Midwest Topology Seminar

May 17-18, 2025 IU Indianapolis

Building and Room: Plenary talks will be held at IT 252 (aka ICTC 252) and first floor lounge for refreshments. We will have two simultaneous sessions of contributed talks at IT 157 and IT159. Click here to locate the building on a map.

Lodging:

• Hampton Inn Indianapolis Downtown IUPUI
• Residence Inn Indianapolis Downtown on the Canal
• Courtyard Indianapolis at the Capitol

Parking: The Gateway Garage is the best place to park. (Quite possibly the department can provide parking passes to those who need them. One parks, gets a ticket from the machine, then pays using the pass instead of using a cc.)

Travel Information: Indianapolis International Airport is the closest airport to the venue. O’Hare International Airport in Chicago is a ~3 hour drive from the conference venue.

Lunch and coffee: There’s a group of restaurants about a 15 minute walk down Indiana Ave, including Parlor Donuts which has good coffee. Map

There are also a few places along the canal (5-10 min walk): Fresco (small sandwich place), Burgeezy (excellent vegan burgers; fairly large), Black Leaf Vegan (small café). Map

There are lots of restaurants on Mass Ave, a 20 minute walk.

Dinner: India Garden (Saturday May 17, 6:30pm)

Schedule

Indianapolis is on Eastern Daylight Time (EDT)

Abstracts

Talks (one hour each)

Soren Galatius (Columbia University)

Title: The Grothendieck-Teichmüller group and the suspension spectrum of real projective space.

Abstract: In joint work with Brown, Chan, and Payne, we defined a filtered space reminiscent of the Waldhausen construction, but with graphs instead of projective modules.  The $E^1$-page of the associated rational homology spectral sequence was related to the Grothendieck-Teichmüller group.  I will review the definition of this space and discuss its integral homotopy type.

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Fedor Manin (University of Toronto)

Title: Persistent homology of function spaces

Abstract: Suppose you’re given two homotopic maps between spaces.  Does a homotopy between them have to pass through much more complicated maps?  More generally, if you have a homologically trivial cycle in a function space, is it possible to trivialize it in the subspace of maps that are not too complicated?  This can be thought of as studying the “Morse landscape” of a height function on the mapping space.  In joint work with Jonathan Block and Shmuel Weinberger, we make this precise using the language of persistent homology and give some first results.

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Daniel Minahan (University of Chicago)

Title: The second rational homology of the Torelli group

Abstract: The Torelli subgroup of the mapping class group of a surface is the subgroup acting trivially on the first homology of the surface. We will discuss recent joint work with Andrew Putman where we compute the second rational homology of the Torelli group for closed, orientable surfaces of genus at least 6.

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Morgan Opie (UCLA)

Title: A motivic approach to efficient generation of projective modules

Abstract: A classical question in commutative algebra is the following: given a finitely generated projective module M over a ring R, what is the minimal number of generators of M as an R-module? A classical theorem of Forster and Swan implies that, if R is of dimension d over a field k and M is of rank r, then M can always be generated by r+d elements. Work of Murthy shows that, if k is algebraically closed, the only obstruction to r+d-1 generation of M is vanishing of the top Segre class of M. I will report on an approach to this problem using motivic obstruction theory. This approach recovers and improves these classical bounds: we prove results depending only on the homotopy dimension of R over k, we remove hypotheses on the base field, and we study r+d-2 generation in certain cases. We also prove a symplectic Forster–Swan theorem. This is joint work with Aravind Asok, Brian Shin, and Tariq Syed

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Natalia Pacheco-Tallaj (MIT)

Title: Twisted string bordism in 7 dimensions with applications to anomaly cancellation

Abstract: In upcoming joint work with I. Basile, C. Krulewski and G. Leone, we use homotopic and topological techniques to study anomaly cancellation in 6d supergravity theories. In this talk, I will give a quick overview of anomalies of quantum field theories and their classification using twisted bordism groups. Then, I will overview how we leverage the unstable classification of vector bundles over CP^2 to construct explicit generators for 7-dimensional twisted string bordism groups, and I will show how to construct a complete family of index-theoretic invariants to detect the torsion in these groups.

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Dmitri Pavlov (Texas Tech)

Title: The classification of two-dimensional extended conformal field theories

Abstract: I will start by reviewing my recent work with Dan Grady on the geometric cobordism hypothesis and locality of fully extended nontopological functorial field theories. I will then apply these results to explicitly compute, in terms of homotopy coherent higher representations of Lie groups, the space of 2-dimensional fully extended conformal field theories.  If time permits, I will discuss further constructions of field theories (ongoing work with Dan Grady) that involve differential characteristic classes, index and eta-invariants, and quantization. No knowledge of physics or conformal field theory is assumed in this talk.

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Contributed talks (20 minutes each)

Hassan Abdallah (Wayne State University)

Title: Some Fivebrane Bordism Groups at the Prime 2

Abstract: Given a smooth n-dimensional manifold M, a tangential G-structure is a lift of the classifying map of the tangent bundle of M to BG. When G=String or G=Spin , this recovers the familiar notions of string and spin structures. A higher analog of these is a fivebrane structure, associated with the group G=Fivebrane = O<9>, named based on its relationship to certain ideas in physics. While much has been written about the bordism groups of spin and string manifolds, far less is known about the bordism groups of fivebrane manifolds. These groups are isomorphic, via the Pontryagin-Thom construction, to the homotopy groups of the spectrum MO<9>, which has connections to interesting phenomena in homotopy theory. For instance, MO<9> is speculatively a home for a higher-height analog of the Atiyah-Bott-Shapiro and Ando-Hopkins-Rezk orientations. In this talk, I will present calculations of the fivebrane bordism groups through dimension 22 at the prime 2, along with partial information through dimension 30, using the Adams spectral sequence.

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Fabio Capovilla-Searle (Purdue University)

Title: On the 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 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) \rightarrow 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.

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Ekansh Jauhari (University of Florida)

Title: On the cohomology and homotopy of symmetric products of surfaces

Abstract: Symmetric products of closed orientable surfaces are classically well-studied objects in topology and geometry. In this talk, I will discuss several new topological and geometric properties of these manifolds that arise from the study of their rational cohomology ring and second homotopy group. This is joint work with L. F. Di Cerbo and A. Dranishnikov.

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Johnathon Taylor (Case Western University)

Title: Fibrant Replacement, Distributive Series, and Infinity Groupoids

Abstract: Dimitri Ara, in his PhD thesis  wrote down a fully algebraic definition of infinity groupoid based on the sketch provided by Grothendieck in Pursuing Stacks. Grothendieck hypothesized that these fully algebraic infinity-groupoids model spaces, a statement now known as the Homotopy Hypothesis.  I will describe how to construct a free and contractible theory for infinity groupoids using distributive series of monads, as developed by Eugenia Cheng, and the standard tools of factorization systems.

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Yi Wang (Purdue University)

Title: Adams’ Cobar Construction and Chen’s Iterated Integrals Revisited

Abstract: Adams’ cobar construction (1950s) and Chen’s iterated integrals (1970s) provide two classical algebraic models for the based loop space and path space of a simply connected space. Adams’ original map goes from the cobar construction of the dg coalgebra of singular chains on a topological space to the cubical chains on its based loop space, while Chen’s original map goes from the bar construction of the dg algebra of differential forms on a differentiable space to differential forms on its based loop space. In this talk, I will explain how a formal dual of Chen’s map yields a chain homotopy inverse to Adams’ map for simply connected topological spaces. I will also present a new streamlined proof that Adams’ map is a quasi-isomorphism even when the space is not simply connected, as established by Rivera–Zeinalian in 2018. This is joint work with Manuel Rivera.

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David White (Denison University)

Title: Homotopical recognition of diagram categories

Abstract: Building on work of Marta Bunge in the one-categorical case, we characterize when a given model category is Quillen equivalent to a presheaf category with the projective model structure. This involves introducing a notion of homotopy atoms, generalizing the orbits of Dwyer and Kan. Apart from the orbit model structures of Dwyer and Kan, our examples include the classification of stable model categories after Schwede and Shipley, isovariant homotopy theory after Yeakel, and Cat-enriched homotopy theory after Gu. As an application, we give a classification of polynomial functors (in the sense of Goodwillie calculus) from finite pointed simplicial sets to spectra, and compare it to the previous work by Arone and Ching. This is joint work with Boris Chorny.

Organizers: Jeremy Miller (Purdue), Dan Ramras (IU Indianapolis), Manuel Rivera (Purdue University)