Purdue Topology Seminar
In Fall 2025, the Purdue Topology Seminar will be held on Thursdays 11:00am - 12:00pm EST at SC G014 (if we meet in person) unless otherwise noted. Some of the talks will be online through Zoom. 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 2025
August 28 (online)
Andrea Bianchi (MPI Bonn)
Title: String topology and graph cobordisms
Abstract: String topology, introduced by Chas-Sullivan, is the study of the homology of mapping spaces of the form M^X, where M is a closed oriented d-dimensional manifold and X is a space. Fixing M and letting X vary, the homology groups H_*(M^X) carry additional algebraic structure coming from (contravariant) functoriality in X and from Poincare' duality of M; the most famous example is the Chas-Sullivan product.We introduce a symmetric monoidal infty-category GrCob of "graph cobordisms between spaces"; we define compatible local coefficient systems xi_d on the morphism spaces of GrCob, and use the twisted homology of the morphism space GrCob(Y,X) to define higher string operations H_*(M^X)--->H_*(M^Y). We assemble all such operations into a "graph field theory" associated with M, i.e. a contravariant symmetric monoidal functor out of the linearisation of GrCob given by xi_d. We recover some basic operations, including the Chas-Sullivan product, as special cases.The construction of the graph field theory can in fact be carried out for any oriented Poincare' duality space M, and is natural in M with respect to orientation-preserving equivalences; in particular all string operations we obtain are automatically homotopy invariant.The main technical input to the construction is a recent result by Barkan-Steinebrunner, giving a universal property for the category of graph cobordisms between finite sets in terms of commutative Frobenius/Calabi-Yau algebras.
September 4 (in-person)
Yun Liu (Indiana University)
Title: Quasi-flag manifolds and moment graphs
Abstract: 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. They solved the realization problem in the rank one case using the fiber-cofiber construction introduced in topology by T. Ganea. In this talk, we will introduce a new class of topological G-spaces generalizing the classical flag manifolds G/T of compact connected Lie groups. These spaces --- which we call the m-quasi-flag manifolds F_m(G,T) --- are topological realizations of the rings Q_m(W) of m-quasi-invariant polynomials of finite reflection groups.
September 9 (in-person)
Time: 2:00-3:30
Location: Math 942
Note: This is a Tuesday
Fabio Capovilla-Searle (Purdue)
Title: Top degree cohomology of congruence subgroups of symplectic groups
Abstract: The cohomology of arithmetic groups has connections to many areas of mathematics such as number theory and diffeomorphism groups. Classifying spaces of congruence subgroups of symplectic groups have an algebro-geometric interpretation as the moduli space of principally polarized abelian varieties with level structures. These congruence subgroups Sp_2n(Z,L) are the kernel of the mod-p reduction map Sp_2n(Z) to Sp_2n(Z / L). By work of Borel-Serre, H^i(Sp_2n(Z / L)) vanishes for i > n^2. I will report on lower bounds in the top degree i = n^2. The key tools in the proof are the theory of Steinberg modules and highly connected simplicial complexes.
September 11 (online)
Thomas Tradler (City Tech CUNY)
Title: Massey products for homotopy inner products
Abstract: I will review some basic properties of Massey products, both for DGAs and A infinity algebras. The definition of Massey products extends to the setting of A infinity modules, and an appropriate variation also gives Massey products for A infinity algebras with homotopy inner products. Massey products can be used to give higher level invariants coming from the cochain level of a space. I will give examples coming from links and low dimensional manifolds. This is joint work with Kate Poirier and Scott O. Wilson.
September 18 (in-person)
Marian Mrozek (Jagiellonian University)
Title: Links between Persistent Homology and Combinatorial Dynamics
Abstract: Ties between classical topology and dynamics go back to the seminal work of Marston Morse in the the 1930s. Links on the combinatorial level started with the discrete Morse theory by Robin Forman in the late 1990s. In the talk I will present some recent observations indicating that such links may lead to fruitful interactions between Persistent Homology and gradient combinatorial dynamical systems. In particular, the topological invariants in dynamics such as Conley-Morse graph and connection matrix may be used to enhance the methods of Persistent Homology.
September 25 (in-person)
Jeremy Miller (Purdue University)
Title: Duoidal bi-algebras and congruence subgroups
Abstract: I will describe an exotic algebraic structure called "duoidal bi-algebras" and describe some applications of it to the cohomology of congruence subgroups of special linear groups of number rings. This is joint work with Avner Ash and Peter Patzt.
September 29 (in-person)
Joint with the Geometry Seminar, Monday 11:30 in MATH 731
Andrew Putman (Notre Dame)
Title: The second rational homology group of the Torelli group
Abstract: After giving an introduction to finiteness properties of the mapping class group and its subgroups, I will discuss a recent theorem I proved with Minahan calculating the second homology group of the Torelli group.
October 2 (online)
Michelle Bucher (Université Genève)
October 9 (in-person)
Nick Gurski (Case Western Reserve University)
Title: Diagrammatics for Picard categories
Abstract: Picard categories are symmetric monoidal categories in which every object is invertible with respect to the tensor product, and every morphism is invertible in the usual sense. These arise in classical constructions like the ideal class group of a ring or the first cohomology group of a complex manifold using the sheaf of invertible holomorphic functions. It has long been known that Picard categories encode essentially the same information as stable homotopy 1-types, and that the free Picard category on one object "is" the 1-truncation of the sphere spectrum. In this talk, I will explain some recent work with Niles Johnson in which we give a complete treatment of the free Picard category on one object that was inspired by three fun pictures.
October 16 (online)
Anna Fokma (Utrecht University)
October 23 (in-person)
Susan Hermiller (University of Nebraska)
October 30 (in-person)
Urshita Pal (University of Michigan)
November 4 (in-person)
Department Colloquium, Tuesday 3:00 in Math 175
Daniil Rudenko (University of Chicago)
November 13 (in-person)
Francis Bischoff (University of Regina)
November 20 (joint with GGA seminar and in-person)
Egor Shelukhin (University of Montreal)
December 11 (in-person)
Lucy Yang (Columbia University)
Purdue topology group:
- Lvzhou Chen
- Xingshan (Shawn) Cui (Fall 2025 seminar organizer)
- Colleen Delaney
- Ralph Kaufmann
- Ben McReynolds
- Jeremy Miller
- Sam Nariman
- Yash Lodha
- Manuel Rivera
- Eric Samperton