Spring 2024
January 17 (in-person)
Vijay Higgins (Michigan State)
Title: Central elements in the SL(d) skein algebra
Abstract: The skein algebra of a surface is spanned by links in the thickened surface, subject to skein relations which diagrammatically encode the data of a quantum group. The multiplication in the algebra is induced by stacking links in the thickened surface. This product is generally noncommutative. When the quantum parameter q is generic, the center of the skein algebra is essentially trivial. However, when q is a root of unity, interesting central elements arise. When the quantum group is quantum SL(2), the work of Bonahon-Wong shows that these central elements can be obtained by a topological operation of threading Chebyshev polynomials along knots. In this talk, I will discuss joint work with F. Bonahon in which we use analogous multi-variable 'threading' polynomials to obtain central elements in higher rank SL(d) skein algebras. Time permitting, I will discuss how a finer version of the skein algebra, called the stated skein algebra, can be used to show that the threading operation yields a well-defined algebra embedding of the coordinate ring of the character variety into the root-of-unity skein algebra in the case of SL(3).
February 7 (in-person - different location: STON 215)
Morgan Opie (UCLA)
Title: Enumerating stably trivial topological vector bundles with higher real K-theories
Pre-talk 1:30-2:20pm
Abstract: The goal of the pre-talk will be to set up some basic notions and ideas for the talk "Enumerating stably trivial topological vector bundles with higher real K-theories", to make the main talk more accessible. I will discuss vector bundles (and how I view them as a homotopy theorist); will talk about some historical methods for enumerating vector bundles; and will also talk a bit about "higher real K-theories" and why these generalized cohomology theories come into play. I aim for this can be an interactive talk, so if you have questions about specific things in the abstract please feel free to ask!
Main talk 2:30-3:30pm
Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes.
Building on work of Hu, we use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. Our main result is to compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. The talk will include a gentle discussion of the tools involved. This is joint work with Hood Chatham and Yang Hu.
February 13, 14, and 15 (in-person)
Ajay Ramadoss (Indiana)
This week, we will have a special three-lecture series on representation homology. Please note the unusual times and locations.
First Lecture: Tuesday, February 13, 2:30-4:00. Location: MATH 731
Title: Representation homology of associative algebras
Abstract: Classical representation varieties of associative algebras, deriving the representation functor (and definition of representation homology), cyclic homology and derived character maps, the stabilization theorem, combinatorial applications.
Second Lecture: Wednesday, February 14, 2:30-3:30. Location: SCHM 308
Title: Representation homology of Lie algebras
Abstract: Classical representation varieties of Lie algebras, derived representation schemes of Lie algebras and their representation homology, Hodge decomposition of cyclic homology (of universal enveloping algebras), derived character maps and the Drinfeld homomorphism, the derived Harish-Chandra homomorphism, an (open) conjecture on derived commuting schemes.
Third Lecture: Thursday, February 15, 2:30-4:00. Location: MATH 731
Title: Representation homology of spaces
Abstract: Representation varieties of groups, derived representation schemes of simplicial groups (and hence, spaces), basic properties of representation homology of spaces, analogy with and relation to Pirashvili-Hochschild homology, computations for Riemann surfaces (time permitting, knot complements and Lens spaces), representation homology of simply connected spaces, topologically interpreting the Strong Macdonald Conjectures via representation homology.
February 21 (in-person)
Jake Rasmussen (Illinois)
Title: Heegaard Floer homology, sutures, and functoriality
Abstract: Heegaard Floer homology is a powerful invariant of 3 and 4-manifolds, which fits (roughly) into the framework of a 3+1 dimensional TQFT. The "hat" version of this theory should fit into an extended TQFT in dimensions 2,3, and 4; in dimensions 2 and 3 this theory is described by the bordered Floer homology of Lipshitz, Ozsvath, and Thurston. To make the bordered theory into a 2-functor, one needs an appropriate 2-category of sutured manifolds, I'll discuss this category and what the 2-functor should look like.
February 28 (in-person)
Soren Galatius (Copenhagen)
Title: The action of Aut(C) on symplectic K-theory of the integers
Abstract: I will discuss aspects of the groups Sp_{2g}(Z), the integral symplectic groups, and their homology. On the one hand, Charney proved homological stability for these groups and Karoubi defined a flavor of K-theory based on symplectic groups instead of general linear groups, and also studied its relationship to ordinary algebraic K-theory. On the other hand, the relationship to principally polarized abelian varieties gives rise to actions of certain Galois groups on this flavor of K-theory, at least after completing at a prime p. I will discuss joint work with Feng and Venkatesh, in which we study the resulting Galois representations.
March 6 (online)
Title: New types of homotopy bialgebras in geometry and topology
Abstract: It is well known from the PhD thesis of Jim Stasheff that the homotopy theory of associative algebras is encoded by homotopy associative algebras, aka A_infini-algebras, since this latter notion carries infini-morphisms and satisfies a homotopy transfer theorem, for instance. A_infini-algebra structures encode the topological data of a space on the level of cochain complexes. When one wants to encode more data, like the Poincaré duality of manifolds, string topology, or non-commutative derived geometry, then one has to consider further structural operations, like symmetric bitensors or double brackets. The purpose of this talk will be to present the associated new types of homotopy bialgebras, to explain their relationship, and to show that they admit suitable homotopy properties like infini-morphisms and homotopy transfer theorem. To mention them, we will treat pre-Calabi—Yau algebras, homotopy double Poisson bialgebras, and homotopy infinitesimal balanced bialgebras. This is based on a joint work with Johan LERAY available at arXiv:2203.05062
March 20 (online)
Carlos De La Cruz Mengual
Title: The Degree-Three Bounded Cohomology of Complex Lie Groups of Classical Type
Abstract: The theory of continuous bounded cohomology was developed in the early 2000s by Burger and Monod as a topological refinement of the corresponding notion for discrete groups, and has powerful applications in geometry, dynamics and rigidity theory. Despite its power, the computation of continuous bounded cohomology remains a notorious challenge. In the setting of Lie groups, the problem can be reduced to the connected semisimple case, in which the so-called “isomorphism conjecture” predicts the isomorphism between continuous bounded and continuous cohomology. Open for more than twenty years, this conjecture is so far known to hold only in very few cases.
In this talk, I will discuss recent work that establishes the isomorphism conjecture in degree three for all complex Lie groups of classical type. The main ingredient of the proof is a bounded-cohomological stability theorem with optimal range, which itself relies on the occurrence in our setting of a phenomenon reminiscent of secondary homological stability.
April 3 (online)
Filippo Baroni (Oxford)
Title: Algorithmic classification of surface homeomorphisms
Abstract: Up to homotopy, homeomorphisms of closed surfaces come in three guises: periodic, reducible, and pseudo-Anosov. Among these three categories, pseudo-Anosov homeomorphisms exhibit qualitatively different topological, dynamical, and geometric properties.
The aim of this talk is to present an algorithm to decide if a surface homeomorphism is pseudo-Anosov, with a good theoretical upper bound on the running time. In particular, the algorithm runs in polynomial time in the genus of the surface and in the amount of information required to represent the input homeomorphism.
The inner workings of the algorithm rely on the combinatorics of splitting sequences of train tracks, together with a criterion of Masur and Minsky to estimate distances in the curve graph.
April 10 (in-person)
Michael Monaco (Purdue)
April 24 (in-person)
Jeremy Miller (Purdue)
Title: Hopf algebras, Steinberg modules, and the cohomology of GL_n(Z)
Abstract: I will review known results about the rational cohomology SL_n(Z) and GL_n(Z) and then discuss new joint work with Avner Ash and Peter Patzt that allows us to produce new families of unstable cohomology classes. We construct these new classes by proving that the direct sum of all homology groups of the integral general linear groups with Steinberg module coefficients form a commutative Hopf algebra, in particular a free graded commutative algebra.
FALL 2023
August 23 (online)
Yu Leon Liu (Harvard University)
Title: E_n-algebras in m-categories
Abstract: E_n-algebras are associative algebras with various levels of commutativity. They play a crucial in many parts of mathematics, from homotopy theory to topological field theories. In this talk, we are motivated by the following question: how to explicitly construct E_n algebras in a m-categories?
While natural, E_n-algebras have infinite coherence and are hard to construct by hand. However, by the Eckmann-Hilton argument, many of the coherences are redundant. In this talk, we will use a generalization of the Eckmann-Hilton argument to derive a ''minimal'' construction of E_n-algebras. Along the way we will use the interplay of arity-restriction and connectness for operads, as well as a general Blakers-Massey theorem for operadic algebras.
September 6 (in person)
Yilong Zhang (Purdue University)
Title: Monodromy of vanishing cycles
Abstract: A vanishing cycle on a compact complex surface is represented by a topological 2-sphere that contracts to a point as the surface deforms and acquires a nodal singularity. As an example, a vanishing cycle on a cubic surface is the difference of the cohomology classes [L_1]-[L_2] of two disjoint lines. As a cubic surface acquires a node, L_1 and L_2 become the same line and the vanishing cycle becomes the node on the line. As a complex surface varies in the universal family of smooth hyperplane sections of a complex 3-fold, the monodromy of vanishing cycles carries topological information of the 3-fold. We investigate this problem for smooth complex hypersurfaces in CP^4.
September 13 (in person)
Matthew Scalamandre (Notre Dame)
Title: A Solomon-Tits Theorem for Rings
Abstract: The classical Solomon–Tits theorem states that a spherical Tits building over a field is homotopy equivalent to a wedge of spheres of the appropriate dimension. In this talk, we’ll define a Tits complex that makes sense for an arbitrary ring, and prove a Solomon-Tits theorem when R either satisfies a stable range condition, or is the ring of S-integers of a number field. We will discuss applications to the cohomology of principal congruence subgroups of SL_n(Z), and some results about the top homology of this complex (an analogue of the classical Steinberg representation)
September 20 (two in-person talks)
2:30-3:00 Daniel Tolosa (Purdue University)
Title: An algebraic model for the free loop space as an S^1-space
Abstract: The free loop space of a topological space has a canonical circle action given by rotating loops, making it an S1-space. The work of Jones, Goodwillie, and others, relates the equivariant homology of the free loop space to the cyclic homology of the algebra of singular chains on the topological monoid of based loops. Recently, M. Rivera described a construction that models the free loop space in terms of the chains on the underlying space considered as a “categorical coalgebra”, a notion Koszul dual to a non-negatively graded dg category. This construction is "as small as possible”, has no hypotheses on the underlying space, and is suitable for computations in string topology. I will present a cyclic theory for categorical coalgebras and dg-categories extending the theory of cyclic homology for (dg) algebras and coalgebras. In particular, the cyclic chains of the categorical coalgebra of chains on a simplicial set provides a model for the S1-equivariant chains on the free loop space that is suitable for computations. Time permitting, we establish a comparison between Goodwillie's and Rivera's models, which can be understood in terms of the combinatorics of certain polytopes.
3:30-4:30 Yi Wang (Purdue University)
Title: Loop spaces, cyclic homology, and the Fukaya A-infinity algebra
Abstract: Firstly, I will describe a unified construction of chain models of the based loop space / free loop space / path space of a path-connected topological space X, defined using the fundamental groupoid of X, which can be viewed as a generalization of classical theorems of J. F. Adams and K. T. Chen. Secondly, I will combine this model with a Jones' type theorem in cyclic homology, as well as K. Irie's work in string topology and B. Ward's work in Deligne's conjecture, to describe chain level gravity algebra structures in string topology. (B. Ward is a former student of R. Kaufman at Purdue.) Lastly, time permitted, I will discuss an application of this chain model to formal aspects of Lagrangian Floer theory – lifting the Fukaya A-infinity algebra of a Lagrangian submanifold L to a Maurer-Cartan element in the dg Lie algebra of cyclic invariant chains on the free loop space of L.
