Topology at IU

Seminars During Academic Year 2023-24

Fall 2023

Monday August 28, 2023 in SE 010: William Balderrama (Univ. of Virginia), "The equivariant J-homomorphism and RO(G)-graded periodic phenomena"; I will describe how the G-equivariant J-homomorphism can be "desuspended" in a way that produces nontrivial RO(G)-graded periodicities in equivariant stable homotopy theory, such as in the G-equivariant stable stems. When G = C_2, these are essentially versions of James periodicity, and I will explain how this recovers and unifies theorems of Bredon, Araki and Iriye, and Behrens and Shah.
Wednesday September 13, 2023 in SE 140: Manuel Rivera (Purdue), "Coalgebras, loop spaces, and homotopy theory"; A central problem in algebraic topology is to determine how much information about topological spaces considered up to a specified notion of equivalence is preserved by a particular functorial invariant. In this talk, I will discuss a version of this problem for spaces up to “$\pi_1$-R-equivalence” by means of R-coalgebras, for a fixed commutative ring R. A continuous map of spaces is called a $\pi_1$-R-equivalence if it induces an isomorphism on fundamental groups and an R-homology equivalence between universal covers. I will describe a corresponding notion of weak equivalence for simplicial R-coalgebras using ideas from differential graded Koszul duality. When R is an algebraically closed field, the R-chains functor embeds fully and faithfully the homotopy theory of spaces up to $\pi_1$-R-equivalence in the corresponding homotopy theory of simplicial R-coalgebras. One of the key ingredients is the new (somewhat surprising) observation that the coalgebra of chains on a space, appropriately considered, completely determines the fundamental group (and much more). This is based on a series of joint projects with F. Wierstra, M. Zeinalian, and G. Raptis.
Wednesday September 20, 2023 in SE 140: Ningchuan Zhang (IU), "Equivariant algebraic K-theory and Weil L-functions of finite fields"; In 1972, Quillen computed higher algebraic K-groups of finite fields. The orders of these groups are very closely related to their Weil L-functions. In this talk, I will generalize this connection to Weil L-functions of abelian characters of Galois groups of finite fields. On the homotopy theory side, we compute Galois equivariant algebraic K-groups of finite fields with coefficients in those characters. If time allows, I will explain how this generalization works for function fields and number fields. This is joint work in progress with Elden Elmanto.
Wednesday September 27, 2023 in SE 140: Michael Mandell (IU), "Factorization Homology and HHR Norms"; This talk is nominally about a project with Blumberg and Hill to generalize the Hill-Hopkins-Ravenel norm to compact Lie groups, but it will mostly consist of an introduction to factorization homology and some of its properties as background, motivation, and introduction to the HHR norm problem.
Wednesday October 4, 2023 in SE 140: Yun Liu (IU), "Relative Join Construction and Towers of Borel Fibrations"; In a recent work, Yu. Berest and A. C. Ramadoss formulated and studied the realization problem for rings of quasi-invariants of finite reflection groups in terms of classifying spaces of compact Lie groups The main tool used in their work is the fiber-cofiber construction introduced in topology by T. Ganea.
In the first part of this talk, which is based on joint work with Yu. Berest and A. C. Ramadoss, we will describe a natural generalization of the fiber-cofiber construction in the context of abstract homotopy theory, which we call relative join construction. Then, in the second part, we will show how to apply this relative join construction to obtain analogues of spaces of quasi-invariants which can be identified with homotopy quotient of certain $G$-spaces for a compact Lie group $G$. The rational cohomology rings of these generalized spaces of quasi-invariants have nice algebraic properties that are similar to those of classical quasi-invariants. Moreover, the equivariant K-theory of those $G$-spaces exhibits similar properties.
Wednesday October 11, 2023 in SE 140: Dylan Thurston (IU), “Floer Homology beyond Borders” -- postponed
