Purdue Topology Seminar
Purdue Topology Seminar
In Spring 2025, the Purdue Topology Seminar will be held on Wednesdays 11:30am - 12:30pm EST at BRNG B206 (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 Manuel Rivera (manuelr at purdue.edu).
We have begun to record some of our online seminars. We publish them on our YouTube-Channel.
Title: An equivariant pair-of-paints product from higher Hochschild cochains
Abstract: Hochschild homology and cohomology are both generally defined in the settings of A-infinity algebras/categories. In general, Hochschild homology does not inherently possess a ring structure, unlike its cohomological counterpart. Rather, it is a module over Hochschild cohomology. However, when a (homologically) smooth A-infinity algebra/category, introduced by Kontsevich-Soibelman, is equipped with a smooth Calabi-Yau structure, a product structure emerges on Hochschild homology, introducing a richer algebraic framework. In the presence of group actions (for instance, Z/2 in our case), a natural question arises: ``Can this (non-equivariant) product structure, called pair-of-pants product, often be extended or adapted to its equivariant counterpart?’'.
In this talk, I will show that there is a canonical equivariant refinement, called algebraic equivariant pair-of-pants product, of the (non-equivariant) product structure provided that a canonical lift, of a Calabi-Yau map, induced by a (strong) smooth Calabi-Yau structure exists. I will also give the motivations behind the question via a conjectural diagram related to open-closed maps, fixed point Floer cohomology and Hochschild homology of Fukaya categories and Smith-type inequalities (modulo technical details) as long as time permits. This is joint work with Sheel Ganatra.
Maximilian Stegemeyer (Freiburg)
Title: Intersection multiplicity in loop spaces and the string topology coproduct
Abstract: String topology is the study of algebraic structures on the free loop space of a closed oriented manifold. The Goresky-Hingston coproduct is geometrically defined by cutting loops apart at points of self-intersection. A result by Hingston and Wahl makes this intuitive description precise and sates that if a homology class can be represented by loops without basepoint self-intersections then the coproduct of this class vanishes. In this talk I will talk about some results where the concept of intersection multiplicity is used to conclude the triviality of the string topology coproduct. We will also encounter the relation between the homology of the free loop space and the closed geodesics in the underlying manifold. The triviality results for the coproduct open up a number of questions.
Shuhei Maruyama (Kanazawa University)
Title: McDuff's secondary class and the Euler class of foliated sphere bundles
Abstract: Tsuboi proved that the Calabi invariant transgresses to the Euler class of foliated circle bundles. The Calabi invariant is a first cohomology class of the group of area-preserving diffeomorphisms of a closed disk that are identity on the boundary.
In this talk, I will introduce a group cocycle representative of McDuff's secondary class: a (2n−1)-dimensional cocycle of the group of volume-preserving diffeomorphisms of a 2n-dimensional closed ball that are identity on the boundary. I will also discuss a higher-dimensional analog of Tsuboi's theorem, which states that this cohomology class transgresses to the Euler class of foliated sphere bundles.
Liam Morgan Ashton (Purdue University)
Title: The algebra of strings
Abstract: One may associate to any surface a Lie bialgebra structure constructed by transversally intersecting, concatenating, and splitting free homotopy classes of loops. Deep geometric results have been proven using this structure; however there are still many mysteries and open questions around it. In this talk, I will describe the Goldman bracket and the Turaev cobracket on homotopy classes of free loops (or strings) and give a few examples to show how they work and interact. I will then discuss several statements and open questions that relate these operations to minimal intersections and minimal self-intersections of strings. This is my advanced topics exam.
Title: A pleasant surprise of the cyclotomic character
Abstract: Abstract: In 1990, V. Drinfeld introduced the Grothendieck-Teichmueller group GT. This group receives a homomorphism from the absolute Galois group G_Q of rational numbers, and this homomorphism is injective due to Belyi's theorem. In his 1990 ICM talk, Y. Ihara posed a very hard question about the surjectivity of this homomorphism from G_Q to GT. In my talk, I will introduce the groupoid of GT-shadows and show how this groupoid is related to the group GT. I will formulate a version of Ihara's question for GT-shadows and describe a family of objects of the groupoid for which this question has a positive answer. My talk is loosely based on the joint paper https://arxiv.org/abs/2405.11725 with I. Bortnovskyi, B. Holikov and V. Pashkovskyi.
Ben McReynolds (Purdue University)
Title: Nilpotent representations and fundamental groups
Abstract: I will discuss a construction for producing groups with the same nilpotent representation theory. The construction is motivated by a theorem of Stallings. I will discuss some applications to Kleinian groups and a conjectural result on finite groups that could have important implications.
Calvin McPhail-Snyder (Duke University)
Title: Geometric quantum invariants and the volume conjecture
Abstract: This talk is about recent joint work with N. Reshetikhin defining a new family of geometric quantum link invariants. Here "geometric" means that they depend on an additional choice of a flat 𝔰𝔩₂ connection (i.e. a generalized hyperbolic structure) on the link complement. When this structure is trivial we recover Kashaev's invariants, which are particular values of the colored Jones polynomials. In this talk I will explain how our geometric invariants naturally appear in the context of the Volume Conjecture, then sketch their construction and how they relate to both classical and quantum Chern-Simons theory for the complex group SL₂(ℂ).
Martin Tancer (Charles University)
Title: Simpler algorithmically unrecognizable 4-manifolds
Abstract: Markov proved that there exists an unrecognizable 4-manifold, that is, a 4-manifold for which the homeomorphism problem is undecidable. In this talk we consider the question how close we can get to S^4 with an unrecognizable manifold. We sketch a way to remove so-called Markov's trick from the proof of existence of such a manifold. This trick contributes to the complexity of the resulting manifold. We also show how to decrease the deficiency (or the number of relations) in so-called Adian-Rabin set which is another ingredient that contributes to the complexity of the resulting manifold. Altogether, our approach allows to show that the connected sum #_9(S^2 x S^2) is unrecognizable while the previous best result is the unrecognizability of #_12(S^2 x S^2) due to Gordon.
Amit Kumar (Louisiana State University, IMSc)
Title: HL Cone, Foams, and Graph Coloring
Abstract: We begin with a review of the modern perspective on graph coloring which appeared in the work of Kronheimer-Mrowka and Khovanov-Robert. Next, we outline how the work of Treuman-Zaslow and Caslas-Zaslow lead to seeing graph coloring as topological defects labelled by the elements of Klein-Four Group. This highlights the quantum nature of graph coloring, namely, it satisfies the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. This TFT has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Coloring of it. Defects can be considered as a generalization of groups. With the Klein-four group as a 1-defect condition, we reinterpret graph coloring as sections of a certain cover, distinguishing a coloring (global-sections) from a coloring process (local-sections), and give a new formulation of some of Tait's work.
Peter Patzt (University of Oklahoma)
Title: Uniform twisted homological stability of braid groups and moments of quadratic L-functions
Abstract: A conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith aims to describe the asymptotics of moments of quadratic L-functions. In joint work with Miller, Petersen, and Randal-Williams and in combination with a paper by Bergström–Diaconu–Petersen–Westerland, we proved a version of this conjecture for function fields. Using the Grothendieck-Lefschetz trace formula, Bergström–Diaconu–Petersen–Westerland established a connection between the conjecture and the twisted homology of the braid groups. In our paper, we showed what was needed to make this connection. Homological stability says that the k-dimensional homology groups are all isomorphic for a large enough number of strands of the braid groups. This is even known for twisted coefficients pulled back from polynomial representations of the symplectic groups. We proved that the starting point of stability is independent of which irreducible polynomial representation of the symplectic groups one uses. In the talk, I will explain the connections between number theory, the braid groups, the symplectic groups, and homological stability.
Title: Defining extended TQFTs via handle attachments
Abstract: Extended Topological Quantum Field Theories model some physical theories where time evolution doesn't depend on any metric, but only on the topology of space-time. Using Morse theory, one can decompose any space-time into a series of "events" corresponding to handle attachments. We will reformulate this idea into a presentation of the bicategory of cobordisms, i.e. space-times. If time allows, I will also discuss some applications in dimension 4 coming from skein theory.