September 27 (in person)
Adrian Clough (NYU Abu Dhabi)
Title: The homotopy theory of differentiable sheaves
Abstract: Many of the topological spaces involved in geometric topology, such as spaces of embedded manifolds, have complicated topologies that are difficult to specify and to manipulate. As already observed in the work of Galatius-Madsen-Tillmann-Weiss as well as Kupers, these spaces are often much easier to construct in the topos Diff_{≤ 0} of set-valued sheaves on manifolds, and may moreover be endowed with naturally occurring smooth structures.
Viewing Diff_{≤ 0} as a subcategory of the infinity topos Diff of homotopy-type valued sheaves — the eponymous differentiable sheaves— we will give conceptual proofs of how Diff_{≤ 0} provides a model for the theory of homotopy types, and exhibit many good formal properties of Diff_{≤ 0}, such as the fact that all filtered colimits are homotopy colimits in Diff_{≤ 0}. By endowing Diff with certain homotopical calculi, we are moreover able to obtain a generalisation of Berwick-Evans, Boavida de Brito, and Pavlov’s result that for any (paracompact Hausdorff) manifold A, and homotopy-type valued Sheaf X the mapping sheaf Diff(A,X) computes the mapping space of the underlying homotopy types of A and X.
October 4 (online)
Jan Dymara (Instytut Matematyczny Uniwersytetu Wroclawskiego)
Title: Tautological characteristic classes
Abstract: The tautological cochain of a simplicial complex maps a simplex to the same simplex treated as an element of the chain group. Passing to a suitable quotient of the chain group one can turn this cochain into a cocycle. If the simplicial complex is acted upon by a group $G$, one can further produce a tautological $G$-invariant cocycle. If the complex is contractible, one can get a group cohomology class, i.e. a characteristic class of flat $G$-bundles. We investigate this procedure for several linear groups and their actions on complexes spanned by ``generic'' configurations of points in projective spaces. For $SL(2,K)$ we recover the Witt class of Nekovar; for $PGL_+(2n,K)$ we obtain the Euler class for an arbitrary ordered field $K$.
This is a joint work with Tadeusz Januszkiewicz.
October 11 (in person)
Ben Antieau (Northwestern University)
Title: Integral models for spaces
Abstract: Generalizing and building on the work of Kriz, Ekedahl, Goerss, Lurie, Mandell, Mathew, Mondal, Quillen, Sullivan, Toën and Yuan, I will describe an integral cochain model for nilpotent spaces of finite type. A binomial ring is a lambda-ring in which all Adams operations act as the identity. A derived binomial ring is a derived Λ-ring equipped with simultaneous trivializations of the commuting Adams operations. For example, if X is a space, then ZX, the integral cochains on X, is naturally a derived binomial ring. The induced contravariant functor from spaces to derived binomial rings is fully faithful when restricted to nilpotent spaces of finite type. This is related, closely, to recent work of Horel and of Kubrak—Shuklin—Zakharov.
October 18 (in person)
Ajay Ramadoss (Indiana University)
Title: Topological realizations of algebras of quasi-invariants.
Abstract: In this talk, based on joint work with Yuri Berest, I shall formulate a topological realization problem for the algebras of quasi-invariants of Weyl groups and give its solution in the rank one case ($G=SU(2)$). The solution to this realization problem could be seen as generalizing Borel's theorem that realizes the ring of invariant polynomials of a Weyl group as the cohomology ring of the classifying space $BG$ of the associated Lie group $G$. We call the resulting $G$-spaces $F_m(G,T)$ $m$ quasi-flag manifolds, and their Borel homotopy quotients $X_m(G,T)$ spaces of $m$-quasi-invariants. I shall talk about the equivariant K-theory and equivariant elliptic cohomology of these spaces, identifying them with exponential and elliptic quasi-invariants respectively. Time permitting, I shall also talk about the higher rank case.
October 25 (in person)
Bena Tshishiku (Brown University)
Title: Group actions on homotopy hyperbolic manifolds
Abstract: In this talk we consider group actions on smooth manifolds that are homotopy equivalent to a hyperbolic manifold. Such actions were first studied by Farrell-Jones, motivated by a question of Schoen-Yau. We discuss progress toward classifying these actions. The main techniques include geometric and topological rigidity together with smoothing theory. This is joint work with Mauricio Bustamante.
November 1
November 6 (in person, on a Monday in room Math 431)
Aaron Landesman (Harvard)
Title: Homological stability for generalized Hurwitz spaces with an application to number theory
Abstract: We describe a new homological stability result for a generalized version of Hurwitz spaces. This builds on previous work of Ellenberg-Venkatesh-Westerland, showing that homology groups of certain Hurwitz spaces stabilize. We generalize this in two directions. First, we work with covers of arbitrary punctured Riemann surfaces instead of just the disc. Second, we generalize the result to "coefficient systems," which are essentially a sequence of compatible local systems on configurations spaces. After detailing the above homological stability result, we will then explain how both these generalizations are employed to prove versions of numerous conjectures from number theory relating to the distributions of ranks of elliptic curves and Selmer groups of elliptic curves.
November 8 (online)
Violeta Borges Marques (University of Antwerp)
Title: The category of Necklaces is Reedy Monoidal
Abstract: We further the study of the interactions between Reedy and monoidal structures on a small category, building upon the work of Barwick. In the second part, we study the category Nec of necklaces, as defined by Baues and Dugger-Spivak, showing that the category of A-valued presheaves on Nec is model monoidal when equipped with the Day convolution product, for any symmetric monoidal model category A. If time permits, I will explain an application of this result to the construction of a model structure on templicial chain complexes, which is a first step in the construction of a model structure of dg-Segal categories.
November 15 (online)
Michael Borinsky (ETH-ITS Zürich)
Title: On the top-weight cohomology of the moduli space of curves
Abstract: I will present new results on the asymptotic growth rate of the Euler characteristic of Kontsevich's commutative graph complex. By a work of Chan, Galatius and Payne, these results imply the same asymptotic growth rate for the top-weight Euler characteristic of M_g, the moduli space of curves, and establish the existence of large amounts of unexplained top-weight cohomology in this space.
November 29 (online)
Tobias Hartnick (KIT)
Title: The isomorphism conjecture in bounded cohomology
Abstract: In this survey talk I will discuss one of my favourite conjectures in the topology of Lie groups, the so-called isomorphism conjecture in bounded cohomology. While there is still not a single connected finite-dimensional Lie group whose bounded cohomology ring is both known and non-trivial, this bold conjecture proposes an explicit formula for the bounded cohomology rings of all connected finite-dimensional Lie groups (and in fact, all connected locally compact groups).
I will discuss some of the progress towards the conjecture that has been made over the last decade, focussing on three main aspects:
* Partial surjectivity results based on characteristic classes of flat bundles
* Partial injectivity results in rank one based on analytic methods
* Computations in low degree based on primary and secondary bounded cohomological stability
The goal is to give a flavour of the large variety of approaches inspired by the conjecture.
December 6 (online)
Tomasz Maszczyk (University of Warsaw)
Title: Foliations with rigid Godbillon-Vey class
Abstract: We show (under the technical condition of separability of some topological space of cohomology) that the Godbillon-Vey number of a foliation of codimension one on a compact orientable 3-fold is topologically rigid (i.e. constant under infinitesimal singular deformations) iff it admits a projective transversal structure (i.e. a foliation arises by gluing of level sets of local functions with fractional linear transition maps).
spring 2023
January 18, 2023
Xin Li (University of Glasgow) video
Title: Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
Abstract: Topological groupoids describe orbit structures of dynamical systems by capturing their local symmetries. The group of global symmetries, which are pieced together from local ones, is called the topological full group. This construction gives rise to new examples of groups with very interesting properties, solving outstanding open problems in group theory. This talk is about a new connection between groupoids and topological full groups on the one hand and algebraic K-theory spectra and infinite loop spaces on the other hand. Several applications will be discussed. Parts of this connection already feature in work of Szymik and Wahl on the homology of Higman-Thompson groups.
January 25, 2023
Dev Sinha (University of Oregon) (online)
Title: Geometric cochains and the phenotypics of homotopy theory.
Abstract: (joint with Greg Friedman and Anibal Medina) Interplay between discrete and continuous, combinatorial and geometric, digital and analog has always been at the heart of topology. This was expressed by Sullivan, who likened homotopy types with genetic codes, both being discrete data with continuous expressions, as he made remarkable progress in both homotopy theory and smooth topology. We are developing geometric cochains for among other reasons as a way, conjecturally, to provide an E-infinity algebra model - and thus homotopy model, by Mandell’s Theorem - for smooth manifolds through their differential topology. This would provide a phenotypical determination of the genetics of a manifold.
Geometric cochains are a smooth version of Chow theory, first developed just in the last decade by gauge theorists such as Lipyanskiy and Joyce. The theory could have been defined alongside all of the classical cohomology theories, but there are technical obstacles. With manifolds with boundaries being needed for chain complex structure, and the natural product being intersection or more generally fiber product, one is quickly led to working with manifolds with corners. A satisfactory theory of these with transversality was only in the last decade developed by Joyce. We set the foundations of this theory. Our original motivation for studying such a theory was pedagogical - teaching basic and intermediate algebraic topology - and we indicate some of those applications.
In our first application (posted before the foundational work) we bind the combinatorially defined cup product to the geometrically defined fiber product when both are in play - namely on a manifold with a smooth cubulation or triangulation. We do this through an almost-canonical vector field on a cubulated manifold, whose flow interpolates between the usual geometric diagonal and the Serre diagonal.
In the combinatorial setting, choices for resolving lack of commutativity at the cochain level give rise to an E-infinity structure. We posit that choices for resolving lack of transversality give rise to a partially defined E-infinity structure on geometric cochains. In particular, we have an explicit conjecture for a partially defined action of the Fulton-MacPherson operad on geometric cochains. We hope to connect with experts on partially defined algebras and related matters to help resolve technicalities in this program.
February 1, 2023
Fabio Capovilla-Searle (Purdue) (In-person)
Topics Exam: The Steinberg Module
Abstract: In this talk I will recall the Steinberg module definition and Borel-Serre Duality. I will discuss the Solomon-Tits Theorem proving that the Tits Building is highly connected.
February 8, 2023
Muriel Livernet (IMJ-PRG) (online)
Title: Homotopy theory of spectral sequences
Abstract: Spectral sequences arise as algebraic tools in many areas of mathematics. Of particular interest are spectral sequences of filtered complexes and multicomplexes. Questions addressed during the talk: how to compare spectral sequences? Can we invert some class of weak equivalences, as e.g. morphisms realizing an isomorphism from a given page and afterwards? This is a long project with S. Whitehouse and I will explain some model category structures relying on these weak equivalences, for filtered complexes, multicomplexes and show that for spectral sequences we have a weaker notion of model category structure.
February 15, 2023 *special time - 1:00 pm*
Geoffroy Horel (Paris 13) (online)
Title: Binomial rings and homotopy theory
Abstract: In a famous paper, Sullivan showed that the rational homotopy theory of finite type nilpotent spaces can be encoded in a fully faithful manner by mapping it to the homotopy category of commutative differential graded algebras over the rational numbers. For integral homotopy theory, a result of Mandell shows that it is faithfully captured by the integral cochains equipped with their E-infinity structure. This functor is however not full. I will explain a way of fixing this problem inspired by work of Toën, using cosimplicial binomial rings instead of E-infty differential graded algebras.
February 22, 2023
Tara Brendle (University of Glasgow) (In-person)
Title: Semi-direct product structures in mapping class groups of 3-manifolds
Abstract: There is a natural map from the mapping class group of any manifold to the (outer) automorphism group of its fundamental group. In general, this map is neither injective nor surjective. In this talk, we will show that an associated short exact sequence splits, which gives rise to a semi-direct product structure on the mapping class group when the 3-manifold contains copies of S^2 x S^1. (This is joint work with Nathan Broaddus and Andrew Putman.)
