# Homotopy Seminar @OSU

The Homotopy Seminar @OSU is currently co-organized by myself and Zeshen Gu. The Seminar meeting time is 3-4pm Thursdays in MW152 unless otherwise advertised.

## Autumn 2024

Date | Details |
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Sep 5, 2024 | Fields in Equivariant AlgebraNoah Wisdom (Northwestern) |

Tambara functors generalize finite Galois extensions and are of independent interest to equivariant homotopy theorists, appearing as the homotopy groups of ring spectra. Nakaoka defined the notion of a field-like Tambara functor (for which Galois extensions provide an example). In this talk, we’ll introduce Tambara functors and classify field-like Tambara functors for the group \(C_{p^n}\). Additionally, in collaboration with Ben Spitz and Jason Schuchardt, we will study what it means for such a Tambara functor to be algebraically closed, or “Nullstellensatzian” (in the sense of Burklund-Schlank-Yuan), with an eye towards an equivariant chromatic Nullstellensatz. | |

Sep 19, 2024 | Fiber of the cyclotomic trace of the sphere spectrum and K-theoretic Tate-Poitou duality at prime 2.Myungsin Cho (Indiana Bloomington) |

Understanding the algebraic K-theory of the sphere spectrum has long been recognized as a fundamental problem in algebraic and differential topology. Since the homotopy fiber of its p-completed cyclotomic trace depends only on the zeroth homotopy group, we can apply algebraic methods to study it. Blumberg and Mandell’s work demonstrates that, for odd primes, Tate-Poitou duality can be enhanced to an Anderson duality between the homotopy fiber and the K(1)-local K-theory of the integers. In this talk, I will present this connection and extend the result to the case where p=2. | |

Oct 3, 2024 | Model structures on operads and algebras from a global perspectiveDavid White (Denison) |

I will report on joint work with Michael Batanin and Florian De Leger studying the homotopy theory of the Grothendieck construction, given a category B and a functor F from B^op to CAT. We introduce powerful new techniques for constructing transferred model structures on Grothendieck constructions, such as the category of pairs (P,A) where P is a (symmetric or non-symmetric) colored operad and A is a P-algebra or a left P-module. From such a (semi-)model structure on pairs, we produce a “horizontal” (semi-)model structure on the base B and “vertical” (semi-)model structures on the fibers F(P). This recovers in a unified framework all known (semi-)model structures on categories of operads and their algebras, and produces new model structures, e.g., on the category of twisted modular operads. Additionally, we study when these model structures are left proper, and when a weak equivalence in the base B gives rise to a Quillen equivalence of fibers. Applications include change of rings, rectification of operad-algebras, and strictification for categorical structures. The relevant arXiv papers are at arXiv:2311.07320 and arXiv:2311.07322. | |

Dec 5, 2024 | TBDLucas Williams (Binghamton) |

TBD |

## Past Semesters

Spring 2024 Talks

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Date | Details |
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Jan 30, 2024 | The homotopy theory of ideals structured by operadsDavid White (Denison) |

In 2006, Jeff Smith gave a talk explaining how to define ideals in the context of ring spectra, and laying out applications to algebraic K-theory computations. In 2014, Mark Hovey posted a preprint working out the homotopy theory of such ideals. Since every map is homotopic to an inclusion, ideals must be defined diagrammatically rather than as subobjects. In this talk, I will explain the definition, and then discuss joint work with Donald Yau that extends Hovey’s work to develop a homotopy theory of ideals with additional structure encoded by an operad. If there’s time, I’ll discuss ongoing work in progress to carry out Smith’s plan regarding algebraic K-theory. | |

March 5, 2024 | Equivariantly enriched curve countingCandace Bethea |

Enumerative geometry asks for integral solutions to geometric questions, such as how many genus g, degree d rational curves with n marked points lie on a given surface. Recently, motivic and equivariant homotopy have been used to generalize classical enumerative results to non-closed fields and under the presence of a group action respectively. In this talk, we will discuss the connections between equivariant homotopy theory and equivariant curve counting, specifically giving a count of nodal orbits in an invariant pencil of plane conics enriched in the Burnside Ring of a finite group. Time permitting, we will discuss joint work in progress with Kirsten Wickelgen on defining a local degree using stable equivariant homotopy theory and an application to Euler numbers. | |

March 19, 2024 | Synthetic cyclotomic spectra and the even filtrationNoah Riggenbach (Northwestern) |

Synthetic spectra, as defined by Pstrągowski, are infinity categories which act as deformations of the usual stable homotopy category. Having such deformations has made it possible to push our current techniques developed in the stable category much farther by applying them in synthetic spectra and forgetting down to spectra. In this talk I will discuss work, joint with Ben Antieau, which defines a similar deformation of cyclotomic spectra. As applications I will discuss the unification of the different filtrations, all of which are called the even filtration, constructed by Hahn-Raksit-Wilson. | |

March 26, 2024 | ell-adic topological Jacquet-Langlands dualityAndrew Salch (Wayne State) |

I will describe the construction of an “ell-adic topological Jacquet-Langlands (TJL) dual” of each finite CW-complex. One effect of the TJL dual is that the representations of the Morava stabilizer group, which arise in the height n Morava E-theory of a finite spectrum X, get turned into continuous actions of GL_n(Z_p) on the ell-adic TJL dual of X. On the level of homotopy groups, these actions of GL_n(Z_p) correspond, via the classical ell-adic Jacquet-Langlands correspondence, to the representations of the Morava stabilizer group on the Morava E-theory groups of X. When n=1, these representations of GL_1 have associated automorphic L-factors. I will explain how the product of those L-factors, i.e., the “automorphic L-function of a finite CW-complex X,” analytically continues to a meromorphic function on the complex plane whose special values in the left half-plane recover the orders of the KU-local stable homotopy groups of X. Part of the construction of the TJL dual is a solution to an old problem on construction of E_\infty-ring realizations of the Lubin-Tate tower. I will describe that solution, which is joint work with Matthias Strauch. | |

