Topology at IU

Seminars During Academic Year 2024-25

Fall 2024

Wednesday September 11, 2024 in SE 105: Jeremy Miller (Purdue), “Scissors congruence automorphism groups”; Two polytopes are called scissors congruent if they can be cut up into isometric pieces. The question of classifying polytopes up to scissors congruence dates back to Hilbert. In joint work with Lemann, Malkiewich, and Sroka, we study an orthogonal question, investigating the group of scissors congruence automorphisms of a fixed polytope. We prove a homological stability result for these groups. This yields a new model of Zakharevich’s scissors congruence spectrum. Combining our result with work of Malkiewich on Tits building models of the scissors congruence spectra yields an approach to calculating the homology of these groups and their variants.
Wednesday September 18, 2024 in SE 105: David Boozer (IU), “The combinatorial and gauge-theoretic foam evaluation functors are not the same”; Kronheimer and Mrowka have outlined a new approach that could potentially lead to the first non-computer based proof of the four-color theorem. Their approach relies on a functor J-sharp, which they define using gauge theory, from a category of webs in R^3 to the category of finite-dimensional vector spaces over the field of two elements. They have also suggested a possible combinatorial replacement J-flat for J-sharp, which Khovanov and Robert proved is well-defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor J-sharp to the subcategory of planar webs is not the same as J-flat.
Wednesday September 25, 2024 in SE 105: Myungsin Cho (IU), “Fiber of the cyclotomic trace of the sphere spectrum and K-theoretic Tate-Poitou duality at the prime 2”; Understanding the algebraic K-theory of the sphere spectrum has long been recognized as a fundamental problem in algebraic and differential topology. Since the homotopy fiber of its p-completed cyclotomic trace depends only on the zeroth homotopy group, we can apply algebraic methods to study it. Blumberg and Mandell’s work demonstrates that, for odd primes, Tate-Poitou duality can be enhanced to an Anderson duality between the homotopy fiber and the K(1)-local K-theory of the integers. In this talk, I will present this connection and extend the result to the case where p=2.
Wednesday October 2, 2024 in SE 105: Paul Kirk (IU), “A lagrangian correspondence on traceless SU(2) character varieties induced by a marking”; I’ll describe a non-trivial endofunctor on the Fukaya A_infty category of the SU(2) character variety of a Conway sphere (S^2,4) induced by a cobordism obtained by adding a marking to the product (S^2,4) x I. (You don’t need to know what any of these words mean.)
Wednesday October 9, 2024 in SE 105: Juan Moreno (CU Boulder), “Duals of higher real K-theories at p=2”; Morava E-theories are higher chromatic analogs of complex K-theory, KU. They come equipped with cyclic group actions generalizing the complex conjugation action on KU. Taking fixed-points in a suitable sense gives rise to generalizations of KO called higher real K-theories. The higher KO’s form a large class of interesting cohomology theories that are great at detecting elements in the stable homotopy groups of spheres. Interestingly, one often finds that the higher real K-theories are self-dual up to a suspension. To the speaker’s knowledge, whether this is the case in general is an open question.
In this talk, we will explore a connection between the self-duality of higher real K-theories and equivariant periodicities of E-theories generalizing the classical Bott periodicity. In particular, we establish the self-duality of a particularly interesting higher real K-theory known to detect the Kervaire invariant one elements.
Wednesday October 16, 2024 in SE 105: Ben Spitz (UVA), “Algebraically Closed Tambara Functors”; In equivariant stable homotopy theory, objects called "Tambara functors" play the role of commutative rings. Tambara functors are abstract algebraic objects: they consist of sets with certain operations satisfying certain axioms; however, the theory of Tambara functors is much less developed than the theory of commutative rings, in part because it is not clear exactly how to define the "Tambara analogues" of many classical notions.
In their proof of the Redshift Conjecture, Burklund-Schlank-Yuan introduced the notion of a "Nullstellensatzian object" in a category C. When C is the category of commutative rings, this recovers precisely the notion of an algebraically closed field, so we therefore define an "algebraically closed Tambara functor" to be a Nullstellensatzian object in the category of Tambara functors.