March 1, 2023
Deepam Patel (Purdue) (In-person)
Title: Algebro-Geometric constructions of some representations of the Braid group
Abstract: In this talk, i will explain an "algebro-geometric" construction of some well known representations of the Braid group (including the Burau and Gassner representations). The algebraic perspective allows one to equip these representations with additional structures. If there is time, I will explain that these are in fact examples of Gamma Motives. This is partially based on joint work with M. Nori.
March 8, 2023
Joshua Sussan (CUNY) (In-person)
Title: Non-semisimple Hermitian TQFTs
Abstract: Topological quantum field theories coming from semisimple categories build upon interesting structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give some applications.
March 15, 2023
Andy Qingyun Zeng (UPenn) (zoom talk)
Title: Lie infinity groupoids and algebroids in singular foliations
Abstract: We use Lie infinity groupoids and L infinity algebroids in studying higher structures arising in differential geometry. We study the homotopy coherent representations of Lie infinity groupoids and L infinity algebroids, and apply them to (singular) foliations. Finally, we prove an A infinity version of de Rham theorem, and a higher Riemann-Hilbert correspondence for foliated infinity local systems.
March 20, 2023 (Monday at 10)
Peter Patzt (University of Oklahoma) (In-person)
Title: Congruence subgroups of braid groups
Abstract: The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation B_n -> GL_n(Z). We find the image of such congruence subgroups in GL_n(Z) - an open problem by Dan Margalit. Additionally, we characterize the quotients of braid groups by their congruence subgroups in terms of symplectic congruence subgroups.
March 22, 2023 *special time - 9:30 am*
Alexander Berglund (Stockholm University) (online)
Title: Algebraic models for classifying spaces of fibrations
Abstract: We construct an algebraic model for the rational homotopy type of Baut(X), the classifying space of fibrations with fiber a simply connected finite CW-complex X. As an application, this reduces the computation of the rational cohomology ring of Baut(X) to the computation of cohomology of arithmetic groups and dg Lie algebras. Another corollary is that the higher homotopy groups of Baut(X) are arithmetic in a suitable sense, which extends a classical result of Sullivan and Wilkerson. Our results also improve and generalize certain earlier results of Madsen and myself on Baut(M) for highly connected manifolds M. This is joint work with Tomas Zeman.
March 29, 2023
Alexander Martsinkovsky (Northeastern) (online)
Title: The ubiquity of stable functors
Abstract: We consider additive functors on module categories. The word "stable" in the title refers to the functors defined on stable categories, also known as categories modulo projectives (respectively, injectives). Such categories are still additive but rarely abelian. They were introduced by Eckmann and Hilton in the early 1950s in "preparation" for their work on duality in homotopy theory. Stable functors came into prominence in the late 1960s in the foundational works of M. Auslander "Coherent functors” and (with M. Bridger) "Stable module theory". Those sources are nowadays frequently cited but not necessarily, at least explicitly, in connection with stable functors. The work on stable module theory spun off an entire new industry, called Gorenstein homological algebra, which can be viewed as a distant algebraic relative of stable homotopy (via various generalizations of Tate (co)homology). Eventually, stable functors have been all but forgotten.
What has transpired very recently is the remarkable ubiquity and usefulness of stable functors. The goal of this lecture is to try and survey some of the new related results together with their applications. Perhaps, the most significant one is the introduction of cotorsion, a notion dual to torsion. Unlike torsion, it does not have a classical prototype.
Yet another remarkable aspect of the recent developments is the persistence of topological overtones: the comparison of Vogel homology and the asymptotic stabilization of the tensor product is a startling look-alike of the comparison map between Steenrod-Sitnikov homology and Čech homology, derived functors of (co)torsion beg for a connection with the work of Greenlees-May on the derived functors of adic completion and local homology, the universal coefficient formulas for both homology and cohomology are picture-perfect examples of stabilization of an additive functor, etc. Could this possibly be related to the fact that several topologists were among the main contributors to the field in the second half of the last century: B.Eckmann, P.Hilton, D.B. Fuchs, P. Vogel, G. Mislin, … ?
This will be an expository talk; no prior experience with functor categories is assumed.
April 5, 2023
April 12, 2023
Thilo Kuessner (online)
Title: Topology of representations
Abstract: We will explain some examples of how topology can be used to get information about the variety of representations of a finitely presented group. For 3-manifold groups we will use the fundamental class in the extended Bloch group - an explicit description of the third group homology for the special linear group - to distinguish components in the representation variety and to show, for example, that the number of components can become arbitrarily large for 2-bridge knots. For surface groups, where the components of the representation variety are known, we will use a non-Hausdorff version of the Fock-Goncharov coordinates to parametrize the open set of Anosov representations, extending the work of Bonahon-Dreyer who used Fock-Goncharov coordinates to parametrize the Hitchin components.
April 19, 2023
Michael Monaco (Purdue) (in-person)
Title: Equivariant bimodule algebras and their representations
Abstract: Bimodules (or distributors) are a very general notion suitable for describing many different combinatorial, algebraic, and categorical structures. Our bimodules are groupoid bimodules and we consider different representation categories of these for which we introduce monoidal structures. The different versions are related by so-called element constructions. As an application, this allows us to describe a variety of structures such as operads, properads, and hyper modular operads as so-called plethysm monoids, and formulate generalizations. We will highlight several of the important ideas through examples.
April 26, 2023
Title: Equivariant Trees and Partition Complexes
Abstract: Given a finite set, the collection of partitions of this set forms a poset category under the coarsening relation. This category is directly related to a space of trees, which in turn has interesting connections to operads. But what if the finite set comes equipped with a group action? What is an "equivariant partition"? And what connection is there to equivariant trees? We will explore possible answers to these questions in this talk, based on joint work with J. Bergner, P. Bonventre, D. Chan, and M. Sarazola.
FALL 2022
August 17, 2022 (in-person)
Andrea Bianchi (University of Copenhagen)
Title: Symmetric groups, Hurwitz spaces and moduli spaces of surfaces
Abstract: Let d = 2g ≥ 2 be even. There is a graded commutative ℚ-algebra, denoted A(d), arising as the conjugation invariants of the associated graded of a multiplicative filtration on the group algebra ℚ[S_d] of the symmetric group S_d. In joint work with Alexander Christgau and Jonathan Pedersen, I computed a minimal system of generators for A(d) and a formula for the lowest degree relation among minimal generators. There is a simply connected topological space B(d) enjoying the following two properties:
the rational cohomology ring of B(d) is isomorphic to A(d);
a component of the double loop space of B(d) is homotopy equivalent to a component of the group completion of a certain topological monoid Hur(S_d^geo) of Hurwitz spaces.
I will briefly describe the topological monoid Hur(S_d^geo) and its relation to the moduli space M_{g,1} of Riemann surfaces of genus g with one boundary curve. Time permitting, I will describe how the lowest degree relation gives rise to a cohomology class in H^{2g-1}(M_{g,1}; ℚ), which I conjecture to be non-trivial.
August 31, 2022 (in-person)
Eric Samperton (Purdue)
Title: Topological quantum computation is hyperbolic
Abstract: We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams one compiles are hyperbolic. Furthermore, the diagrams can be arranged to have additional nice properties, such as being alternating with minimal crossing number. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.
September 7, 2022
Title: Classification of Consistent Systems of Handlebody Group Representations
Abstract: In my talk I will explain how the theory of modular and cyclic operads can be used to establish a bridge between representations of three dimensional handlebody groups and categorical algebra. The crucial result, building on work of Giansiracusa, is a description of the modular envelope of the framed $E_2$-operad in terms of handlebodies and a classification of cyclic framed $E_2$-algebras in terms of ribbon Grothendieck-Verdier categories. I will focus on the operadic and topological aspects. The talk is based on joint work with Lukas Woike.
September 14, 2022
Title: Mod p homology of unordered configuration spaces via spectral Lie algebras
Abstract: Labeled configuration spaces B_k(M;X) of a manifold M with labels in a spectrum X are generalizations of the unordered configuration spaces B_k(M)=B_k(M;S^0). When M is framed, Knudsen identified labeled configuration spaces in M with the topological Quillen objects of certain spectral Lie algebras. This equivalence allows us to extract information about the mod p homology of B_k(M;X) via a bar spectral sequence and knowledge about the structure of the homology of spectral Lie algebras. For p>2, some immediate results include that the mod p homology groups of B_k(M) depend on and only on the cohomology ring H^*(M; F_p) for small k when M is of even dimension; and that for a closed genus g surface with g>0, the dimensions of the mod p homology groups of B_k(\Sigma_g) are the same as its Betti numbers when k is at most p.
September 21, 2022
Dan Petersen (Stockholm University)
Title: Hyperelliptic curves, the scanning map, and moments of quadratic L-functions
Abstract: I will explain a calculation of the stable cohomology of the hyperelliptic mapping class group with coefficients in an arbitrary symplectic representation. The result is closely related to, and provides a geometric interpretation of, a series of conjectures on asymptotics of moments of families of quadratic L-functions. (Joint with J. Bergström, A. Diaconu and C. Westerland).
September 28, 2022
Joana Cirici (Universitat de Barcelona)
Title: A-infinity structures on almost complex manifolds
Abstract: Dolbeault cohomology is a fundamental cohomological invariant for complex manifolds. This analytic invariant is connected to de Rham cohomology by means of a spectral sequence, called the Frölicher spectral sequence. In this talk, I will explore this connection from a multiplicative viewpoint: using homotopy-theoretical methods, I will describe how products (and higher products) behave in the Frölicher spectral sequence. I will also review an extension of the theory to the case of almost complex manifolds and talk about some open problems in complex geometry that may be addressed using homotopy theory.
October 5, 2022 (in-person)
Anh Trong Nam Hoang (University of Minnesota)
Title: Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups
Abstract: In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra heuristics. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture over function fields by employing a topological observation which identifies the homology of the braid groups with coefficients arising from braided vector spaces with the cohomology of a quantum shuffle algebra, using the Fox-Neuwirth cellular stratification of configuration spaces of the plane. In this talk, we will extend their techniques to study configuration spaces of the punctured plane and prove a similar result for the homology of the Artin groups of type B. As an application, we will discuss computations when the braid representations are one-dimensional over a field, which shed light on a special case of a conjecture about the homology of mixed braid groups due to Ellenberg-Shusterman.
October 12, 2022 (in-person)
Title: Computing high-dimensional group cohomology via duality
Abstract: In recent years, duality approaches have yielded new results about the high-dimensional cohomology of several groups and moduli spaces, such as SL_n(Z) and M_g. I will use the example of SL_n(Z) to explain the strategy of these approaches, presenting results of Church-Farb-Putman and Brück-Miller-Patzt-Sroka-Wilson. I will put this into a more general context by giving an overview of similar results for related families of groups.
Title: Partial bases and homological stability for general linear groups revisited
Abstract: While the rational cohomology of the general linear group GL_n(R) of a ring R remains mysterious in high cohomological degrees, it can often be completely computed if the cohomological degree is small compared to n. The reason for this is a phenomenon called (co-) homological stability. In this talk I will discuss work in progress with Calista Bernard and Jeremy Miller in which we are establishing a generic slope-1 homological stability result for the general linear groups GL_n(R) of a large class of rings. The class of admissible rings is expected to include all euclidean rings. This extends the classical slope-1/2 stability results due to van der Kallen and Maazen, and builds on recent ideas of Galatius--Kupers--Randal-Williams and Kupers--Miller--Patzt.
October 19, 2022
Anton Galaev (University of Hradec Králové)
Title: Losik classes and Reeb foliations
Abstract: Developing ideas of Gelfand’s formal geometry, Losik suggested to consider characteristic classes of foliations as elements of cohomology of certain bundles over the leaf spaces of foliations. These classes come from the Gelfand-Fuchs cohomology. In this way, for a codimension-one foliation appear two characteristic classes modifying the classical Godbillon-Vey class. We study these classes for the case of the Reeb foliations on the 3-dimensional sphere. The Godbillon-Vey class of all these foliations is trivial. In contrast, one of the classes under consideration is non-trivial for all Reeb foliations, and it detects the compact leaf with non-trivial holonomy. The other characteristic class is more delicate: it is non-trivial for some Reeb foliations and it is trivial for some other Reeb foliations, i.e., this class is very sensitive to the dynamics of the non-compact leaves and it distinguishes non-diffeomorphic Reeb foliations. The talk is based on joint work with Yaroslav Bazaikin and Pavel Gumenyuk.
