**AMTRaK** (**A**tlantic **M**eeting on **T**opology, **R**epresentation theory, **a**nd **K**-theory) is a day-long seminar aimed at algebraic topologists working in the mid-Atlantic and Northeast region. We encourage participants to carpool or travel via train (although Amtrak is not required) to minimize environmental impact.

This seminar is meant to boost connections within the algebraic topology community in the mid-Atlantic and Northeast region, particularly amongst graduate students. Each meeting will focus on a different topic of contemporary research interest, with two advanced talks in the afternoon and two “pre-talks” in the morning.

We are planning to hold two meetings in Fall 2024, one at the University of Pennsylvania on Friday, Sept. 20th and one at Johns Hopkins University on Friday, Nov. 8th. The proposed topics are parametrized equivariant homotopy theory and the *K*-theory of dualizable categories, respectively. See the Fall 2024 poster

**November 2024 – Dualizable categories and continuous K-theory**

November 8th at Johns Hopkins University

Register for the November meeting

Please register by November 6th to attend the meeting and by October 20th to request funding/lodging. We expect to be able to provide meals and refreshments and have limited funding available for travel expenses.

A more detailed list of talks, speakers, and schedule will be posted at a later date.

Pre-talk 1: Introduction to dualizable categories and continuous K-theory

**Speaker:** TBD

**Abstract:** TBD

Pre-talk 2: Introduction to rigidity and six functor formalism

**Speaker:** TBD

**Abstract:** TBD

Talk 1: TBD

**Speaker:** Liam Keenan

**Abstract: **TBD

Talk 2: TBD

**Speaker:** Marco Volpe

**Abstract:** TBD

**September 2024 – Equivariant and parametrized homotopy theory**

September 20th at the University of Pennsylvania

Register for the September meeting

Please register by September 18th to attend the meeting and by September 13th to request funding. We expect to be able to provide meals and refreshments and have limited funding available for travel expenses.

Pre-talk 1: Introduction to Equivariant Homotopy Theory

**Speaker:** Albert Yang

In this talk, we will review the definition of the stable and unstable categories of genuine *G*-spectra for a finite group *G*, using the perspective of spectral Mackey functors. We will also define the Hill-Hopkins-Ravenel norm on genuine *G*-spectra and (if time permits) the isotropy separation sequence.

References: Sections 1-2 of these notes by Ramzi, Section 9 of this paper by Bachmann-Hoyois, Part 1 of this paper by Mathew-Naumann-Noel, and Appendix A of Nardin's thesis.

Pre-talk 2: Parametrized Equivariant Stable Homotopy Theory

**Speaker:** Maxine Calle

In this talk, we will explain how the ∞-categories of *G*-spaces and *G*-spectra upgrade to the parametrized ∞-categorical setting. We will define genuine *C _{p}*-

*E*-operads after Nardin-Shah and discuss how being a genuinely

_{∞}*G*-commutative monoid gives rise to norm maps.

References: Section 2 of this paper by Yang, Sections 2.1 and 4.1 of this paper by Cnossen, and Sections 2.1-2.4 of this paper by Nardin-Shah.

Talk 1: *G*-commutative monoids and sheaves on *G*-manifolds

**Speaker:** Andres Mejia

**Abstract: **In this talk we will explain how one can adapt (semi)-recent techniques from parametrized homotopy theory to the setting of sheaves on equivariant manifolds. More specifically, one can recover the category of spaces via the category of sheaves on manifolds. An interesting question is what commutative monoids are in this setting. One guess is that it’s those that possess a covariant functoriality known as a transfer with respect to bundles with compact manifold fiber, but in fact we need much less: it suffices to have transfers along finite covering maps. Quillen conjectured that this was the case and equipped with the tools of Bachmann-Hoyois, the proof of this statement is relatively formal. Recent advances in parametrized equivariant homotopy theory as well as the properties of equivariant sheaves on *G*-manifolds allows one to assemble these proofs together equivariantly. This talk aims to tell this story in a way that makes a case for the formalism of parametrized equivariant homotopy theory. 🙂

Talk 2: Normed E_{∞}-rings in genuine equivariant *C _{p}*-spectra

**Speaker:** Lucy Yang

**Abstract:** Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an *E _{∞}*-algebra. Our work concerns the

*C*-

_{p}*E*-algebras of Nardin–Shah with respect to a cyclic group

_{∞}*C*of prime order. We show that many of the higher coherences inherent to the definition of parametrized algebras collapse; in particular, they may be described more simply and conceptually in terms of ordinary

_{p}*E*-algebras as a diagram category which we call normed algebras. Our main result provides a relatively straightforward criterion for identifying

_{∞}*C*-

_{p}*E*-algebra structures. We visit some applications of our result to real motivic invariants.

_{∞}**Contact**

Please contact the organizers with any questions, comments, or concerns:

Maxine Calle (**call **m**e** **m**aybe AT** sas** DOT **upenn** DOT** edu**)

Anish Chedalavada (**a**xolotl **c**arnivorous **hedal1 **AT **jh **DOT **edu**)

This seminar is made possible thanks to the generous support of the Mathematics Departments at UPenn and JHU. We would especially like to thank Nir Gadish, David Gepner, and Mona Merling for their support.