Purdue Topology Seminar

In Fall 2026, the Purdue Topology Seminar will be held on Wednesdays 10:30am - 11:30am Eastern time in BRNG B212 (if we meet in person) unless otherwise noted. If you want to be added to our email list please contact Shawn Cui (shawn.cui at purdue.edu) or Manuel Rivera (manuelr at purdue.edu). 

Some recorded talks may be found on our YouTube Channel.

 

Fall 2026

September 2 (in-person)

Jeremy Miller (Purdue)

Title: Nontriviality of the Morita classes

Abstract: I will describe work in progress with Kupers and Patzt on the nontriviality of the Morita classes for \(\mathrm{Aut}(F_n)\).

 

September 9 (in-person)

Kevin Piterman (Purdue)

Title: Spherical complexes associated to buildings 

Abstract: The Solomon–Tits theorem states that a spherical building of dimension d has the homotopy type of a wedge of d-spheres. One proof of this theorem uses the natural CAT(1) metric on the building, under which each apartment is isometric to a unit sphere. In this talk, we will see how the intrinsic geometry of a building can be used to show that certain natural complexes associated to it are spherical, and how these complexes relate to familiar constructions in the GL_n case. In particular, I will propose definitions of a common basis complex and a partial decomposition complex for an arbitrary spherical building. Using the intrinsic metric, we will show that these two complexes are homotopy equivalent and that the latter is highly connected. This is joint work with J. Shareshian and V. Welker.

 

September 16 (in-person)

Lucas Williams (Purdue)

Title: Invariants for Families of Periodic Points

Abstract: In this talk we investigate invariants that count periodic points of a map. Given a self map f of a compact manifold we could detect n-periodic points of f by computing the Reidemeister trace of f^n or by computing the equivariant Fuller trace. In 2020 Malkiewich and Ponto showed that the collection of Reidemeister traces of f^k for varying k|n and the equivariant Fuller trace are equivalent as periodic point invariants, and they conjecture that for families of endomorphisms the Fuller trace will be a strictly richer invariant for n-periodic points.


In this talk we will explain our new result which confirms Malkiewich and Ponto's conjecture. We do so by proving a new Pontryagin-Thom isomorphism between equivariant parameterized cobordism and the spectrum of sections of a particular parametrized spectrum and using this result to carry out geometric computations.

Time permitting, we will discuss how this homeomorphism of a manifold gives rise to an element of the kernel of the ghost map on \pi_1(-) of topological restriction homology.

 

September 23 (in-person)

Sarah Anderson (Purdue)

Title: Stability Patterns for Spherical Braid Groups

Abstract: Stability patterns for the homology of configuration spaces has been studied for decades, starting with homological stability in the 1970's, then representation stability in 2010, and stable periodicity in 2015. When the configuration spaces are Eilenberg-MacLane spaces, these stability results extend to the group homology of surface braid groups. However, configuration spaces on S^2 and RP^2 are not Eilenberg-MacLane spaces and so the stability patterns for their group homology had not yet been established. In this talk, I will use a result from Fred Cohen and Jonathan Pakianathan as well as the theory of FI-homology to establish stability patterns for the group homology of braid groups on S^2 and RP^2. 

 

September 30 (in-person)

Sam Nariman (Purdue)

Title: The continuous cohomology of the diffeomorphism group of compact manifolds and a question of Bott and Haefliger

Abstract: For finite-dimensional Lie groups, there is an isomorphism due to van Est that identifies the continuous group cohomology of a Lie group G with the relative Lie algebra cohomology of the Lie algebra g relative to its maximal compact Lie group. The maximal compact Lie subgroup does not exist in general for diffeomorphism groups. Bott formulated a statement and stated it as a belief of his and Haefliger that identifies the continuous cohomology of the diffeomorphism group of compact manifolds as the real cohomology of the homotopy quotient of certain mapping space. We reinterpret the van Est isomorphism as "the homotopy quotient" of the Lie algebra cohomology to realize Bott-Haefliger's belief. 

 

October 7 (in-person)

Miguel Barata (Purdue)

Title: A new proof of Dunn-Lurie additivity and the Gray tensor product of operads

Abstract: An important tool in modern developments of homotopy theory is the notion of an operad, and among these the E_n-operad is of particular relevance. Roughly speaking, for each natural number n these correspond to a symmetric sequence of spaces interpolating between homotopy coherent associative and homotopy coherent commutative multiplications. Although originating many years ago through the study of iterated loop spaces, it has also had a recent meaningufl impact in areas such as higher algebra, (un)stable homotopy theory and the study of embedding spaces of manifolds. In this talk I want to present a new proof of Dunn-Lurie additivity, a cornerstone result in the operad theory which relates the E_n-operads for different balues of n. The main idea of the proof comes from a construction of a new tensor product for operads, using ideas of Balteanu, Fiedorowicz, Schwaenzl and Vogt. If time permits, I will explain how the formalism of dendroidal spaces can be used to prove this result in the derived setting of infinity-operads. Joint work with Ieke Moerdijk.