October 26, 2022
Cary Malkiewich (Binghamton University)
Title: Higher scissors congruence
Abstract: Hilbert's Third Problem asks for sufficient conditions that determine when two polyhedra in three-dimensional Euclidean space are scissors congruent. Classically, the attempts to solve this problem (in this and other geometries) lead into group homology and algebraic K-theory, in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that the presence of K-theory here is not ad-hoc, but is integral to the definition of scissors congruence itself. This leads to a natural notion of "higher" scissors congruence groups, or higher algebraic K-theory of scissors congruence. In this talk I'll describe an exciting, ongoing program to better understand these higher groups, and to compute them in new cases. The main results so far are a trace map to group homology, a Farrell-Jones isomorphism, a Solomon-Tits theorem, and a new description of scissors congruence K-theory as a Thom spectrum. Much of this is joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.
November 2, 2022 (in-person)
Title: Bases, Nerves, and Homotopy Groups
Abstract: There a variety of ways to recover the homotopy type of a space X from the combinatorics of a nice open cover, the first and most famous being Borsuk’s Nerve Theorem. In the 1960’s, McCord deduced a Nerve Theorem for weak homotopy by studying certain base-like open covers. Much more recently, inspired by ideas from Algebraic Quantum Field Theory, G. Ruzzi showed that the fundamental group of a space can be computed from the combinatorics of a base, so long as each set in the base is simply connected. I’ll explain a variation on Ruzzi’s technique that recovers the homotopy n-type of a space from the combinatorics of a sufficiently fine cover by n-connected sets. In addition to extending Ruzzi’s result, this leads to a new generalization of the Nerve Theorem for homotopy n-type, generalizing McCord’s result.
November 9, 2022 (in-person)
Title: Homology of big mapping class groups via homological stability
Abstract: Certain inclusions of finite-type surfaces induce inclusions of the corresponding mapping class groups. These maps between mapping class groups are known to induce isomorphisms on homology groups in low degrees (compared to the complexity of the surfaces). This is known as the homological stability, relating the homology groups of different mapping class groups. There are some recent studies of mapping class groups of infinite-type surfaces, such as a finite-type surface minus a Cantor set or a surface of infinite genus, some of which have self-similarity. We use homological stability tools to show that certain self-similar inclusions induce homological isomorphisms for the mapping class groups (of homeomorphic surfaces) in all degrees. That is, we have (non-surjective) endomorphisms of such a big mapping class group that induce isomorphisms on homology. We use this as a tool to compute the homology groups of certain big mapping class groups (e.g. the disk minus a Cantor set). This is on-going joint work with Danny Calegari and Nathalie Wahl.
November 16, 2022 (in-person)
Najib Idrissi (Université Paris Cité and IMJ-PRG)
Title: Formality and non-formality of Swiss-Cheese operads and variants
Abstract: Configuration spaces consist in ordered collections of points in a given ambient manifold. Kontsevich and Tamarkin proved that the configuration spaces of Euclidean n-spaces are rationally formal, i.e., that their rational homotopy type is completely encoded by their cohomology. Their proofs use ideas from the theory of operads, and they prove the stronger result that the operads associated to configuration spaces of Euclidean n-spaces, called the little n-cubes operads and denoted E_n, are formal. Based on these two proofs, we have computed (partially in joint work with Campos, Ducoulombier, Lambrechts, and Willwacher) the rational homotopy types of configuration spaces and framed configuration spaces of compact manifolds.
Voronov’s Swiss-Cheese operads encode the action of an E_n-algebra on an E_(n-1)-algebra. Livernet and Willwacher proved that an enlarged version of this operad which encodes morphisms (rather than actions) is not formal. In this talk, I will present two results: 1. the higher codimensional version of the Swiss-Cheese operad which encodes a central derivation from an E_m-algebra to an E_n-algebra, is formal for n-m is at least two; 2. Voronov’s original version of the codimension 1 Swiss-Cheese operad is not formal (in joint work with Vieira).
November 23, 2022
Thanksgiving Vacation: no seminar
November 30, 2022 (in-person)
Title: An algebraic model for the free loop space
Abstract: I will describe an algebraic construction that models the passage from a topological space to its free loop space, without imposing any restrictions on the underlying space. The input of the construction is a “curved coalgebra" over an arbitrary ring R equipped with additional structure and satisfying certain properties. The output is an R-chain complex equipped with a “rotation” operator. The construction is a modified version of the coHochschild complex of a conilpotent coalgebra and is invariant with respect to a notion of weak equivalence for coalgebras drawn from Koszul duality theory. When this construction is applied to a suitable model for the coalgebra of chains of an arbitrary simplicial set X (possible non-simply connected and non-fibrant) one obtains a chain complex that is quasi-isomorphic to the singular chains on the free loop space of the geometric realization of X. The construction is as small as it can be. Time permitting, we also discuss the relationship with Ed Brown’s twisted tensor product model in terms of the holonomy of the free loop space fibration given by the conjugation action of (a suitable model for) the based loop space on itself. This model for the free loop space is potentially useful in studying invariants arising in string topology of non-simply connected manifolds, some of which are able to distinguish homotopy equivalent non-homeomorphic manifolds.
December 7, 2022 (in person)
Martina Rovelli (UMass Amherst)
Special format. Everything will take place in room MTH 731.
9:30-10:30 Limits in categories and ∞-categories
Abstract: In this pre-talk, we will revise the theory of limits for diagrams valued in a category or an ∞-category, presented by a quasi-category or a topological category. We will introduce the universal property of (∞-)limits in terms of the (∞-)functor they represent, and discuss the characterization in terms of the (∞-)categories of cones. We will also go over a few examples to highlight the different behavior between the notion of limit and ∞-limit.
10:30 - 11:30: Questions, coffee, pastries
11:30 - 12:30 Limits in 2-categories and (∞,2)-categories
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 n-category or an (∞,n)-category. In this talk, we will focus on recent developments in the formalization of the theory of limits for diagrams valued in a 2-category or an (∞,2)-category. We will introduce the universal property for 2- and (∞,2)-limits in terms of the 2-functor or (∞,2)-functor they represent, and discuss current work in progress (joint with Moser and Rasekh) towards a characterization of (∞,2)-limits in terms of double ∞-categories of cones.
Spring 2022
May 11, 2022
Dima Roytenberg (Utretch and Nijmegen)
Title: Equivalences of differential graded manifolds
Abstract: Classical Lie theory can be encapsulated in an adjunction between the category of finite-dimensional real Lie algebras and that of finite-dimensional real Lie groups. Higher Lie theory aims to extend this to an infinity-adjunction between the infinity-category of non-negatively graded dg-manifolds (or higher Lie algebroids), and that of Kan simplicial manifolds (or higher Lie groupoids). To make this work, one of the necessary steps is to define and characterize the weak equivalences in the category of dg-manifolds, localizing with respect to which will yield the sought-after infinity category. I will explain the setup and formulate some conjectures concerning equivalences of dg-manifolds, together with some partial results in their support.
May 4, 2022:
Title: Deformations of Lie groupoids
Abstract: In this talk we will discuss deformations of Lie groupoids and present the cohomology which controls them. We will study some of its properties, and look at relations with other cohomologies: deformation cohomologies for classical examples, such as Lie group actions and foliations, and deformation cohomology of Lie algebroids. The deformation cohomology is used to prove rigidity and normal form results for compact and for proper Lie groupoids, which can provide results in foliation theory. The talk is based on joint work with Marius Crainic and Ivan Struchiner.
April 27, 2022
Title: Models for topological spaces in $I$-chain complexes
Abstract: In joint work with Steffen Sagave we find a strictly commutative model for the cochain algebra of a space over an arbitrary ground ring. This model is built in the category of $I$-chain complexes, where $I$ is the category of finite sets and injections. In this talk we review this result and explain how it determines the integral homotopy type of (nilpotent) spaces (of finite type). We also describes features of the category of $I$-chain complexes and discuss how these might affect the chances of finding algebraic models for spaces via comonoids in $I$-chain complexes.
April 20, 2022
Oishee Banerjee (University of Bonn)
Title: Filtration of cohomology via symmetric semisimplicial spaces
Abstract: Inspired by Deligne's use of the simplicial theory of hypercoverings in defining mixed Hodge structures, we replace the indexing category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$ and study (a class of) $\Deltainj S$-hypercoverings, which we call \emph{spaces admitting symmetric (semi)simplicial filtration}- this special class happen to have a structure of a module over a graded commutative monoid of the form $\Sym M$ for some space $M$. For $\Delta S$-hypercoverings we construct a spectral sequence, somewhat like the Čech-to-derived category spectral sequence. The advantage of working on $\Delta S$ is that all of the combinatorial complexities that come with working on $\Delta$ are bypassed, giving simpler, unified proof of results like the computation of (in some cases, stable) singular cohomology (with $\Qb$ coefficients) and \etale cohomology (with $\Qb_{\ell}$ coefficients) of the moduli space of degree $n$ maps $C\to \P^r$, $C$ a smooth projective curve of genus $g$, of unordered configuration spaces, of the moduli space of smooth sections of a fixed $\grd$ that is $m$-very ample for some $m$ etc. In the special case when a $\Deltainj S$-object $X_{\bullet}$ \emph{admits a symmetric semisimplicial filtration by $M$}, the derived indecomposables of $H^*(X_{\bullet})$ as a $H^*(\Sym M)$-module (in the sense of Galatius–Kupers–Randal-Williams) give the cohomology of the space of \emph{$M$-indecomposables}.
April 13, 2022:
Luciana Basualdo Bonatto (University of Oxford)
Title: Grothendieck-Teichmüller theory and modular operads
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 well-known 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.
April 6, 2022:
Title: The noncommutative Legendre transform and Calabi-Yau structures
Abstract: I will discuss a noncommutative version of the Legendre transform, in the formalism of A-infinity algebras/categories. This operation gives maps between certain classes in negative cyclic homology, seen as an nc version of differential forms, and an nc version of an integrable polyvector field. This allows one to compute certain TQFT operations on the Hochschild and negative cyclic homology of a smooth CY algebra/category, such as, for example, the algebra of chains on the based loop space of an orientable compact manifold, with the product induced by loop composition. This is joint work with Maxim Kontsevich and Yiannis Vlassopoulos.
March 30, 2022
Title: Combinatorial structures in string topology
Abstract: String topology makes use of spaces of graphs to define algebraic structures on the chains of the loop space of a manifold as well as on the Hochschild cochains of an algebra. Isomorphisms on homology are shown to respect at least some of these structures and one question is what the richest algebraic structure respected by such an isomorphism should be. As a first step toward an answer, we investigate the spaces of operations themselves—these spaces of graphs. On both the loop space side and on the Hochschild side of the discussion, the building blocks for the graphs are trees. The trees on each side satisfy conditions that appear at face value to be completely different but produce spaces of trees which are in fact equivalent. In this talk, we describe these spaces of trees and of graphs in detail. Along the way we’ll introduce polyhedra called “assocoipahedra,” whose combinatorics determine the structure of a V-infinity algebra analogous to how combinatorics of associahedra determine the structure of an A-infinity algebra. This is joint work with Thomas Tradler.
March 23, 2022:
Erik Lindell (Stockholm University)
Title: Abelian cycles in the homology of the Torelli group
Abstract: The mapping class group of a compact and orientable surface of genus g has an important subgroup called the Torelli group, which is the kernel of the action on the homology of the surface. In this talk we will discuss the stable rational homology of the Torelli group of a surface with a boundary component, about which very little is known in general. These homology groups are representations of the arithmetic group Sp_{2g}(Z) and we study them using an Sp_{2g}(Z)-equivariant map induced on homology by the so-called Johnson homomorphism. The image of this map is a finite dimensional and algebraic representation of Sp_{2g}(Z). By considering a type of homology classes called abelian cycles, which are easy to write down for Torelli groups and for which we can derive an explicit formula for the map in question, we may use classical representation theory of symplectic groups to describe a large part of the image.
March 9, 2022:
Three short talks:
Zach Himes (Purdue University)
Title: On not the dualizing module for Aut(F_n)
Abstract: Bestvina--Feighn proved that Aut(F_n) is a rational duality group, i.e. there is a Q[Aut(F_n)]-module, called the rational dualizing module, and a form of Poincare duality relating the rational cohomology of Aut(F_n) to its homology with coefficients in this module. Bestvina--Feighn's proof does not give an explicit combinatorial description of the rational dualizing module of Aut(F_n). But, inspired by Borel--Serre's description of the rational dualizing module of arithmetic groups, Hatcher--Vogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that Hatcher--Vogtmann's module is not the rational dualizing module.
Michael Monaco (Purdue University)
Title: The plus construction for monoidal categories
Abstract:A Feynman category is a symmetric monoidal category $\mathcal{F}$ with a particular set of properties that make it suitable for encoding operad-like structures. These structures often organize themselves into a hierarchy. For example, we can think of algebras as being subordinate to operads. The plus construction allows one to go ``up'' in the hierarchy. There are important consequences when a Feynman category comes from something below. In other words, when there exists an $\mathcal{M}$ such that $\mathcal{F} \cong \mathcal{M}^+$.
For some Feynman categories the ``equation'' $\mathcal{F} \cong \mathcal{M}^+$ does not have a solution in Feynman categories. However, we will show that the plus construction can be defined for any symmetric monoidal category. Moreover, when $\mathcal{M}$ is what we call a unique factorization category (UFC), the plus construction $\mathcal{M}^+$ is a Feynman category. The typical example of a UFC is the category of cospans. Modifications of this example already represent a large class of UFCs, so we get many more pairs $(\mathcal F, \mathcal M)$ such that $\mathcal{F} \cong \mathcal{M}^+$.
Yang Mo (Purdue University)
Title: Simply Colored coalgebra
Abstract: In this short talk, we will discuss the construction of simply colored coalgebra that appears in the work with Kaufmann. The coalgebra generalizes the notion conilpotent coaugmented coalgebra over a unital commutative ring by allowing more than one (possibly infinite) group-like element and is equipped with new reduced comultiplication based on different group-like elements. It is the right framework in which one can address the antipode for bialgebra naturally appearing in topology, number theory and physics.
February 23, 2022:
Alexis Aumonier (University of Copenhagen)
Title: An h-principle for complements of discriminants
Abstract: In classical algebraic geometry, discriminants appear naturally in various moduli spaces as the loci parametrising degenerate objects. The motivating example for this talk is the locus of singular sections of a line bundle on a smooth projective complex variety, the complement of which is a moduli space of smooth hypersurfaces.
I will explain how one can study the homology of these algebraic objects using homotopy theory. This is done by proving an h-principle comparing algebraic sections to continuous ones. Using a bit of rational homotopy theory, one may then prove a homological stability result for moduli spaces of smooth hypersurfaces of increasing degree.
February 16, 2022:
Anna Marie Bohmann (Vanderbilt University)
Title: Free Loop Spaces and Topological coHochschild Homology
Abstract: Free loop spaces arise in many areas of geometry and topology. Simply put, the free loops on a space X is the space of maps from the circle into X. This is a main object of study in string topology and has important connections to geodesics on manifolds. In this talk, we discuss a new approach to computing the homology of free loop spaces via topological coHochschild homology, which is an invariant of coalgebras arising from homotopy theory techniques. This approach produces a spectral sequence for the homology of free loop spaces that has an algebraic structure allowing for new computations. This is joint work with Teena Gerhardt and Brooke Shipley.
February 9, 2022:
José Cantarero (CIMAT) video
Title: Configuration spaces of commuting elements
Abstract: The rational cohomology of the space of configurations of commuting elements in a compact Lie group is determined by the action of the Weyl group on the configuration space of its maximal torus. This fact can be used to determine (co)homology stability phenomena, and other unstable computations. In this talk I will begin with some motivation for the study of these spaces and the case of SU_2, where the homotopy type can be determined completely. Then I will describe the stability results mentioned previously and other interesting cohomology calculations. This is joint work with Angel R. Jimenez.
February 2, 2022:
Jeremy Miller (Purdue University)
Title: Rognes’ connectivity conjecture
Abstract:Rognes’ connectivity conjecture concerns the connectivity of a simplicial complex called the common basis complex. The equivariant homology of this complex is the E^1 page of a spectral sequence converging to the homology of K-theory spectra. I will describe joint work in progress with Patzt and Wilson where we prove the connectivity conjecture for fields. I will also explain a connection between the homology the common basis complex and the André–Quillen homology of a certain equivariant ring built out of Steinberg modules.
January 26, 2022:
Arun Debray (Purdue University)
Title: Constructing the Virasoro groups using differential cohomology
Abstract: The Virasoro groups are a family of central extensions of Diff+(S^1) by the circle group T. In this talk I will discuss recent work, joint with Yu Leon Liu and Christoph Weis, constructing these groups by beginning with a lift of the first Pontrjagin class to "off-diagonal" differential cohomology, then transgressing it to obtain a central extension. Along the way, I will discuss what the Virasoro extensions are and how to recognize them; a brief introduction to differential cohomology; and lifts of characteristic classes to differential cohomology.
January 19, 2022:
Sam Nariman (Purdue University)
Title: Bounded and unbounded cohomology of diffeomorphism and homeomorphism groups
Abstract: Bounded cohomology for groups and spaces was originally defined by Gromov in the 80's and it is intimately related to the geometric and dynamical properties of the groups. For example Ghys used the bounded Euler class to classify certain group actions on the circle up to (semi)conjugacy. However, unlike the group cohomology, it is notoriously difficult to
calculate bounded cohomology of groups. And in fact there is no countably generated group known for which we can completely calculate the bounded cohomology unless it is trivial in all positive degrees like the case of amenable groups. In this talk, I will talk about joint work with Nicolas Monod on the bounded cohomology of certain homeomorphisms and diffeomorphism groups. In particular we show that the bounded cohomology of Diff(S^1) and Diff(D^2) are polynomial rings generated by the Euler class. If time permits, I also discuss our solution to Ghys' question about generalizing Milnor-Wood inequality to flat $S^3$-bundles. In particular, we show that the Euler class for flat S^3-bundles is an unbounded class
January 12, 2022:
Marius Dadarlat (Purdue University) slide
Title: Topological obstructions to matrix stability of discrete groups
Abstract: A discrete countable group is matricially stable if its finite dimensional approximate unitary representations are perturbable to genuine representations in the point-norm topology. We aim to explain in accessible terms why matricial stability for a group G implies the vanishing of the rational even cohomology of G for large classes of groups, including all linear groups.
FALL 2021
December 15, 2021:
Title: Keller admissible triples and Duflo theorem
Abstract: The Hochschild cohomology of (associative) algebras can be generalized to dg algebras in two different ways. While the first kind of Hochschild cohomology of dg algebras admits a natural description in terms of a derived category of dg modules and is therefore preserved by quasi-isomorphisms of dg modules, the second kind of Hochschild cohomology is not. B. Keller proved that certain triples, which can be understood as a sort of `Morita equivalences’ of dg algebras, induce isomorphisms of Hochschild cohomology of the first kind. In this talk, we will define another class of triples, which we call `Keller admissible triples,’ that induce isomorphisms of Hochschild cohomology of the second kind. As an application, given a Lie algebra, we construct a Keller admissible triple and obtain an alternative proof of Duflo's theorem. This is a joint work with Hsuan-Yi Liao.
December 8, 2021:
Daniel Lopez Neumann (Indiana University, Bloomington) video
Title: Reshetikhin-Turaev invariants from twisted Drinfeld doubles and Reidemeister torsion
Abstract: As shown by Reshetikhin and Turaev in the 90’s, a braided monoidal category produces topological invariants of knots and tangles in the three-sphere, the Jones polynomial being the most famous example. If the categories admit a grading/action by a group G, then called G-braided categories, an extension of this construction due to Turaev produces invariants of knots together with a representation of the knot group into G. This setting is very natural, for instance invariants coming from covering spaces such as Reidemeister torsion depend on such additional structure. However, this extension is not so well-understood, in particular, it is not known how torsion fits into it (except for special cases).
This talk will be about a special case of Turaev’s construction, namely, the invariants of knots in the three-sphere obtained from the G-category of modules over a “twisted Drinfeld double" of a Hopf algebra. We will show that these invariants have some Fox calculus on it and that they specialize to the SL(n,C)-twisted Reidemeister torsion of the knot complement at an appropriate Hopf algebra. These invariants are the Reshetikhin-Turaev version of the twisted Kuperberg invariants introduced in arXiv:1911.02925. This is work in progress.
December 1, 2021:
Xiaolei Wu (Fudan University) video
Title: Homological stability for the ribbon Higman--Thompson groups
Abstract: I will start the talk with basics about Higman--Thompson groups and then introduce its braided version and ribbon version.I will build a geometric model for the ribbon Higman--Thompson groups, namely as a nice subgroup for the mapping class group of a disk minus a Cantor set. We use this model to prove that the ribbon Higman--Thompson groups satisfy homological stability. This can be treated as an extension of Szymik--Wahl's work on homological stability for the Higman--Thompson groups to the surface setting.This is a joint work with Rachel Skipper.
November 17, 2021:
Behrang Noohi (Queen Mary) slides
Title: Categorical calculus and representation theory
Abstract: Using basic category theory, one can rephrase concepts of representation theory of (discrete) groups in a geometric language, allowing one to import ideas from geometry to prove results in representation theory. For instance, a categorical analogue of Stokes' Theorem, which is almost tautological, gives rise to interesting non-trivial formulas in representation theory. I will give an overview of this method and discuss an application of it in my joint work with Matthew Young.
November 10, 2021:
Shun Wakatsuki (Shinshu University) video
Title: BV exactness and string brackets
Abstract: The negative cyclic homology for a differential graded algebra over the rational field has a quotient of the Hochschild homology as a direct summand if the S-action is trivial. With this fact, we can reduce the string bracket in the sense of Chas and Sullivan to the loop product followed by the BV operator on the loop homology provided the given manifold is BV exact. In this talk, we will explain BV exactness and its application together with examples and non-examples, which are found by a computer-assisted method. If time permits, the higher BV exactness is also discussed featuring the cobar-type Eilenberg-Moore spectral sequence.
This is a joint work with Katsuhiko Kuribayashi, Takahito Naito, and Toshihiro Yamaguchi.
November 3,2021:
George Raptis (University of Regensburg)
Title: Some homotopy-theoretic aspects of bounded cohomology
Abstract: I will discuss some recent results in connection with two well-known theorems about bounded cohomology: Gromov's Mapping Theorem and the Vanishing/Covering Theorems of Gromov and Ivanov. First, I will present some characterizations of bounded cohomology equivalences (with coefficients) in terms of properties of their homotopy fibers. Second, I will discuss a homotopy-theoretic approach to the study of the comparison map from bounded cohomology to singular cohomology. If time permits, I will also briefly discuss Gromov's intriguing question about the Euler characteristic of aspherical manifolds with vanishing simplicial volume.
October 27, 2021:
Georg Frenck (University of Augsburg) video
Title: Characteristic classes of manifold-bundles over spheres
Abstract: Manifold bundles with fiber M are classified by the classifying space of it diffeomorphism group Diff(M). Hence, the cohomology of this classifying space is the ring of characteristic classes for manifold bundles. In this talk I will describe which rational characteristic classes are attained by manifold bundles over spheres. As applications, this allows to investigate the question, which classes of the (rational) oriented cobordism ring contain a representative fibering over the sphere and one can deduce the existence of many fiber bundles where base and fiber both admit positive scalar, Ricci and even sectional curvature, whereas the total space does not.
October 20, 2021:
Ka Ho Wong (Texas A&M University) video
Title: Asymptotics of the relative Reshetikhin-Turaev invariants
Abstract: In a series of joint works with Tian Yang, we made a volume conjecture and an asymptotic expansion conjecture for the relative Reshetikhin-Turaev invariants for a closed oriented 3-manifold with a colored framed link inside it. We propose that their asymptotic behavior is related to the volume, the Chern-Simons invariant and the adjoint twisted Reidemeister torsion associated with the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring.
In this talk, I will first discuss how our volume conjecture can be understood as an interpolation between the Kashaev-Murakami-Murakami volume conjecture of the colored Jones polynomials and the Chen-Yang volume conjecture of the Reshetikhin-Turaev invariants. Then I will describe how the adjoint twisted Reidemeister torsion shows up in the asymptotic expansion of the invariants. Especially, we find new explicit formulas for the adjoint twisted Reidemeister torsion for the fundamental shadow link complements and for the 3-manifold obtained by doing hyperbolic Dehn-filling on those link complements. Those formulas cover a very large class of hyperbolic 3-manifold and appear naturally in the asymptotic expansion of quantum invariants. Finally, I will summarize the recent progress of the asymptotic expansion conjecture of the fundamental shadow link pairs.
October 13, 2021:
Kai Cieliebak (Ausburg university) video
Title: Poincare duality and bialgebra structures for loop spaces.
Abstract: This talk is about ongoing joint work with Nancy Hingston and Alexandru Oancea.
I will explain how various puzzles in string topology get resolved in terms of symplectic geometry: Loop space homology and cohomology are merged into a larger space, Rabinowitz Floer homology, which is an infinitesimal bialgebra in the sense of Joni-Rota and Aguiar and satisfies Poincare duality.
October 6, 2021:
Andreas Stavrou (University of Cambridge) video
Title: Cohomology of configuration spaces of surfaces as mapping class group representations
Abstract: Recently there has been intense study of the homology of configuration spaces of manifolds and many extra structures have been imposed on it to extract more information. We focus on the action of the mapping class group and will present a result for general manifolds. In the case of compact oriented surfaces with one boundary component, we will present the complete answer for the cohomology of its configuration spaces as mapping class group representations, motivate the answer pictorially and give an overview of the steps involved in the proof.
September 29, 2021:
Florian Kranhold (University of Bonn) video, slide
Title: Parametrised moduli spaces of surfaces as infinite loop spaces
Abstract: The Madsen–Weiss theorem states that the group completion of the
collection M of moduli spaces of Riemann surfaces with one
parametrised boundary component is equivalent to the infinite loop
space assigned to the affine Thom spectrum MTSO(2).
We address the following generalisation: for each space X we consider
the mapping space M^X, the space of surface bundles over X, and show
that its group completion splits as the product of the infinite loop
space assigned to MTSO(2) and a certain free infinite loop space.
The proof of this result combines two seemingly unrelated
ingredients: on the one hand, we show a structure theorem for
centralisers of mapping classes in mapping class groups of surfaces,
and on the other hand, we develop some operadic machinery which
enables us to understand group completions of relatively free
algebras over coloured operads with homological stability.
This is joint work with Andrea Bianchi (Copenhagen) and Jens Reinhold
(Münster).
September 22, 2021:
Ismael Sierra (University of Cambridge) video
Title: Homological stability of diffeomorphism groups and splitting complexes of manifolds
Abstract: The most substantial step in the proof of homological stability of diffeomorphism groups is to show that certain complexes, called splitting complexes, are highly connected. In this talk I will define these and give a detailed outline of the proof of their high connectivity, explaining the different techniques that appear along the way.
September 15, 2021:
Bruno Kahn (CNRS - IMJ-PRG) video
Title: A rank spectral sequence for algebraic K-theory
Abstract: I will describe a spectral sequence converging to the homology of Quillen’s Q-construction on the category of coherent sheaves on an integral scheme X, inspired by his proof of the finite generation of algebraic K-groups of rings of integers. The computation of its d1differentials involves the universal modular symbols of Ash-Rudolph, which are certain generators of the Steinberg modules associated to Tits buildings of GLn
of the function field of X.
Spring 2021
April 28, 2021:
David Reutter (Max Planck Institute for Mathematics)
Title: Semisimple topological quantum field theories and stable diffeomorphisms
Abstract: A major open problem in quantum topology is the construction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. More generally, how much manifold topology can a TQFT see?
In this talk, I will answer this question for `semisimple’ field theories in even dimensions — I will sketch a proof that such theories can at most see the stable diffeomorphism type of a manifold and conversely, that if two sufficiently finite manifolds are not stably diffeomorphic, then they can be distinguished by semisimple field theories. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of linear algebraic TQFTs, including `unitary field theories’, and `once-extended field theories' which assign algebras or linear categories to codimension 2 manifolds. I will discuss implications in dimension 4, such as the fact that oriented semisimple field theories cannot see smooth structure, while unoriented ones can. This is based on arXiv:2001.02288 and joint work with Christopher Schommer-Pries.
April 21,2021:
Kathryn Hess (EPFL)
Title: A comonadic machine for creating calculi
Abstract: Abstracting the framework common to most flavors of functor calculus, one can define a calculus on a category M equipped with a distinguished class of weak equivalences to be a functor that associates to each object x of M a tower of objects in M that are increasingly good approximations to x, in some well defined, Taylor-type sense. Such calculi could be applied, for example, to testing whether morphisms in M are weak equivalences.
In this talk, after making the definition above precise, I will describe a machine for creating calculi on functor categories Fun (C,M) that is natural in both the source C and the target M. Our calculi arise by comparison of the source category C with a tower of test categories, equipped with cubical structure of progressively higher dimension, giving rise to sequences of resolutions of functors from C to M, built from comonads derived from the cubical structure on the test categories. The stages of the towers of functors that we obtain measure how far the functor we are analyzing deviates from being a coalgebra over each of these comonads. The naturality of this construction makes it possible to compare both different types calculi on the same functor category, arising from different towers of test categories, and the same type of calculus on different functor categories, given by a fixed tower of test categories.
(Joint work with Brenda Johnson)
April 14, 2021:
Nils Prigge (ETH Zurich)
Title: Self-embedding calculus and automorphisms of manifolds
The space of diffeomorphisms of a closed manifold coincides with the space of self-embeddings and can thus be studied via the homotopy theoretic approximations from embedding calculus. This perspective has led to much recent progress in understanding the space of diffeomorphisms and it is believed that the approximation is quite close. In this talk, I will discuss this approach and how one might detect the difference between the approximation and the space of diffeomorphisms using classical invariants of fibre bundles.
April 7, 2021:
Behrooz Mirzaii (Universidade de São Paulo)
Title: Homology of general linear groups over local rings.
March 31, 2021:
Xiang Tang (Washington University, Saint Louis)
Title: Universal Transgressed Chern Character and Noncommutative geometry
Abstract: In this talk, we will explain the idea of studying secondary invariants for flat bundles using noncommutative geometry. This point of view gives leads to homotopy invariance results for certain characteristics numbers.
March 24, 2021:
Mikala Ørsnes Jansen (University of Copenhagen)
Title: The reductive Borel-Serre compactification and unstable algebraic K-theory
Abstract: The reductive Borel-Serre compactification, introduced by Zucker in 1982, is a stratified space which is well-suited for the study of L2-cohomology of arithmetic groups and which has come to play a central role in the theory of compactifications of locally symmetric spaces. We determine its stratified homotopy type (exit path ∞-category) to be a 1-category defined purely in terms of parabolic subgroups, their unipotent radicals and a conjugation action. This category makes sense in a much more general setting; in fact, we can associate such a category to any ring R and any integer n and interpret it as a "compactification" of GLn(R). We propose these categories as models for unstable algebraic K-theory and investigate them in detail for finite fields and commutative local rings with infinite residue field. This is joint work with Dustin Clausen.
March 17, 2021:
James Conant (Art of Problem Solving)
Title: On the rational cohomology of automorphism groups of free groups
Abstract: Ever since Kontsevich and Morita's seminal work on the rational cohomology of Out(F_n), the techniques of graph homology and infinite dimensional lie algebras have been essential in its study. The speaker, along with Karen Vogtmann, and Martin Kassabov, wrote many papers from this perspective, one outcome of which were several constructions of nice combinatorially defined cohomology classes. In recent work with Hatcher, the four of us have simplified the approach and completely eliminated references to graph homology of infinite dimensional Lie algebras, constructing all of the classes that came from the old approach directly through a geometric "assembly map." In this talk I will give a sketch about how this approach works.
March 10, 2021:
Camille Combe (Université de Paris, IRIF)
Title: Hochschild lattices: a geometric and combinatorial approach
Abstract: Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these lattices leads to some cell complexes introduced by Saneblidze, called the Hochschild polytopes. After recalling their constructions, we will see two precise results among those satisfied by these posets. Thus, we will see that these lattices are constructible by interval doubling, then we will present a way to enumerate their $k$-chains. We will end this talk by exposing new properties of these lattices and by raising open questions.
March 3, 2021:
Boris Tsygan (Northwestern)
Title: Operations on Hochschild and cyclic complexes.
Abstract: Hochschild and cyclic complexes are invariants of associative algebras that have geometric flavour. When the algebra is a commutative algebra of functions on a space such as a manifold, a variety, etc., these complexes recover geometric objects on the underlying space, such as De Rham complex or multi vector fields.
Algebraic structures on Hochschild and cyclic complexes, often generalizing classical structures on geometric objects to noncommutative case but sometimes new, had been extensively studied for the last forty years. Their applications include formality theorems for deformation quantization, generalized index theorems, string topology and its uses in symplectic topology, etc. In my talk I will review current developments in the subject and pose some questions.
February 24, 2021:
Claudio Jacobo Gonzales (UC Irvine)
Title: Topology of non-degenerate functions of projective spaces
Abstract: The topology study of holomorphic mapping spaces as finite-dimensional approximations of continuous mapping spaces has developed immensely since Segal's seminal paper in 1979–and has been enriched by concurrent developments in both number theory and motivic theory. I will present a family of subspaces indexed by degeneracy, along with a new method for studying spaces of maps into P^n in terms of these varieties, and detail some of the unexpected phenomena that have arisen thus far. We will review various topological results that are recovered or extended into new contexts using these new tools, especially cases in which the domain has complex dimension greater than 1. The talk will also mention ongoing work which uses topology to predict arithmetic and vice versa, together with a brief overview of the dictionaries between these contexts.
February 17, 2021:
Benjamin Brück (ETH Zürich)
Title: Between Tits buildings and free factor complexes
Abstract: Much of the modern treatment of automorphism groups of free groups is motivated by analogies with arithmetic groups – this is especially true if it comes to homological questions. I will present a new family of complexes interpolating between two well-studied objects associated to these classes of groups: the free factor complex and the Tits building of GL_n(Q). Each of the new complexes is associated to the automorphism group Aut(A_Γ) of a right-angled Artin group and has the homotopy type of a wedge of spheres. The dimension of these spheres forms a new invariant associated to Aut(A_Γ). These complexes can also be seen as an Aut(A_Γ)-analogue of the curve complex.
February 10, 2021:
Nicholas Wawrykow (University of Michigan)
Title: Secondary Representation Stability and the Ordered Configuration Space of the Once-Punctured Torus
Abstract: In this talk we discuss a notion of secondary representation stability introduced by Miller and Wilson. They proved that there was a stability pattern in the homology of the ordered configuration space of noncompact manifolds in a range beyond the traditional representation stability range of Church, Ellenberg, and Farb. We discuss their result, and describe an example of secondary representation stability, namely the k-th homology of the ordered configuration space of 2k-2 points on the once-punctured torus, the first known example where the FIM^+ structure is neither free nor stably zero.
