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)
Wednesday February 12, 2025 in SE 105: Noah Wisdom (Northwestern)
Wednesday February 19, 2025 in SE 105: Noah Riggenbach (Northwestern)
Wednesday February 26, 2025 in SE 105: Maru Sarazola (Minnesota)
Wednesday March 12, 2025 in SE 105: Matthew Kahle (OSU)
Wednesday March 26, 2025 in SE 105: Cary Malkiewich (Binghamton)

IU/PU/IUPUI Joint Topology Seminar

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