Wednesday October 18, 2023 in SE 140: Daniel Tolosa (Purdue), "An algebraic model for the free loop space as an S^1-space"; The free loop space of a topological space has a canonical circle action given by rotating loops, making it an S^1-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. One can model 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 (non-simply connected) 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 S^1-equivariant chains on the free loop space that is suitable for computations. The proofs of these results can be understood in terms of a combinatorial model for the unit of the Bar-Cobar adjunction.
Wednesday October 25, 2023 in SE 140: Ishan Levy (MIT), “Telescopic stable homotopy theory”; Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
Wednesday November 1, 2023 in SE 140: Hana Kong (Harvard), “A deformation of Borel equivariant homotopy”; The real motivic stable homotopy category has a close connection to the $C_2$-equivariant stable homotopy category. From a computational perspective, the real motivic computation can be viewed as a simpler version which “removes the negative cone” in the $C_2$-equivariant stable homotopy groups. On the other hand, by work of Burklund–Hahn–Senger, one can build the completed Artin–Tate real motivic category from the completed $C_2$-equivariant category using the deformation construction associated to the $C_2$-effective filtration.
In work with Gabriel Angelini-Knoll, Mark Behrens, and Eva Belmont, we try to build an analog of this deformation story for a general finite group $G$. We give a new interpretation of the $C_2$-effective filtration in the Borel equivariant category which generalizes for $G$. Using this new interpretation, the deformation construction gives a deformation of the Borel equivariant stable homotopy category for general finite groups.
Wednesday November 8, 2023 in SE 140: David Mehrle (UKY), “Hochschild homology of polynomial Tambara functors for cyclic p-groups”; Hochschild homology is a ubiquitous invariant that carries both geometric and algebraic information. One of the first calculations in Hochschild homology is the Hochschild homology of $k[x]$ as a $k$-algebra. Inspired by recent progress in equivariant trace methods, we carry out an equivariant version of this calculation, where the polynomial ring $k[x]$ is replaced by a polynomial Tambara functor for a cyclic $p$-group. The key tool in this calculation is the Koszul resolution of a polynomial Tambara functor, which becomes a multicomplex in the equivariant case. This is joint work with J.D. Quigley and Michael Stahlhauer.
Wednesday November 15, 2023 in SE 140: Dylan Thurston (IU), “Floer Homology Beyond Borders”; Heegaard Floer Homology is an invariant of 3-manifolds and 4-manifolds, providing striking topological information. Bordered Floer Homology extends that theory to apply to 3-manifolds with boundary, allowing computation for 3-manifolds from elementary pieces, developed by Robert Lipshitz, Peter Ozsvath, and the speaker.
In this talk we will survey the construction, structure, applications, and future directions of the subject.
Wednesday November 29, 2023 in SE 140: Eva Belmont (Case Western), “Equivariant homological stability for unordered configuration spaces”; In foundational work, McDuff and Segal proved "homological stability" in the setting of unordered configuration spaces $C_n(M)$ of $n$ points in an open manifold $M$; that is, $H_d(C_n(M)) \cong H_d(C_{n+1}(M))$ for $n\gg d$. In joint work with J.D. Quigley and Chase Vogeli, we prove an equivariant generalization of McDuff and Segal's result which applies to Bredon homology of unordered configuration spaces on G-manifolds for a finite abelian group G.
Wednesday December 6, 2023 in SE 140: Hari Rau-Murthy (Notre Dame), “An HKR theorem for factorization homology”; This talk will be on a generalization of the so-called “HKR theorem”. The classical HKR theorem concerns two different ways of generalizing the notion of differential forms on a ring – the theorem asserts that they agree. I will give a generalization of this theorem for ring spectra which will allow us to recalculate the factorization homology of rational KU. A written version of this work is available on arXiv 2305.00658.

