Koytcheff 
16 February 2018
Graph complexes, formality, and configuration space integrals for spaces of braids, part III
Robin Koytcheff 
23 February 2018 (1:302:30 TIME CHANGE)
Graph complexes, formality, and configuration space integrals for spaces of braids, part IV
Robin Koytcheff
Abstract: I will continue to talk about cohomology of the space of loops in a space X, in terms of the bar construction on the cochains (or cohomology) of X. I will briefly discuss the notion of a Koszul algebra, as well as Chen’s iterated integrals for the case where X is a manifold. If time permits, I will begin discussing configuration space integrals and related cochain complexes of graphs. 
16 March 2018
Graph complexes, formality, and configuration space integrals for spaces of braids, part V
Robin Koytcheff
Abstract: I will define Chen’s iterated integrals, which are a way of realizing the cohomology of the loop space of a manifold via differential forms. I will also discuss configuration space integrals. One variant of these was used to prove the formality (over the rationals) of configuration spaces. Another variant produces cohomology classes in spaces of knots and links. In joint work with Volic and Komendarczyk, we establish a relationship involving these three types of integrals. 
23 March 2018
Homotopy fixed points for trivial group actions
Daniel Davis 
19 April 2018
GoodwillieWeiss manifold calculus
Victor Turchin
Kansas State
Abstract: The goal of this talk is to explain a bit more deeply an approach to studying embedding spaces which will be briefly mentioned in the Colloquium talk and which is called GoodwillieWeiss calculus of functors. 
27 April 2018
Another homotopy theory: the category of complete Segal spaces
Daniel Davis
Fall 2017
For the Fall 2017 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

8 September 2017
An introduction to the GoerssHopkins Linearization Hypothesis and a connection to continuous Gspectra.
Daniel Davis 
15 September 2017
More about the GoerssHopkins Linearization Hypothesis and a connection to continuous Gspectra.
Daniel Davis 
22 September 2017
The GoerssHopkins Linearization Hypothesis and continuous Gspectra, part 3.
Daniel Davis 
29 September 2017
The GoerssHopkins Linearization Hypothesis and continuous Gspectra, part 4.
Daniel Davis 
13 October 2017
The GoerssHopkins Linearization Hypothesis and continuous Gspectra, part 5.
Daniel Davis 
20 October 2017
Finitetype invariants of knots, links, and string links
Robin Koytcheff
Abstract: Finitetype knot invariants (a.k.a. Vassiliev invariants) are an important class of invariants in that they conjecturally approximate all knot invariants and hence separate knots. They may also be defined for (closed) links and string links, and they are known to separate string links up to link homotopy. In other words, they are a complete invariant of string links where each component may pass through itself. This parallels (and is related to) a story about the kappa invariant, which conjecturally separates closed links up to link homotopy. In joint work with F. Cohen, Komendarczyk, and Shonkwiler, we showed that the kappa invariant separates string links up to link homotopy. In this talk, we will focus on the elementary, purely combinatorial description of finitetype invariants. 
27 October 2017
Homotopy string links, configuration spaces, and the kappa invariant
Robin Koytcheff
Abstract: A link is an embedding of disjoint circles in space. A link homotopy is a path between two links where distinct components may not pass through each other, but where a component may pass through itself. In the 1990s, Koschorke conjectured that link homotopy classes of ncomponent links are distinguished by the kappa invariant. This invariant is essentially the map that a link induces on configuration spaces of n points. In joint work with F. Cohen, Komendarczyk, and Shonkwiler, we proved an analogue of this conjecture for string links. A key ingredient is a multiplication on maps of configuration spaces, akin to concatenation of loops in a space. This approach is related to recent joint work with Budney, Conant, and Sinha on finitetype knot invariants and the Taylor tower for the space of knots. 
17 November 2017
The Simplicial Model Category Structure on Symmetric Spectra
Thomas Credeur 
1 December 2017
Symmetric Spectra and the Quillen Equivalence with BousfieldFriedlander Spectra
Thomas Credeur
Spring 2017
For the Spring 2017 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

27 January 2017
Genuine equivariant operads
Luis Pereira
University of Virginia
Abstract: 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 CisinskiMoerdijkWeiss. 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. 
3 February 2017
No meeting 
10 February 2017
No meeting
Fall 2016
For the Fall 2016 semester we will meet at 1:00 on Fridays in 208 Maxim Doucet Hall.

16 September 2016
An introduction to infinitycategories and the example of small categories.
Daniel Davis 
23 September 2016
Infinitycategories: more on the example of small categories and infinitycategorical versions of basic categorytheoretic notions.
Daniel Davis 
30 September 2016
Some examples of infinitycategorical concepts that build on categorytheoretic notions.
Daniel Davis 
14 October 2016
The notions of join and overcategory in the setting of infinitycategories.
Daniel Davis 
21 October 2016
Simplicial nerves and the homotopy category of an infinitycategory.
Daniel Davis 
28 October 2016
The homotopy category of an infinitycategory and a nicer formulation of it.
Daniel Davis 
4 November 2016
Given an \inftycategory C, there is an isomorphism h(C) \to \pi(C) of categories.
Daniel Davis 
11 November 2016
Given an \inftycategory C, more on the category \pi(C), and equivalences in C.
Daniel Davis
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