Sofía Martínez (Purdue University)
Title: Coalgebraic Models for Equivariant Homotopy Types
Abstract: This thesis investigates and continues a line of mathematical research dating back to the 1960's. Algebraic topology studies the algebraic invariants associated to topological spaces, and Quillen was the first to describe an algebraic invariant which allows one to completely recover the space up rational homotopy equivalence. In the ninties, Goeress showed that the simplicial coalgebra of $\F$-chains associated to a simplicial set could recover the space up to $\F$-homology equivalence, for $\F$ an algebraically closed field. More recently, Raptis and Rivera introduced another notion of equivalence between simplicial sets which remembers information about the classical invariant known as the fundamental group and showed that the simplicial coalgebra of $\F$-chains could be used to reconstruct the simplicial set. The work in this thesis explores what occurs equivariantly, i.e., when the action of a group, $G$, is present on the simplicial set, and promotes the results of Raptis and Rivera to the equivariant setting. The first we will provide some motivation and historical context for this thesis work then we will review some nonequivariant definitions and notions, most of which can be found in Raptis and Rivera's article. Furthermore, we will introduce the relevant equivariant notions and model structures which will be used to explain equivariant homotopical results. Finally, we provide some partial results towards describing a notion of equivariant Koszul Duality, motivating future research directions.
Title: The Borel Conjecture for manifolds with boundary
Abstract: The Borel Conjecture for closed manifolds implies that two closed aspherical manifolds with isomorphic fundamental group are homeomorphic. The Borel Uniqueness Conjecture for compact aspherical manifolds with boundary states that a homotopy equivalence which is homeomorphism on the boundary is homotopic rel boundary to a homeomorphism. The Borel Existence Conjecture states that a Poincare pair with aspherical total space and closed manifold boundary is homotopy equivalent to a compact manifold pair.
Jonathan Hillman and I prove the Borel Existence Conjecture in many cases. We prove the Borel Conjecture for all compact aspherical four manifolds with boundary with good (= elementary amenable) fundamental group. We classify all possible fundamental groups and all possible 3-manifold boundaries.
Kevin Piterman (Vrije Universiteit)
Title: A categorical approach to study posets of decompositions into subobjects.
Abstract: Given a sequence of groups G_n with inclusions G_n -> G_{n+1}, an important question in group (co)homology is whether there is homological stability. That is, if for a given integer j, there is some m such that for all n>m, the map H_j(G_n) -> H_j(G_{n+1}) is an isomorphism. To detect this behaviour, one usually constructs a family of highly connected simplicial complexes K_n on which the groups G_n naturally act, with stabilisers involving the "smaller groups" G_m, m<n. For example, for the linear groups GL_n or SL_n, K_n can be the Tits building or the complex of unimodular sequences, while for the automorphism group of the free groups F_n one can take the complex of free factors. Another closely related question is whether the homology of a highly connected space computes (in some way) the homology of another object (e.g. a group).
In this talk, we discuss a categorical framework that describes some of the constructions used to approach these problems in a unified way. More precisely, for an initial symmetric monoidal category C, we take an object X and consider the poset of subobjects of X. From this bounded poset, we take only those subobjects which are complemented, i.e. x \vee y = 1 and x \wedge y = 0, and the join operation coincides with the monoidal product. The monoidal product should be interpreted as the "expected" coproduct of the category. Thus, for the free product in the category of groups, if we start with a free group of finite rank, then the complemented subobject poset is exactly the poset of free factors, and for the category of vector spaces with the direct sum, we obtain the subspace poset. From this construction, we define related combinatorial structures, such as the poset of (partial) decompositions or the complex of partial bases, and establish general properties and connections among these posets. Finally, we specialise these constructions to matroids, modules over rings, and vector spaces with non-degenerate forms, where there are still many open questions.
Thomas Wasserman (Oxford University)
Yash Lodha (University of Hawaii)
Andrea Bianchi (MPI Bonn)