In this talk, I will present a classification of algebraically closed G-Tambara functors (for any finite group G). As a corollary, we reduce the K-theory of algebraically closed Tambara functors to the ordinary K-theory of algebraically closed fields. This is joint work with Jason Schuchardt and Noah Wisdom.
Wednesday October 23, 2024 in SE 105: Nikolai Konovalov (Chicago), “Algebraic Goodwillie spectral sequence”; I will discuss two tools for computing homotopy groups of a space: Goodwillie and Adams spectral sequences. The first one is more fundamental, however less known and less understood. Vice versa, the second one is less fundamental, but far more investigated. I will also talk about how these two spectral sequences interact with each other.
Wednesday October 30, 2024 in SE 105: Hana Jia Kong (Zhejiang/Harvard), “Motivic Milnor formulas”; In the latest research, Lin, Wang, and Xu have advanced the Adams computation of stable homotopy groups using computer programs. Developing these algorithms requires a deep understanding of the Steenrod algebra. While Milnor's work provides a foundation in the classical case, the motivic case is much more complex. In this talk, I'll share my work with Weinan Lin on the motivic formulas. This is part of a larger project with Lin, Wang, and Xu, aiming to compute real motivic stable stems over a large range with the assistance of computers.
Wednesday November 6, 2024 in SE 105: Zhonghui Sun (MSU), “Twisted bicategorical shadows and traces”; Bicategorical shadows, defined by Ponto, provide a framework which generalizes (topological) Hochschild homology. Bicategorical shadows have important properties, such as Morita invariance, and allow one to generalize the symmetric monoidal trace to a bicategorical trace. Topological Hochschild homology (THH), which is an essential component to the trace methods approach for algebraic K-theory, is a key example of a bicategorical shadow.
In recent years, equivariant versions of topological Hochschild homology have emerged. In particular, for a C_n-ring spectrum there is a theory of C_n-twisted THH, constructed via equivariant norms. However, twisted THH fails to be a bicategorical shadow. In this talk, we will explain a new framework of equivariant bicategorical shadows and explain why twisted THH is a g-twisted shadow. We also explore g-twisted bicategorical traces.
Wednesday November 13, 2024 in SE 105: Yunze Lu (UCSD), “RO(G)-graded homotopy fixed point spectral sequence for height 2 Morava E-theory”; The homotopy fixed points of Lubin-Tate theories are central objects in chromatic homotopy theory, as they are building blocks of K(n)-local spheres. This talk will describe the use of equivariant structure and vanishing line result to compute the RO(Q_8)-graded homotopy fixed points spectral sequence at height 2 and prime 2. This is joint work with Zhipeng Duan, Hana Jia Kong, Guchuan Li, and Guozhen Wang.
Wednesday November 20, 2024 in SE 105: Daniel Lopez Neumann (Universidad de Concepción, Chile), “Obstructions to knot fibering from quantum groups”; The quantum invariants of knots and 3-manifolds are a wide family of topological invariants coming from the theory of braided monoidal categories, and in particular, quantum groups. The Jones and HOMFLY polynomials of knots are the most studied special cases of this theory. However, it is not known how these invariants are related to classical geometric properties of knots, such as the Seifert genus or fibredness.
In this talk, I’ll propose a partial solution to this problem: quantum invariants coming from "non-semisimple" quantum groups do provide genus bounds and obstructions to fibredness. More precisely, I’ll state and show a q-deformation of the classical fibering obstruction for the Alexander polynomial. The talk doesn't assumes familiarity with quantum groups. This is joint work with Roland van der Veen.
Thursday November 21, 2024 in SE 105: Seymour Sherman Memorial Lecture: Adam Levine (Duke), “New Constructions of Exotic 4-Manifolds”; The search for exotic manifolds — pairs or families of smooth manifolds that are homeomorphic but not diffeomorphic — has been a motivating theme in topology for decades, particularly in dimension 4. In this talk, we will discuss some new constructions of exotic 4-manifolds and new methods for distinguishing between them. We exhibit several novel phenomena, including the first known examples of exotic 4-manifolds with definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.

Spring 2025

Wednesday January 29, 2025 in SE 105: Maximillien Peroux (MSU), “Topological homology of rings with a twisted G-action”; Topological Hochschild homology (THH) is a crucial invariant as it approximates the algebraic K-theory via the Dennis trace. One of its major characteristics is its circle action that leads to cyclotomic structures that allow refinements of THH which approximates K theory more accurately.
THH is the topological analogue of Hochschild homology and both are built from Connes’ cyclic category that extends the usual simplex category. Fiedorowicz and Loday introduced analogues of these categories called crossed simplicial groups. They defined homologies associated to these categories for rings with group actions that are compatible with the multiplication on the rings, modulo some twisting.
In this talk, I introduce their topological analogues in stable homotopy theory, we recover THH but also real THH that is defined on rings with involution which is a form of twisted actions. I introduce new invariants such as quaternionic THH (which has a Pin(2)-action instead of a circle action), topological symmetric homology and topological hyperoctohedral homology which are new invariants for rings. I will present some computations of these invariants on monoid rings. This is joint work with Gabriel Angelini-Knoll and Mona Merling.
Wednesday February 5, 2025 in SE 105: J. D. Quigley (UVA), “On the Tambara affine line”; Tambara functors are the analogue of commutative rings in equivariant algebra. A little over a decade ago, Nakaoka defined the Zariski spectrum of prime ideals of a Tambara functor, and Nakaoka and Calle-Ginnett computed the analogue of $Spec(\mathbb{Z})$ over finite cyclic groups. In this talk, I will survey some new techniques for studying the spectra of Tambara functors. The main application is a description of the Tambara affine line, the equivariant analogue of $Spec(\mathbb{Z}[x])$, over cyclic groups of prime order. This is joint work with David Chan, David Mehrle, Ben Spitz, and Danika Van Niel.
Wednesday February 12, 2025 in SE 105: Noah Wisdom (Northwestern), “Real global group laws and global MR”; In the 1960s, Quillen showed that the complex cobordism spectrum MU supports the universal formal group law, establishing a connection between algebra and topology which paved the way for chromatic homotopy theory. Much later, celebrated work of Hausmann proved an equivariant generalization of Quillen's theorem, in a stunning use of global homotopy theory. In this talk, based on joint work with Jack Carlisle and Guoqi Yan, we will review the history of the relationship between orientations and formal group laws, and discuss a Real global refinement of this story, heuristically obtained by letting Z/2 act by complex conjugation everywhere.
Wednesday February 19, 2025 in SE 105: Noah Riggenbach (Northwestern), “Synthetic cyclotomic spectra”; Cyclotomic spectra arise in various contexts, including in the study of algebraic K-theory via work of Dundas-Goodwillie-McCarthy, and in p-adic Hodge theory via work of Bhatt-Morrow-Scholze and Hahn-Raksit-Wilson. Cyclotomic spectra, and in particular topological Hochschild homology, have several results who's proof is due to algebraic topology considerations. These results have led to conjectures and results for syntomic cohomology (and related theories such as F-gauges and prismatic cohomology), the proofs of which usually need to go through purely algebraic methods. In this talk I will discuss how the category of synthetic cyclotomic spectra, developed in joint work with Antieau, can be used to provide a unified framework for algebraic topology proofs to directly prove results in arithmetic geometry.
Tuesday February 25, 2025 in RH 104: Runjie Hu (Texas A&M), “Galois symmetry on the underlying manifold structures of varieties”; It is known that the underlying profinite homotopy type of a complex variety is invariant under Galois automorphisms of the ground field. However, some transcendental information is not Galois invariant. I will explain how the underlying manifold structures are changed under Galois automorphisms. For this, the concept of manifolds needs to be formalized and then further profinite completed.
Wednesday February 26, 2025 in SE 105: Maru Sarazola (Minnesota), “Squares K-theory and 2-Segal spaces”; It's a well-known result that a simplicial set satisfies the Segal condition precisely if it's the nerve of a category; then, categories represent all possible 1-Segal sets. But what about 2-Segal? This question was studied by Bergner, Osorno, Ozornova, Rovelli and Scheimbauer, who proved that 2-Segal sets are in turn related to a certain class of double categories via a "K-theory construction". In this talk, I will discuss recent work joint with Maxine Calle where we explore the connection between 2-Segal spaces and a recently introduced K-theory framework based on double categories called "Squares K-theory".
Wednesday March 12, 2025 in SE 105: Matthew Kahle (OSU), “Configuration spaces of particles: homological solid, liquid, and gas”; We will discuss configuration spaces of particles with positive thickness in a metric space. These spaces generalize configuration spaces of points, and also arise naturally as energy landscapes or phase spaces in statistical physics. We will survey recent progress in understanding the topology of these spaces, and suggest some open problems and future directions.
Wednesday March 26, 2025 in SE 105: Cary Malkiewich (Binghamton), “Higher scissors congruence”; Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. In the 2010s, Zakharevich defined a "higher" version of scissors congruence, where we don't just ask whether two polytopes are scissors congruent, but also how many scissors congruences there are from one polytope to another.
Zakharevich's definition is a form of algebraic K-theory, which is famously difficult to compute, but I will discuss a surprising result that makes the computation of the higher K-groups possible, at least for low-dimensional geometries. In particular, this gives the homology of the group of interval exchange transformations, and a new proof of Szymik and Wahl's theorem that Thompson's group V is acyclic. Much of this talk is based on joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich, and also with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin Sroka.
Wednesday April 2, 2025 in SE 105: Hyungseop Kim (Université Paris-Saclay), “Thomason filtration via T(1)-local TC”; Algebraic K-theory can be studied by relating it to other invariants, such as topological cyclic homology, as well as through cohomology theories. In this talk, I will discuss an instance of their interactions, namely that how one can reconstruct Thomason’s filtration on T(1)-localized algebraic K-theory of the generic fiber of p-adic rings through topological cyclic homology, and how this recovers the étale-prismatic comparison theorem.
Wednesday April 9, 2025 in SE 105: Liam Keenan (Brown), “Rethinking the Taylor tower of TC”; In his 1990 ICM address, Goodwillie conjectured a very close relationship between the algebraic K-theory and topological cyclic homology (TC) of ring spectra. Namely, that they have equivalent Taylor towers (in the sense of functor calculus), both of which converge on an infinitesimal neighborhood of a ring spectrum. By 2000, this question was settled (in the p-complete case), and the integral case is the subject of the now-famous book by Dundas, Goodwillie, and McCarthy. In 2009, Lindenstrauss—McCarthy offered a complete description of these Taylor towers in terms of "topological Witt vectors" — an invariant which naturally arises in the study of TC. This description rests on an identification of the topological Hochschild homology (THH) of a "trivial square zero extension" of ring spectra. In this talk, I will introduce a new invariant which naturally recovers THH of a trivial square zero extension and is closely related to the Taylor tower of TC. This is accomplished by making use of "cyclic trace theories" as developed by Kaledin, Nikolaus, and Krause—McCandless—Nikolaus.
Friday April 11, 2025 in SE 105: Departmental Colloquium: Mark Powell (Glasgow), “Applications of smoothing theory to 4-manifolds”; Smoothing theory translates the question of how many smooth or PL structures a high-dimensional manifold admits into algebraic topology. While it does not apply verbatim in dimension four, the high dimensional theory makes partial predictions about smoothings of 4-manifolds. I will discuss when these predictions come true, and some of the resulting exotic phenomena. Based on joint work with Daher, Kasprowski, Orson, and Randal-Williams.
Wednesday April 16, 2025 in SE 105: Mark Powell (Glasgow), “4-manifolds with 3-manifold fundamental group”; I will report on joint work with Hillman, Kasprowski, and Ray, in which we classify 4-manifolds with 3-manifold fundamental group up to homotopy equivalence. I will then specialise to infinite dihedral fundamental group, where we can upgrade to a homeomorphism classification.
Wednesday April 23, 2025 in SE 105: Meng Guo (UIUC), “Spectrification of Khovanov arc algebras and higher representation theory”; Spectrification of Khovanov’s arc algebra, introduced by Lawson, Lipshitz, and Sarkar, has provided a powerful framework for studying link homology at the spectrum level, offering refinements that are strictly stronger than classical invariants. In this talk, we present how this spectrification naturally leads to 2-representations of categorified quantum groups over F_2, which we call spectral 2-representations. These take values in the homotopy category of spectral bimodules, opening the door to a spectrum-level understanding of higher representation theory.
A technical innovation in our work is using a set of canonical cobordisms that we call frames to spectrify algebras. As a step towards extending these spectral 2-representations to integer coefficients, we also work in the gl_2 setting and lift the Blanchet–Khovanov algebra to a multifunctor into a multicategory version of Sarkar–Scaduto–Stoffregen’s signed Burnside category. We also discuss future directions on the integer lifting of these spectral 2-representations.

IU/PU/IUPUI Joint Topology Seminar

Fall 2024: Saturday September 28, 2024: IU/PU/IUPUI Joint Topology Seminar, Kate Ponto (UKY) / David Boozer (IU)