February 3, 2021:
Lei Chen (Caltech)
Title: A new proof of Markovic’s result of the nonrealizability of mapping class group as homeomorphisms
Abstract: Nielsen realization problem asks which subgroups of the mapping class group can be realized as homeomorphisms/diffeomorphisms. In this talk, I will describe a new way to prove Thurston’s Conjecture that the mapping class group itself cannot be realized as homeomorphisms. The strategy uses a translation of this problem to a new problem of finding global fixed-points. Potentially this translation can provide a new way to study Nielsen realization problem for other subgroups. This is a joint work with Nick Salter.
January 27, 2021:
Omar Antolín Camarena (UNAM)
Title: Higher generation by abelian subgroups in Lie groups
Abstract: The poset of cosets of Abelian subgroups of a discrete group is simply-connected if and only if the group is Abelian. I'll explain this result of Cihan Okay's and talk about an analogue for compact Lie groups. Alejandro Adem, Fred Cohen and Enrique Torres Giese asociated to any topological group G a space E(2,G) which plays the role of the abelian subgroup coset poset. Simon Gritschacher and Bernardo Villarreal and I proved that a compact Lie group G is Abelian if and only if πᵢ(E(2,G))=0 for i=1,2,4.
January 20, 2021:
Jan Steinebrunner (Oxford University)
Title: The one-dimensional bordism category
Abstract: The topologically enriched bordism category Bord_d has as objects closed oriented (d-1)-manifolds and as morphism spaces the moduli spaces of oriented d-bordisms. The classifying space B(Bord_d) was computed by Galatius-Madsen-Tillmann-Weiss, and has been used to great success in the study of moduli spaces.
In this talk, after recalling Bord_d, I will focus on its much simpler predecessor: the homotopy category h(Bord_d) where any two diffeomorphic bordisms are identified. Surprisingly little is known about the homotopy type of h(Bord_d). I will explain how to compute the classifying space of h(Bord_1) in terms of CP^\infty_{-1} = MTSO_2. The proof makes use of a new 'reduced' bordism category Bord_1^{red} where all circles are deleted.
As a result of the computation we will see that B(h Bord_1) carries a lot of interesting information. To better understand where this is coming from, I will also show how to construct cocycles for an infinite family of non-trivial cohomology classes kappa_i on h(Bord_1). If time permits I will use this to show that a large subcategory of h(Bord_2) is highly non-trivial..
January 13, 2021:
Maru Sarazola (Cornell University)
Title: Cotorsion pairs and a K-theory localization theorem
Abstract: Cotorsion pairs were introduced in the ’70s as a generalization of projective and injective objects in an abelian category, and were mainly used in the context of representation theory. In 2002, Hovey showed a remarkable correspondence between compatible cotorsion pairs on an abelian category A and abelian model structures one can define on A. These include, for example, the projective and injective model structures on chain complexes.
In this talk, we turn our attention to Waldhausen categories, and explain how cotorsion pairs can be used to construct Waldhausen structures on an exact category, with the usual class of admissible monomorphisms as cofibrations, and some freedom to choose the class of desired acyclic objects. This allows us to prove a new version of Quillen’s localization theorem, relating the K-theory of exact categories A ⊆ B to that of a cofiber, constructed through a cotorsion pair.
FALL 2020
December 16, 2020:
Ozgur Bayindir (Paris 13)
Title: Algebraic K-theory of THH(Fp)
Abstract: In this work, we study THH(Fp) from various perspectives. We start with a new identification of THH(Fp) as an E_2-algebra. Following this, we compute the K-theory of THH(Fp). The first part of my talk is going to consist of an introduction to ring spectra, algebraic K-theory and the Nikolaus Scholze approach to trace methods. In the second part, I will introduce our results and the tools we develop to study the topological Hochschild homology of graded ring spectra. This is a joint work with Tasos Moulinos.
December 9, 2020:
Ryan Budney (University of Victoria)
Title: Isotopy in dimension 4
Abstract: I will discuss an (n-3)-parameter family of diffeomorphisms of S^1 x D^n coming from the high-dimensional analogue of a "crossing change". We sketch a geometric description of how these diffeomorphisms act on the "reducing disc" {1}xD^n, and why it is non-trivial. The techniques we use are relatively simple transversality arguments that could be thought of as encoding the rational homotopy of the Taylor tower for various embedding spaces. The discussion will end with some applications: "almost a counterexample" to the smooth 4-dimensional Schoenflies problem in dimension 4, and some basic information about the component of the trivial knot in the space of embeddings of S^2 into S^4.
December 2, 2020:
Ben Williams (University of British Colombia)
Title: A1 homotopy groups of GL_n and a problem of Suslin's
Abstract: Let F be an infinite field. Andrei Suslin constructed a morphism from the (Quillen) K-theory of F to the Milnor K-theory of F: s_n : K_n(F) -> K_n^M(F). He proved that the image of s_n contains (n-1)! K_n^M(F). He raised the question of whether this accounted for the whole image—it was known to when n is 1,2 or 3. In this talk I will explain how one can partially recover this morphism as a morphism of A1-homotopy groups of down-to-earth objects, and I will show how this tells us some things about Suslin's question when n is 4 and settles it when n is 5. This talk represents joint work with Aravind Asok and Jean Fasel.
November 18, 2020:
Rita Jiménez Rolland (Instituto de Matemáticas, UNAM)
Title: Powers of the Euler class for pure mapping class groups
Abstract: The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this talk, we describe some partial results, in ongoing work with Solomon Jekel, on the vanishing and non-vanishing behaviour of the powers of this class.
November 11, 2020:
Jens Reinhold (WWU Münster)
Title: Bundles with non-multplicative Â-genus and spaces of metrics.
Abstract: For a smooth manifold M, the space of all Riemannian metrics on M is convex and hence contractible. Imposing curvature bounds, however, often results in subspaces with an interesting topology. Whereas there are many results describing the topology of this space for positive scalar curvature, little is known about the case of positive or nonnegative Ricci or sectional curvature. In this talk I will first explain how to construct bundles with base and fiber products of spheres whose total spaces have non-vanishing A^-genus. We will then make use of these bundles to detect nontrivial rational homology groups of the aforementioned spaces in many cases. This is joint work with Georg Frenck.
November 4, 2020:
Ben Knudsen (Northeastern)
Title: Stable and unstable homology of graph braid groups
Abstract: The homology of the configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. The degree of this polynomial is captured by a connectivity invariant of the graph, and its leading coefficient may be computed explicitly in terms of cut counts and vertex valencies. This "stable" (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin, but exotic non-toric classes abound unstably. These mysterious classes are intimately tied to questions about generation and torsion whose answers remain elusive except in a few special cases. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole.
October 28, 2020:
Quoc P Ho (Institute of Science and Technology of Austria)
Title: Factorization homology and the arithmetics and topology of configuration spaces
Abstract: The last decade has witnessed many interesting interplays between homological/representation stability phenomena and questions in arithmetic statistics. In this talk, I will show how the algebro-geometric version of factorization homology provides a unifying framework for studying these phenomena in the case of configuration spaces. In particular, I will explain the relationship between various zeta values coming out of point-counts on configuration spaces and homological stability phenomena exhibited by these spaces, answering questions of Farb--Wolfson--Wood. Time permitting, I will explain how these ideas can be further developed to study representation stability for ordered configuration spaces.
October 21, 2020:
Martin Palmer (Mathematical Institute of the Romanian Academy)
Title: Homology of configuration-section spaces
Abstract: For a manifold M equipped with a bundle E, the configuration-section spaces on (M,E) consist of configurations of points on M together with a section of E on the complement of the configuration. These may be thought of as moduli spaces of "fields" with singularities. One often considers subspaces where the behaviour of the field (section) is constrained in a neighbourhood of the singularities (particles), which may be thought of as restricting the allowed "charges" of the particles. As well as the evident physical interpretation, these spaces also include as examples the classical Hurwitz spaces, through which they have connections with number theory. In particular, Ellenberg, Venkatesh and Westerland [EVW] have recently proven an asymptotic version of the Cohen-Lenstra conjecture for function fields via a certain homological stability result for Hurwitz spaces.
I will talk about another homological stability result for configuration-section spaces, which is in a sense both more and less general than that of [EVW]. It is more general in the sense that it holds for any bundle over any connected, open manifold M (theirs is for certain trivial bundles over the 2-disc), but it is also less general in the sense that we assume a stronger condition on the allowed "charges" of the particles. Along the way I will also discuss the action of the braid group of a manifold M on the fibres of configuration-mapping spaces on M.
This represents joint work with Ulrike Tillmann and is based on the arxiv preprints 2007.11607 and 2007.11613.
October 14, 2020:
Chris Kapulkin (UWO)
Title: Cubical models of (∞,1)-categories
Abstract: I will report on the joint work with B. Doherty, Z. Lindsey, and C. Sattler, establishing a family of new models of (∞,1)-categories in different categories of (marked) cubical sets.
October 7, 2020:
Peter Smillie (Caltech)
Title: The borders of outer space
Abstract: The group Out(F_n) acts properly on a contractible space known as outer space. Motivated by the Borel-Serre bordification of symmetric spaces, Bestvina and Feighn gave a bordification of outer space and used it to prove that Out(F_n) is a virtual duality group. I will define outer space, and show how to realize the Bestvina-Feighn bordification as a deformation retract instead of an enlargement. This leads to a new proof that Out(F_n) is a virtual duality group and gives an explicit polyhedral structure on the boundary of outer space.
September 30, 2020:
Julian Holstein (University of Hamburg)
Title: Categorical Koszul Duality
Abstract: The algebraic analogue of the loop space construction of topological spaces is Adams’ cobar construction.
Together with the bar construction it induces a Koszul duality between algebras and coalgebras,
providing an equivalence of suitable homotopy theories of augmented differential graded algebras and differential graded conilpotent coalgebras.
Interesting things happen as one generalises this result, in particular dropping the augmentation on the dg algebra side corresponds to introducing a curvature term on the coalgebra side.
I will talk about joint work with Andrey Lazarev, in which we generalise this to a categorical Koszul duality and find a category of coalgebras Quillen equivalent to differential graded categories. I will show that this construction is closely related to the coherent nerve construction from simplicial categories to quasicategories.
September 23, 2020:
Pavel Safronov (University of Edinburgh)
Title: Coproduct in string topology, Euler structures and topological field theories.
Abstract: Chas and Sullivan have introduced interesting algebraic operations on the homology of the free loop space of a manifold which go under the name of the string topology operations. Cohen—Gaudin gave a TFT interpretation of the string product. Moreover, Cohen—Klein—Sullivan have shown that the string product is homotopy-invariant. In this talk I will explain a TFT interpretation of the string coproduct by disassembling it into elementary pieces. In particular, I will explain a conjecture that the string coproduct is not homotopy-invariant and changes by the Whitehead torsion. This is a report on work in progress joint with Florian Naef.
September 9, 2020:
Andrea Bianchi (Bonn)
Title: Hurwitz spaces and Moduli spaces of Riemann surfaces
Abstract: Let S_d be a symmetric group, with d>=2. For k>=0, the classical Hurwitz space hur(k,S_d) parametrises d-fold branched covers of the complex plane C with precisely k branch points. We will consider a construction that amalgamates all Hurwitz spaces for all values of k into a single space Hur(S_d).The motivation for the construction is the following. For all g>=0 andn>=1, let M_{g,n} denote the moduli space of Riemann surfaces of genus gwith n boundary components. If d is large enough (with respect to g and n), then there exists a connected component of Hur(S_d) which is homotopy equivalent to M_{g,n}.The space Hur(S_d) carries a natural structure of topological monoid graded by natural numbers h>=0. This topological monoid is not homotopy associative, nevertheless the stable homology of its components agrees with the homology of the double loop space of a certain space B(d). The result is very explicit rationally and in degrees up to d-2. Letting d go to infinity, one can in particular recover the Mumford conjecture on the stable, rational cohomology of moduli spaces of Riemann surfaces, originally proved by Madsen and Weiss.