April 2, 2024 | Detecting exotic spheres and their symmetries with stable homotopy theoryJ.D. Quigley (Virginia) |

An exotic n-sphere is a smooth n-manifold which is homeomorphic, but not diffeomorphic, to the n-sphere with its standard smooth structure. Exotic spheres are fundamental objects in algebraic and geometric topology, but even after decades of intense study, there are many simple-sounding questions about them which are unanswered. In which dimensions do exotic spheres exist? Can you rotate an exotic sphere as you would a standard sphere? Are exotic spheres “round” like the standard sphere? In this talk, I will survey what is known about exotic spheres and their geometry. I will then report on joint work with Mark Behrens and Mark Mahowald, Prasit Bhattacharya and Irina Bobkova, and Boris Bovtinnik on detecting exotic spheres and their smooth symmetries. | |

April 23, 2024 | Talk by Alicia LimaAlicia Lima (Chicago) |

In this talk, I will discuss recent developments in the study of the exotic Picard group of the K(p−1)-local category – one of the building blocks of the category of spectra Sp – at a general odd prime p. Unlike the category of spectra Sp, where shifts of the sphere spectrum are the only invertible objects, the K(n)-local category exhibits more unusual invertible objects. In collaboration with Bobkova, Lachmann, Li, Stojanoska, and Zhang, we have established bounds on the descent filtration of the exotic Picard group of the K(p−1)-local category for prime p>3. Furthermore, we also deduced that the K(p−1)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at 3(p−1)^2+1 on the E_2(p−1)^2+2-page. | |

May 8, 2024 | A classification of stable symmetric monoidal model categoriesBoris Chorny (Haifa) |

Since the 2003 paper by Schwede and Shipley about the classification of stable model categories as categories of modules over a ``ring spectrum with several objects’’, the question of specification of this result for symmetric monoidal model categories remained open, although Lurie solved it in the stable infinity category settings. In this talk we will present a classification of stable symmetric monoidal model categories (with the monoidal unit being a compact generator) as the categories of modules over a commutative ring spectrum, up to Quillen equivalence. Joint work with Ozgur Bayindir. | |

May 15, 2024 | Small functors in homotopy theoryBoris Chorny (Haifa) |

A functor is small if it is a left Kan extension of from a small subcategory of the domain or, equivalently, if it is a small colimit of representable functors. We will present several ways to do homotopy theory on small functors taking values in simplicial sets or spectra. The applications include: equivariant homotopy theory, calculus of funtors, representability theorems up to homotopy. | |

May 22, 2024 | A classification of small linear functorsBoris Chorny (Haifa) |

We will continue the discussion of applications of small functors to representability theorems up to homotopy, calculus of functors, and finish by a classification of small linear functors from spectra to spaces. |

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Autumn 2023 Talks

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Date | Details |
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Oct 10, 2023 | Equivariant Dunn AdditivityBen Szczesny |

The Boardman-Vogt tensor product of operads encodes the notion of interchanging algebraic structures. A classic result of Dunn tells us that the tensor product of two little cube operads is equivalent to a little cube operad with the dimensions added together. As models for \(\mathbb{E}_k\)-operads, this reflects a defining property of these operads. In this talk, we will explore some equivariant generalizations to Dunn’s additivity. Along the way, we will play with little star-shaped operads, question if we really need group representations for equivariant operads, and learn to love (and hate) the tensor product. | |

Oct 19, 2023 | A 3-dimensional TFT with many applications Lorenzo Riva (University of Notre Dame) |

Topological field theories live in the intersection of algebraic topology and mathematical physics. They initially served as a rigorous foundation for some aspects of the path integrals appearing in physics, especially quantum field theory (hence the name), and then were extensively studied in conjunction with mirror symmetry and gauge theories. They’re defined formally as monoidal functors from a higher category of bordisms and they are a source of examples for invariants of manifolds (topological or with geometric structures), are connected to representation theory via tensor categories, and serve as a sort of “testing ground” for our understanding of higher categories. In this talk I will forgo the physics aspect completely and instead focus on the general theory, along with an introduction to the higher categorical framework that allows us to define these objects. We will then see an example of a 3-dimensional TFT which is supposed to compute some so-called Rozansky-Witten invariants of 3-manifolds. | |

Nov 2, 2023 | Mackey and Tambara Functors Beyond Equivariant Homotopy Ben Spitz (University of California, Los Angeles) |

“Classically”, Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. More recently, work by Bachmann has garnered interest in related structures which appear in motivic homotopy theory – these Motivic Mackey Functors and Motivic Tambara Functors do not have anything to do with group-equivariance, but have the same axiomatics. In this talk, I’ll introduce a general context for interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions. The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs of essential facts. To this end, I’ll discuss recent progress in boosting a foundational result about norms from equivariant algebra to this more general context. | |

Nov 16, 2023 | A scanning construction of the self duality of the little disks operad Connor Malin (University of Notre Dame) |

Since the introduction of Koszul duality for operads, algebraists and homotopy theorists alike have wanted to understand its effect on the little disks operad En. Over the course of the last 30 years, our knowledge has steadily advanced. People showed (roughly chronologically) that H_(En),C_(En;Q),C_*(En;Z), and the categories of En algebras themselves were all self dual, finally culminating in the fact that the En operad in spectra is self dual. In this talk, we give a new, particularly tractable construction of the self duality of the spectral En operad and describe how it interacts with factorization homology. | |

Dec 7, 2023 | Telescopic stable homotopy theory Ishan Levy (MIT) |

Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel’s telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski. |

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