Spring 2024

Wednesday January 17, 2024 in SE 140: Tanushree Shah (Vienna), “Tight contact structures on Seifert fibered 3-manifolds”; I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory, which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.
Wednesday January 24, 2024 in SE 140: Daniel Lopez Neumann (IU), “Genus bounds from quantum groups at roots of unity”; Quantum invariants of knots and 3-manifolds form a large class of topological invariants built from the theory of quantum groups, and more generally, Hopf algebras and tensor categories. A big open problem in the field is to understand the geometric content of such invariants. This is at most the subject of several conjectures for the Jones polynomial, which is obtained through quantum SL2 at a generic parameter q. In this talk I’ll explain that, if the parameter q of a quantum group is specialized to a root of unity, a different family of link polynomials arises and the degrees of these give lower bounds to the Seifert genus of the link. This is joint work with Roland van der Veen.
Wednesday January 31, 2024 in SE 140: Danika Van Niel (MSU), “Computational approaches to twisted topological Hochschild homology”; Topological Hochschild homology (THH) is an invariant of ring spectra and is a key component of the trace method approach to algebraic K-theory. One of the main computational tools for THH is the Bökstedt spectral sequence. The study of the algebraic structure of THH and the Bökstedt spectral sequence have advanced our computational ability. In recent years, a generalization of THH for equivariant ring spectra called twisted THH has been developed along with an equivariant version of the Bökstedt spectral sequence. In this talk I’ll introduce THH, twisted THH, and discuss computations of twisted THH using the equivariant Bökstedt spectral sequence as well as work on the algebraic structure of twisted THH. I will also talk about Mackey functor fields, which are an equivariant analog of classical fields, and how they can be used in computations of twisted THH.
Wednesday February 7, 2024 in SE 140: Dror Bar-Natan (Univ. of Toronto), "Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants"
Wednesday February 14, 2024 in SE 140: Jim Davis (IU), “The Borel Conjecture and aspherical 4-manifolds”; The Borel Conjecture states that the homeomorphism type of a closed aspherical manifold is determined by its fundamental group.
Freedman and Quinn proved that the disc embedding conjecture (and hence surgery and the s-cobordism theorem) holds for 4-manifolds with EA (= elementary amenable) fundamental group.
In joint work with Jonathan Hillman, we formulate versions (uniqueness and existence) of the Borel Conjecture for compact aspherical manifolds with boundary, and prove that they hold for 4-manifolds with EA fundamental group. Furthermore, we characterize all EA groups which are the fundamental group of a compact aspherical 4-manifold.
Wednesday February 21, 2024 in SE 140: Vladimir Turaev (IU), “Knots and links in 2-complexes”
Wednesday February 28, 2024 in SE 140: Liam Keenan (Minnesota), “Chromatic vanishing results for topological restriction homology”; Over the past several years, there has been an enormous development in our understanding of how algebraic K-theory, an invariant of ring spectra, interacts with chromatic homotopy theory, a powerful tool for organizing computations of stable homotopy groups. These developments are part of Rognes' chromatic redshift philosophy, which, roughly speaking, asserts that K-theory should increase chromatic height by one. In this talk, I will give an introduction to this circle of ideas and explain some recent work, joint with Jonas McCandless, regarding how topological restriction homology fits into this picture.
Wednesday March 6, 2024 in SE 140: Birgit Richter (Hamburg), “Loday constructions for Tambara functors”
Wednesday March 20, 2024 in SE 140: Jonathan Campbell (CCR-L), “The Many Uses of the K-theory of Endomorphisms”; In this talk I'll discuss work with Kate Ponto and Inna Zakharevich on some properties of the K-theory of endomorphisms and a new approach to the cyclotomic trace in K-theory. To us, the definition of the cyclotomic trace has always seemed ad hoc, and so we try to put it on a cleaner conceptual footing via the K-theory of endomorphisms, K(End). I'll try to convince you that K(End) is a somewhat more natural invariant than K-theory for some purposes, prove some interesting properties of K(End) (for example, that it is a "shadow"), and show how these properties point towards a natural definition of the cyclotomic trace. For folks who aren't homotopy theorists: don't fret! I'll explain all of the terms in this abstract.
