Topology at IU

Loading

Seminars During Academic Year 2025-26

Fall 2025

Wednesday September 3, 2025 in RH 104: Michael Mandell (IU), “A multiplicative analogue of the tom Dieck splitting”; I will review the tom Dieck splitting in equivariant stable homotopy theory and talk about a new multiplicative result of the same flavor in recent joint work with Andrew Blumberg.
Wednesday September 10, 2025 in RH 104: Juanita Pinzon-Caicedo (Notre Dame), “Branched covers and SU(2)-representations”; The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds, and the existence of non-trivial homomorphisms $\pi_1(M)\to SU(2)$ is a great way of measuring the non-triviality of a three-manifold $M$. It is known that if an integer homology 3-sphere is either Seifert fibered or toroidal, then irreducible representations do exist. In contrast, the existence of $SU(2)$-representations for hyperbolic homology spheres has not been completely established. With this as motivation, I will talk about partial progress made in the case of hyperbolic homology spheres realized as branched covers. This is joint work with Sudipta Ghosh and Zhenkun Li.
Wednesday September 17, 2025 in RH 104: Ben Spitz (IU), “The spectrum of the Burnside Tambara functor”; Tambara functors are an "equivariant generalization" of commutative rings, in some precise sense(*). The initial object in the category of commutative rings is $\mathbb{Z}$, and the category of Tambara functors likewise has an initial object, called the Burnside Tambara functor(**), denoted A. In this talk, I will explain what exactly is going on with (*) and (**), and discuss joint work with Calle, Chan, Mehrle, Quigley, and Van Niel, in which we give the first complete computation of Spec(A).
Wednesday September 24, 2025 in RH 104: Alex Waugh (Washington), “Equivariant power operations”; Classically, much of our understanding of power operations stems from explicit computations of the group homology of the symmetric group. Equivariantly, this is also true, and is a main obstruction to the construction of equivariant analogues of the classical operations. In this talk, I will introduce a framework for constructing equivariant power operations with respect to any equivariantly structured ring spectrum and G-universe. As an application, I will show how to use equivariant orientation theory to construct nontrivial classes in the homology of equivariant classifying spaces for the symmetric group. I will then use these to construct explicit (stable) G-equivariant Steenrod and Dyer-Lashof operations for Bredon (co)homology with constant F_p Mackey functor coefficients, where G is any finite group. If time permits, I will discuss properties of these operations and potential applications to studying equivariant infinite loop space theory.
Wednesday October 1, 2025 in RH 104: Shangjie Zhang (UCSD), “Mahowald invariants and $C_{p^n}$- Mahowald invariants”; In this talk, I will review the classical Mahowald invariant (also called the root invariant) and some of the computations and applications. The Segal conjecture allows us to define the equivariant Mahowald invariant for any group $G$, such that the classical Mahowald invariant is simply the case $G ＝ C_2$. I will discuss some of the computations of the $C_{p^n}$-Mahowald invariants and their applications. This contains work joint with William Balderrama, Eva Belmont, Yueshi Hou, and Zhouli Xu.
Wednesday October 8, 2025 in RH 104: Anish Chedelavada (Johns Hopkins), “Higher Zariski Geometry”; Many notions of "geometry" concern themselves with topological spaces equipped with additional structure. For a given space X, this structure is usually an assignment of an open set of X to an appropriate "ring of functions" on X; these will satisfy a gluing property along open sets, and thus form a sheaf. Different geometries will impose different conditions on the requisite sheaves and spaces; for example, the category of locally ringed spaces is the basic geometric structure considered in algebraic geometry, and geometrizes the study of commutative rings via the Zariski spectrum. In DAG V, Lurie explains that such a category of "locally (-)-spaces" is a generic feature of other categories which carry a small amount of structure, known as a "geometry". Based on this, I will present joint work with Aoki-Barthel-Schlank-Stevenson which constructs a category of "locally 2-ringed spaces", which will geometrize the study of tensor triangulated-categories (which we will recall) via a (now classical) construction known as the Balmer spectrum. In this setting, we will also produce an affine spectrum ⊣ global sections adjunction exactly analogous to ordinary algebraic geometry.
I will then explain how to approximate locally 2-ringed spaces by locally ringed spaces via a "relative spectrum" construction. In doing so, I will show how to recover an early result on the reconstruction of schemes as locally ringed spaces originally due to Balmer, and enhance it to provide a universal property for the category of perfect complexes on a qcqs (spectral) scheme. Time permitting, I will use this latter result to indicate an application to torsion-free endotrivial modules over finite groups in modular characteristic.