September 2, 2020:
Ezra Getzler (Northwestern)
Title: The infinity-groupoid of an L-infinity algebra in the cubical formalism
Abstract: Sullivan associates to the dg commutative algebra of Chevalley-Eilenberg cochains C*(g) on a nilpotent (or pro-nilpotent) differential graded Lie algebra g a Kan complex <C*(g)>, using the differential graded commutative algebras Ω(Δn) of polynomial-coefficient differential forms on the simplex. If g is a Lie algebra, with Lie group G, this Kan complex is not isomorphic to BG. In 2004, using Dupont's explicit simplicial homotopy for the de Rham theorem, I showed that <C*(g)> has a natural simplicial subset γ(g) with the following properties:
1) γ(g) is a Kan complex (in fact, the functor takes fibrations to fibrations, and trivial fibrations to trivial fibrations);
2) if g vanishes in degree -k and below, γ(g) is a k-groupoid in the sense of Duskin;
3) the inclusion of γ(g) in <C*(g)> is a homotopy equivalence;
4) if g is a nilpotent Lie algebra, γ(g) is naturally isomorphic to BG;
5) if g vanishes in negative degree, γ(g) is the nerve of the Deligne groupoid of g.
In fact, γ(g) is really a derived stack, but I will focus on the underlying simplicial set, since it exhibits all of the essential ideas of the construction.
In this talk, I give a new approach to γ(g), using differential forms on the cube. The explicit homotopy for the de Rham theorem is much easier to construct for cubes: the main new result is that this homotopy is not just cubical in the sense of Serre, but also in the sense of Brown and Higgins. This is an important refinement, since the analogue of Moore's theorem that a simplicial group is a Kan complex need the enrichment of Brown and Higgins (what they call connections) in order to hold, by the work of Tonks.
Replacing the cube by the cubical complex Qn, associated with straightening/unstraightening over a point, we obtain a new construction of a functor from L-infinity algebras to Kan complexes with the same properties as γ(g).
SUMMER 2020
July 22, 2020:
Anibal Medina (EPFL)
Title: A finitely presented E-infinity prop
Abstract: The Comm operad in chain complexes admits a presentation in terms of finitely many generators and relations, but no such presentation can be given for a sigma-free resolution of it. By passing to the more general setting of props, we are able to describe finitely presented E-infinity props in the categories of chain complexes and of cellular spaces. We relate the operads associated with these to the E-infinity operad models introduced by McClure-Smith, Berger-Fresse and Kaufmann, and describe novel actions on simplicial and cubical sets complementing these authors' work.
April 1:
Andrey Lazarev (Lancaster University)
Title: Differential graded Koszul duality: a global approach
Abstract: Koszul duality is a phenomenon that shows up in rational homotopy theory, deformation theory and other subfields of algebra and topology. It exists on various levels: as a correspondence between operads and cooperads, algebras over operads and coalgebras over cooperads and modules and comodules over associative algebras (the latter, simplest, version will be relevant to this talk). Usually some kind of conilpotence is assumed on the `co' side. In this talk I explain what happens if this condition is dropped; the consequences turn out to be quite dramatic. I will show how this non-conilpotent (or global) version of Koszul duality comes up naturally in the study of derived categories of complex algebraic manifolds and infinity local systems on topological spaces. Time permitting, I will also explain how one can construct a global version of deformation theory for certain deformation problems based on this approach. This is joint work with Ai Guan.
April 8:
Calista Bernard (Stanford University)
Title: Twisted Dyer-Lashof Operations
Abstract: In the 70s, Fred Cohen and Peter May gave a description of the mod p homology of a free E_n algebra in terms of certain homology operations, known as Dyer-Lashof operations, and the Browder bracket. These operations capture the failure of the E_n multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and explain computational results that show the existence of additional operations in the twisted case.
April 15:
Yining Zhang (University of Colorado at Boulder)
Title: Hodge decomposition of string topology
Abstract: Given a closed oriented manifold X, Chas and Sullivan constructed a Lie bracket on the (reduced) S^1-equivariant homology of the free loop space of X, called the string bracket. Furthermore, if the manifold is simply connected, there is a Hodge type decomposition on the S^1-equivariant homology induced by the n-fold coverings of the circle. It is natural to ask whether the string bracket preserve this decomposition. We provide a positive answer under the additional assumption that X is rationally elliptic. Our argument is based on algebraic models of string topology and analogous operations on Hochschild and cyclic homologies. This is a joint work with Yuri Berest and Ajay Ramadoss.
April 22:
Nir Gadish (MIT)
Title: The “generating function” of configuration spaces, as a source for explicit formulas and representation stability
Abstract: As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. In this talk I will apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.
April 29:
Sam Nariman (Copenhagen University)
Title: Diffeomorphisms of reducible three manifolds and bordisms of group actions on torus
Abstract: I first talk about the joint work with K. Mann on certain rigidity results on group actions on torus. In particular, we show that if the torus action on itself extends to a C^0 action on a three manifold M that bounds the torus, then M is homeomorphic to the solid torus.
Motivated by this result, I will talk about certain finiteness results about classifying space of reducible three manifolds which is related to an unfinished project by Allen Hatcher.
May 1:
(Friday)
Søren Galatius (Copenhagen University)
Title: E_\infty algebras and general linear groups of infinite fields
Abstract: Homology of general linear groups of a field F is calculated by the chain complex R(n) = C_*(BGL_n(F)). There are chain maps from the tensor product of R(n) and R(m) to R(n+m), induced by direct sum of F-vector spaces. These products make R into a bigraded differential graded algebra, which is homotopy commutative, and can moreover be given the structure of an E_\infty algebra. In joint work with Alexander Kupers and Oscar Randal-Williams, we study the structure of this E_\infty algebra when F is infinite. We are led to new results about homology of general linear groups, for instance an extension of a relationship between the relative homology of BGL_n relative to BGL_{n-1} and Milnor K-theory, due to Nesterenko and Suslin, as well as new questions.
May 5:
(Tuesday)
Markus Land (Copenhagen University)
Title: Hermitian K-theory
Abstract: I will start by introducing the Grothendieck-Witt group associated to a ring and discuss some examples. I will show how it relates to the algebraic K-group and the Witt group. I will then explain that this relation can be refined to fibre sequence of spaces relating the Grothendieck-Witt space to algebraic K-theory and Ranicki’s algebraic L-theory. Finally, I will indicate that the Grothendieck-Witt space is closely related to an algebraic version of the geometric cobordism category, and highlight some similarities between the geometric and algebraic world. All of this is joint work with Calmès, Dotto, Harpaz, Hebstreit, Moi, Nardin, Nikolaus and Steimle.
May 6:
Camilo Arias Abad (Universidad Nacional de Colombia- Medellín)
Title: Singular chains on Lie groups, the Cartan relations and Chern-Weil theory
Abstract: Let G be a simply connected Lie group. The space C(G) of smooth singular chains on G has de structure of a differential graded algebra, with product induced by the Eilenberg-Zilber map. We will describe how the category of (sufficiently smooth) modules over this algebra can be described infinitesimally. More precisely, we will show that the category of modules over C(G) is equivalent to the category of represenations of the differential graded Lie algebra Tg, which is universal for the Cartan relations. This extends the usual correspondence between representations of G and representations of the Lie algebra g. We also prove that in case G is compact, this equivalence of categories can be extended to an A_∞-equivalence of dg-Categories. The main ingredients in the proof are Gugenheim’s A_∞-version of de Rham’s theorem, the non- commutative Weil algebra of Alekseev-Meinrenken, and the Van-Est map. If time permits, I will describe how this construction relates to a version of Chern-Weil theory for ∞-local systems on classifying spaces. This talk is based on joint work with Alexander Quintero.
May 13:
Jeremy Miller (Purdue University)
Joint with the CUNY Topology, Geometry, and Physics seminar
Title: André–Quillen homology and homological stability for general linear groups of Euclidean domains
Abstract: I will discuss joint work in progress with Kupers and Patzt on improved stable ranges for homological stability for general linear groups of Euclidean domains. The main ingredient is a vanishing result for E_infinity André–Quillen homology. By work of Galatius--Kupers--Randal-Williams, these André--Quillen homology groups are isomorphic to the equivariant homology of certain posets.
May 20:
Edouard Balzin (Centre de Mathématiques Laurent Schwartz, École Polytechnique, France)
Title: Operation-indexing categories and Grothendieck fibrations
Abstract: It has been known since Segal that various small categories can be used as blueprints for algebraic structures in homotopy theory, providing alternatives to operads in such questions as for example delooping. The examples of those categories include finite sets, ordered sets, n- ordinals of Batanin and various exit path categories of configuration spaces, as well as categories of operators of general topological operads.
We would like to offer a definition for such operation-indexing categories, called weak operads or algebraic patterns, and how to describe homotopy-algebraic structures over such things via fibrations of model and higher categories. Depending on time and the interest of the audience, we may attempt to introduce the notion of a weak approximation as a means of establishing certain “Morita”-type equivalences in the world of weak operads.
June 3:
Manuel Krannich (University of Cambridge, UK)
Title: Pseudoisotopies of discs and algebraic K-theory of the integers
Abstract: There is an intimate connection between algebraic K-theory and the space of pseudo-isotopies P(M) of a compact d-manifold M (that is, diffeomorphisms of a cylinder M x I that are the identity on M x 0 and ∂M x I). Classically, the pseudo-isotopy space P(M) is studied in two steps: there is a stabilisation map P(M)-> P(M x I) which is approximately d/3-connected by a result of Igusa, and the colimit has a description in terms of Waldhausen's algebraic K-theory of spaces due to Waldhausen--Jahren--Rognes' stable parametrised h-cobordism theorem. In this talk, I will focus on the case of an even-dimensional disc and explain a new method to relate its space of pseudo-isotopies to the algebraic K-theory of the integers in a range up to roughly the dimension. This approach is independent of the usual route, does not involve stabilising the dimension, and is homological in nature.
June 10:
Mark McConnell (Princeton University)
Title: Computing Hecke Operators for Cohomology of Arithmetic Groups
Abstract: I will describe three projects. The first, which is joint with Avner Ash and Paul Gunnells, concerns arithmetic subgroups Gamma of G = SL(4,R). We compute the cohomology of the locally symmetric spaces Gamma\G/K, focusing on the cuspidal degree H^5. We compute a range of Hecke operators on this cohomology. We find Galois representations that appear to be attached to the Hecke eigenclasses, based on the operators we have computed. We have done this for both non-torsion and torsion classes. The method is to use a cell complex, the well-rounded retract, due to Ash. The second project, joint with Bob MacPherson, is an algorithm for computing the Hecke operators on the cohomology H^d of Gamma in SL(n,R) for all n and all d. For this we introduce a new retract, the well-tempered complex. In the third project, joint with Dylan Galt, we give an algorithm for Hecke operators on H^d of Gamma in Sp(4,R) for all d. The method is to define an appropriate subcomplex of the well-tempered complex for SL(4,R).
June 17:
Manuel Rivera (Purdue)
Title: The fundamental group and homotopy theory over a field
Abstract: I will explain how the fundamental group of a space, as well as its homology groups with coefficients in all possible local systems of vector spaces, are completely determined by chain level algebraic data. This algebraic data may be packaged as the weak equivalence class of the simplicial cocommutative coalgebra of chains under a notion of weak equivalence which involves adjusting the coalgebra structure through the cobar construction and is stronger than quasi-isomorphism. I will outline a theory, of independent interest, of twisting cochains and twisted tensor products at the level of simplicial coalgebras and algebras, which we used to construct the fundamental bialgebra and the universal cover of an abstract connected simplicial cocommutative coalgebra. This opens up the possibility of using the power of abstract algebraic homotopy theory to study geometric spaces with arbitrary fundamental group.