Wednesday March 27, 2024 in SE 140: Maxine Calle (Penn), “Nested cobordisms and TQFTs”; The folklore theorem identifying 2-dimensional topological quantum field theories (TFQTs) with Frobenius algebras is a starting point for a lot of interesting mathematics, from mathematical physics to homotopy theory to higher category theory. In this talk, we will explore what happens if we replace the cobordism category with a category of nested cobordisms, where 2-dimensional surfaces may have embedded 1-dimensional submanifolds, and what kind of algebraic structure the corresponding nested TQFTs pick out. This talk is based on joint work with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.
Wednesday April 3, 2024 in SE 140: Bob Oliver (Paris XIII), “The loop space homology of a small category”; In an article published in 2009, Dave Benson described, for a finite group $G$, the mod p homology of the space $\Omega (BG^{\wedge}_p)$ (the loop space of the p-completion of $BG$) in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we tried to better understand Benson’s result by generalizing it. Among other things, we showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is its geometric realization, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a “plus construction” of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|^+_R); R)$ is the homology of any chain complex of projective $R\mathcal{C}$-modules that satisfies certain conditions. Benson’s theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R = \mathbb{F}_p$ , so that p-completion appears as a special case of the plus construction.
Wednesday April 10, 2024 in SE 140: David Chan (MSU), “Equivariant Algebraic K theory and the stable parametrized h-cobordism theorem”; The stable parametrized h-cobordism theorem is a result which connects the study of cobordisms of manifolds to stable homotopy theory. It is formulated in terms of Waldhausen’s algebraic K-theory of spaces and helps to explain the connection between algebraic K-theory and geometric topology through Wall’s finiteness obstruction and Whitehead torsion. In this talk we will discuss a refinement of Waldhausen’s construction which accounts for symmetries of the underlying manifold, implemented as the algebraic K theory of an equivariant ring spectrum. As an application, we will explain how this framework can be used as a step to proving an equivariant stable parametrized h-cobordism theorem.
Wednesday April 17, 2024 in SE 140: J.D. Quigley (UVA), “Detecting exotic spheres and their symmetries with stable homotopy theory”; An exotic n-sphere is a smooth n-manifold which is homeomorphic, but not diffeomorphic, to the n-sphere with its standard smooth structure. Even after decades of intense study, there are many simple-sounding questions about exotic spheres which are unanswered. In which dimensions do exotic spheres exist? Can you rotate an exotic sphere as you would a standard sphere? Are exotic spheres "round" like the standard sphere? In this talk, I will survey what is known about exotic spheres and their geometry. I will then report on work with Mark Behrens and Mark Mahowald, Prasit Bhattacharya and Irina Bobkova, and Boris Bovtinnik on detecting exotic spheres and their smooth symmetries.
Wednesday April 24, 2024 in SE 140: Noah Snyder (IU), “Towards the Quantum Exceptional Series”; Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding symmetric tensor categories fit into continuous familes GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These are the tensor categories attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don’t fit into any of the classical families: G2, F4, E6, E7 and E8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-parameter family of planar algebras which introduces a variable q, and yields a new exceptional knot polynomial. In work joint with Dylan Thurston and joint in part with Scott Morrison arXiv:2402.03637, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations.
Saturday April 27, 2024: Midwest Topology Seminar; https://mts.math.indiana.edu/2024/
Wednesday May 22, 2024 in SW 220: Jonathan Hillman (Univ. of Sydney), "Hypersurfaces in S^4"; Abstract: I shall outline the contents of a book manuscript that I have been writing, with the title "Locally flat embeddings of 3-manifolds in S^4"
The emphases in this book are on embeddings of Seifert fibred 3-manifolds and on properties of the complementary regions.
The choice of "locally flat" rather than "smooth" reflects Freedman's result on homology spheres, which all have locally flat embeddings, whereas the smooth problem is almost intractable, and on the hope of applying topological surgery when the complementary regions have good fundamental groups.

IU/PU/IUPUI Joint Topology Seminar

Fall 2023: Saturday December 9, 2023: IU/PU/IUPUI Joint Topology Seminar, Ningchuan Zhang (IU) / Manuel Rivera (Purdue)