Wednesday October 15, 2025 in RH 104: Jonathan Hanselman (IU), “Cabling operations and knot Floer homology”; Satellite operations provide a method for constructing complicated knots from simpler ones, with cabling providing a particularly nice example. It is natural to ask how satellite operations act on various knot invariants. For cables, this question turns out to have a satisfying answer when the invariant in question is knot Floer homology, provided this invariant is viewed through the right lens. I will describe how cabling operations induce geometric transformation of knot Floer invariants. This is a joint result with Liam Watson, which was generalized to a broader class of satellite operations in joint work with Wenzhao Chen.
Wednesday October 29, 2025 in RH 104: Andrew Davis (IUI), “The Homotopy Type of Spaces of Flat Connections for Classical Lie Groups”; Given a Lie group G and connected smooth manifold M, one can construct a G-bundle over M using a continuous family of representations from the fundamental group of M to G. I will explain how Chern-Weil theory can be used to show that the rational characteristic classes of such bundles vanish in high degree. By considering families of representations parametrized by spheres, this vanishing allows us to study the weak homotopy type of the space of flat connections for G = U(n), O(n), SO(n), and Spin(n). In particular, we show that for many high dimensional manifolds, the space of flat connections is highly disconnected, contrary to the case of surfaces.
Wednesday November 5, 2025 in RH 104: Sanath Devalapurkar (Chicago), “Calculus, (generalized) cohomology, and (quantum) groups”; I will explain a simple reformulation of the Hochschild–Kostant–Rosenberg theorem, describing the algebra of (crystalline) differential operators on a smooth scheme over a commutative ring k in terms of (higher) Hochschild cohomology. When k is replaced by a commutative ring spectrum, essentially the same construction gives a way of assigning to each 1-dimensional formal group H an algebra of “H-differential operators” on affine space. When H is the additive (resp. multiplicative) formal group, this recovers usual differential operators (resp. q-difference operators). Computing the algebra of global H-difference operators on P^1 leads to a definition of an associative algebra U_H(PGL_2) which recovers the usual enveloping algebra (resp. quantum group) when H is the additive (resp. multiplicative) formal group.
Many aspects of ordinary (and q-) calculus generalize to H-differential operators (and admit nice homotopy-theoretic explanations), like the Gauss–Manin connection, Frobenius-constant quantizations via p-curvature, Lagrangian p-support, Mellin + Fourier transforms, “H-analogues” of various special functions like hypergeometric functions, etc. Since these objects are defined using homotopy theory, one can in fact give topological proofs of various identities involving these special functions! This story has two (mirror dual!) motivations: relative Langlands duality + quantum generalized cohomology; and the theory of prismatic cohomology. If there is time, I will discuss applications to both. Some of this story builds on joint work with various subsets of Jeremy Hahn, Max Misterka, Arpon Raksit, and Allen Yuan.
Wednesday December 3, 2025 in RH 104: Maxine Calle (Penn), “Cut-and-paste K-theory of manifolds and SK-automorphisms”; Given two manifolds M and N, one can ask whether it is possible to cut M up into pieces and reassemble them to obtain N. This “cut-and-paste” (SK) relation fits into the framework of scissors congruence K-theory, which is an extension of higher algebraic K-theory to more general settings. In this talk, we will discuss a new model for the cut-and-paste K-theory of manifolds, modeled on Waldhausen’s S-dot construction, and describe how the first K-group is related to SK-automorphisms of manifolds, i.e. the ways a manifold can be SK-equivalent to itself. This talk is based on joint work with Maru Sarazola.
Wednesday December 10, 2025 in RH 104: Lucy Yang (Columbia), “Involutions and the Brauer group in derived algebraic geometry”; Classical results of Albert and Saltman (extended by Knus–Parimala–Srinivas) have established a connection between the existence of involutions on the Azumaya algebras A used to define the Brauer group and 2-torsion Brauer classes. Moreover, the presence of more general forms of involutions is related to the behavior of Brauer classes under corestriction along quadratic extensions. In this work, we investigate how the data of an involution on A is reflected in additional structure on its (derived) category of modules and introduce the notion of an involution on a generalized Azumaya algebra. Using the theory of Poincaré \infty-categories developed by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, we introduce involutive versions of the derived Picard and Brauer groups and relate them to their classical and non-involutive counterparts. We show that we can recover the involutive Brauer group of Parimala-Srinivas when the former is defined; however, they no longer agree even for closed points due to the existence of shifted perfect pairings. We also compute these invariants for the sphere spectrum and other examples. As a consequence, we deduce a derived enhancement of a classical theorem of Saltman. This is joint work with Viktor Burghardt and Noah Riggenbach.

Spring 2026

Nothing Scheduled

IU/PU/IUPUI Joint Topology Seminar

Spring 2026: Saturday March 7, 2026: IU/PU/IUI Joint Topology Seminar: Jenny Wilson (Michigan) / Ben Spitz (IU)