 

October 14 (online)

Rhuiadi Burke (Oxford) 

Title: PL Recognition After Two \(S^2\times S^2\) Stabilisations is PSPACE-Hard
Abstract: Let \(Z\) be any fixed closed connected PL 4-manifold. We give a reduction from the compressed word problem in Thompson's group \(F\) to fixed-target recognition of \(Z\#_2(S^2\times S^2)\). Once a finite triangulation of \(Z\) has been fixed, the construction sends a straight-line program \(A\) to a closed triangulated PL 4-manifold \(X_{A,Z}\) such that a designated output word \(\operatorname{val}(A)\) is equal to the identity in \(F\) if and only if \(X_{A,Z}\) is PL-homeomorphic to \(Z\#_2(S^2\times S^2)\). The compressed word problem in \(F\) has been shown to be PSPACE-complete, and so every such recognition problem is PSPACE-hard. In particular, fixed-target recognition of \(\#_n(S^2\times S^2)\) is PSPACE-hard for every fixed \(n\geq 2\).

October 21 (in-person)

Matthew Scalamandre (Toronto) 

Title: Koszul duality for symplectic Steinberg modules, and a conjecture about the Torelli map

Abstract: Rognes conjectured that a certain space called the common basis complex CB_n(R) is (2n-3)-connected when R is a local ring or Euclidean domain. His original motivation was to study a rank filtration of the algebraic K-theory spectrum. More recently, it has been shown that this conjecture is equivalent to Koszulness of an algebra built from the Steinberg modules associated to the general linear group. These Steinberg modules have been found to be connected to the cohomology of arithmetic groups and to homological stability of GL_n(R). Miller--Patzt--Wilson proved this conjecture when R is a field. I will discuss work in progress with Jeremy Miller and Robin Sroka, in which we prove Koszulness, and compute the Koszul dual, for an analogous algebra constructed from symplectic Steinberg modules for fields. There are some interesting differences with the general linear group case. For example, this algebra does not appear to be connected to a rank filtration of Hermitian K-theory K^{Sp}, but instead is related to the homotopy fiber of the map K->K^{Sp}. I will conclude by discussing some insights and conjectures about the Torelli map, which relates the integer symplectic group with mapping class groups of surfaces. 

 

October 28 (online)

Oscar Harr (Stockholm University)

Title: Reconstruction of topological spaces from sheaf categories

Abstract: Let X and Y be nice topological spaces (e.g. manifolds or triangularizable spaces). If X and Y have equivalent derived categories of sheaves, does it follow that X and Y are homeomorphic? Although the answer is probably "no", there is no known counterexample. I will explain why any counterexample cannot involve certain low-dimensional spaces; graphs, surfaces, and the 3-sphere are all uniquely determined by their derived category of sheaves (arXiv:2610.13488).

By a fancy version of the duality theory of Poincare and Verdier, the derived category of sheaves on a nice space has a property called dualizability. Recent work of Efimov, Bartels--Nikolaus, and others has found striking applications of the theory of dualizable categories to geometric topology and algebraic geometry. In my talk, I will gently introduce the concept of a dualizable category, introduce two new invariants of dualizable categories --- (1) Tate--Hochschild cohomology, which has previously been defined for a small class of "classical/small" dualizable categories, and (2) the Gorenstein locus, --- and explain their connection to topology and how they can be used to reconstruct some spaces from their derived categories.

 

November 4 (online)

Thomas Willwacher (ETH Zurich)

Title: Homotopy Frobenius structures on the cohomology of a manifold
Abstract:  Rational homotopy theory assigns to each topological space X a homotopy commutative algebra that captures the rational homotopy type of that manifold. We show that if X is a smooth (parallelized) d-manifold, then this homotopy commutative algebra structure can be upgraded to a homotopy involutive d-Frobenius algebra structure, that captures the rational homotopy types of all configuration spaces of points on the manifold.
More precisely, we show that the category of lax involutive d-Frobenius algebras is Quillen equivalent to the category of right comodules of the d-Poisson cooperad.
This is joint work with Florian Naef.

 

November 11 (in-person)

David Chan (Michigan State)

Title:Norms in the algebraic K-theory of group rings  

Abstract: Algebraic K-theory is a rich invariant of rings and ring spectra with connections to geometric topology, algebraic geometry, and number theory. Given a manifold M with fundamental group G, the K-theory of the group ring \(\mathbb{Z}[G]\) is home to important invariants like Whitehead torsion.  Unfortunately, computing the K-theory of group rings is difficult, and complete results are somewhat limited. As an intermediate step, one often simplifies the problem by replacing the integers with a finite field \(\mathbb{F}_p\). In this talk I will discuss joint work with Chase Vogeli on applying tools from equivariant stable homotopy theory to give new computations of the algebraic K-theory of \(\mathbb{F}_p[G]\).

November 18 (online)

Maxine Calle (Bonn)

Title: Morse theory and flow categories

Abstract: Classical Morse theory describes how to encode the homology of a smooth manifold using a chain complex built from the data of critical points of a Morse function and its gradient flow. In the 1990s, Cohen–Jones–Segal proposed a categorification of this complex, called the flow category, with an eye towards developing analogous refinements in the context of Floer homology. In this talk, we will explain why the flow category of a Morse function indeed recovers the homotopy type of the underlying manifold, as long as the gradient flow is suitably "tame". Moreover, we show that this tameness assumption is crucial by constructing a Morse function and Riemannian metric on S2×S1 for which the associated flow category fails to recover the correct homotopy type. This talk is based on joint work with Fangji Liu.

 

December 2 (in-person)

Urshita Pal (Michigan)

 

Spring 2027

 

January 25 (in-person)

Ben Ward (Bowling Green State University)

 

 

Purdue topology group: