AMTRaK

AMTRaK (the Atlantic Meeting on Topology, Representation theory, and 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.

The conference is organized by Riley Shahar (Penn) and Nooria Ahmed (Hopkins). Previous organizers include Maxine Calle, Anish Chedalavada, and Ben Spitz. The conference would not be possible without the generous support of many people. Supporters of past conferences include Nir Gadish, David Gepner, and Mona Merling; the Mathematics Departments at Penn, Hopkins, and UVA; and the K-theory foundation.

Past Conferences

AMTRaK V (April 2026, University of Pennsylvania): Scissors Congruence K-Theory

AMTRaK V was held on April 17, 2026 at the University of Pennsylvania.

The conference was themed around Scissors Congruence K-Theory and featured research talks by Maru Sarazola and Cary Malkiewich, with pretalks by Fangji Liu and Mattie Ji. It was made possible thanks to the generous support of Nir Gadish and Mona Merling, as well as the Mathematics Department at Penn.

Schedule

Time	Event	Location
9-10	Breakfast	DRL A6
10-11	Pretalk 1: Fangji Liu	DRL A6
11:30-12:30	Pretalk 2: Mattie Ji	LRSM 112B
12:30-2	Lunch	—
2-3	Talk 1: Maru Sarazola	DRL A4
3:15-3:45	Department tea	DRL 4E17
4-5	Talk 2: Cary Malkiewich	DRL A4
5 onward	Dinner and hanging out	—

Talks

Maru Sarazola (University of Minnesota): K-theory of manifolds up to cut-and-paste Toggle abstract for K-theory of manifolds up to cut-and-paste1

The classical problem of scissors congruence for polytopes admits a natural translation to the world of manifolds: given two manifolds M and N, one can study whether it’s possible to cut M into pieces and reassemble them to obtain N. Recent advances have made it possible to interpret this problem within the framework of algebraic K-theory, providing an exciting new perspective on this problem that upgrades this classical algebraic invariant to the homotopical realm.

While the Grothendieck group of this K-theory spectrum is well-understood, not much else is known in terms of its higher homotopical information. In this talk, I will describe joint work with Maxine Calle where we provide a new simplicial model for cut-and-paste K-theory in the spirit of Waldhausen’s S-dot construction. We will discuss our progress in computing $K_1$ of this spectrum, and how it encodes information about the cut-and-paste automorphisms of manifolds.
Cary Malkiewich (Binghamton University): Higher scissors congruence revisited Toggle abstract for Higher scissors congruence revisited2

Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. In the 2010s Zakharevich defined a “higher” version of scissors congruence via K-theory, and in the early 2020s my collaborators and I showed that Zakharevich’s scissors congruence K-theory is a Thom spectrum, allowing us to make many new computations of the higher K-groups.

It’s rare for K-theory to have such a nice description. Why does it happen in this case? In this talk I’ll discuss the proof, and a couple of ways that our understanding of the proof has changed in the last few years. In particular, I’ll describe a generalization that restricts to polytopes whose facets lie along any prescribed set of hyperplanes. This makes more computations possible, including a recent computation by Holley for all polygons in the Euclidean plane whose edges have rational slope.

AMTRaK IV (April 2025, University of Pennsylvania): Homological Stability

AMTRaK IV was held on April 4, 2025 at the University of Pennsylvania.

The conference was themed around homological stability and featured research talks by Søren Galatius and Nir Gadish, with pretalks by Fangji Liu and Riley Shahar.

Schedule

Time	Event	Location
9:45-10	Welcome and coffee	DRL A1
10-11	Pretalk 1: Fangji Liu	DRL A1
11-12	Pretalk 2: Riley Shahar	DRL A1
12-2	Lunch	—
2-3	Talk 1: Søren Galatius	DRL A4
3:15-3:45	Tea	DRL 4E17
4-5	Talk 2: Nir Gadish	DRL A4
5 onward	Dinner and hanging out	—

Talks

Fangji Liu (University of Pennsylvania): Homological stability and Quillen's argument Toggle abstract for Homological stability and Quillen's argument1

This talk will introduce the ideas of homological stability and highlight some important examples. The second part of the talk will focus on Quillen’s spectral sequence argument for the homological stability of symmetric groups.
Riley Shahar (University of Pennsylvania): Connectivity of simplicial complexes: A toolbox Toggle abstract for Connectivity of simplicial complexes: A toolbox2

This talk will elaborate on the connectivity arguments that were swept under the rug during pre-talk 1. In particular, we will focus on tools and techniques coming from discrete Morse theory. After introducing the basics of discrete Morse theory, we will use it to prove the Solomon-Tits Theorem.
Søren Galatius (Columbia University): Homological stability for moduli spaces of manifolds Toggle abstract for Homological stability for moduli spaces of manifolds3

I will discuss an instance of homological stability which is not about homology of discrete groups, but whose proof nevertheless is very similar to that for the symmetric groups, discussed in section 2 of the notes by Kupers mentioned in pre-talk 1.
Nir Gadish (University of Pennsylvania): Homological stability from the point of view of cells Toggle abstract for Homological stability from the point of view of cells4

Virtually every homological stability argument proceeds by showing that some (semi-)simplicial complex is highly connected. Finding the right complex to use is somewhat of an art, but they usually follow a certain pattern of being the collection of “destabilizations” of an object. But where do these complexes come from? And what are they measuring?

In this talk I will present a recent observation of Randal-Williams, that the complexes of destabilizations in fact measure the $E_2$ -cells needed to construct the classifying spaces of the groups in question, and thus most examples of homological stability are a consequence of stability for the $E_2$ -operad.

AMTRaK III (February 2025, University of Virginia): Hermitian K-theory and Motivic Homotopy Theory

AMTRaK III was held on February 28, 2025 at the University of Virginia.

The conference was themed around Hermitian K-theory and motivic homotopy theory and featured two research talks by Thomas Brazelton, with pretalks by Albert Yang and Ben Spitz.

Schedule

Time	Event	Location
10-10:50	Welcome and coffee	Kerchof 314
11-12	Pretalk 1: Albert Yang	New Cabell 323
12-1	Pretalk 2: Ben Spitz	New Cabell 323
1-2:30	Lunch	—
2:30-3:30	Talk 1: Thomas Brazelton	New Cabell 323
4-5	Talk 2: Thomas Brazelton	New Cabell 323
5 onward	Dinner and hanging out	—

Talks

Albert Yang (University of Pennsylvania): Introduction to Motivic Homotopy Theory Toggle abstract for Introduction to Motivic Homotopy Theory1

This talk will introduce the basic definitions of motivic homotopy theory, such as Zariski and Nisnevich descent, along with key examples of motivic spaces including Eilenberg–Maclane spaces.
Ben Spitz (University of Virginia): Postnikov Towers and Obstruction Theory Toggle abstract for Postnikov Towers and Obstruction Theory2

This talk will introduce the construction and properties of Postnikov towers, the basic ideas of obstruction theory, and connections to characteristic classes.
Thomas Brazelton (Harvard University): A leisurely introduction to Hermitian K-theory Toggle abstract for A leisurely introduction to Hermitian K-theory3

We will motivate and introduce the basic ideas of Hermitian K-theory, highlighting the similarities and differences with algebraic K-theory of rings. This will be approached from two perspectives: one being a stripped-down low-tech version of the (very high-tech) story of Poincare $\infty$ -categories, and the other being the classical picture of the K-theory of exact categories with duality.

Finally we’ll discuss how Hermitian K-theory can be promoted to a motivic spectrum, which is used as a crucial tool to access stable and unstable motivic homotopy groups of spheres.
Thomas Brazelton (Harvard University): Rank two algebraic vector bundles on smooth affine fourfolds Toggle abstract for Rank two algebraic vector bundles on smooth affine fourfolds4

Given a finite CW complex, it is a folklore result in obstruction theory that it admits only finitely many isomorphism classes of complex vector bundles with some prescribed Chern classes. Given a smooth affine variety of finite Krull dimension over a field, we can ask an analogous question for algebraic bundles, and the answer is mostly unknown.

However Morel’s “affine representability” theorem indicates that these sorts of algebraic questions are amenable to techniques from motivic obstruction theory. Following work of Mohan Kumar, Murthy and others in the 20th century, as well as the contemporary research program of Asok and Fasel, we understand that the complexity of these sorts of questions are governed by two factors: (1) the 2-cohomological dimension of the base field, and (2) the corank (dimension of base minus rank of bundle).

In joint work with Opie and Syed we explore corank two in the first interesting setting. We explore to what extent Chow-valued Chern classes and Chow-Witt-valued Euler classes uniquely classify algebraic vector bundles over smooth affine fourfolds over an algebraically closed field.

AMTRaK II (November 2024, Johns Hopkins University): Dualizable categories and continuous K-theory

AMTRaK II was held on November 8, 2024 at Johns Hopkins University.

The conference was themed around dualizable categories and continuous K-theory and featured research talks by Liam Keenan and Marco Volpe, with pretalks by Nooria Ahmed and Anish Chedalavada.

Schedule

Time	Event	Location
9:30-10	Welcome and coffee	Krieger 413
10-11	Pretalk 1: Nooria Ahmed	Krieger 411
11:30-12:30	Pretalk 2: Anish Chedalavada	Krieger Hall
12:30-2	Lunch	—
2-3:15	Talk 1: Liam Keenan	Krieger 411
3:30-4:45	Talk 2: Marco Volpe	Krieger Hall
5 onward	Dinner and hanging out	—

Talks

Nooria Ahmed (Johns Hopkins University): Introduction to dualizable categories and continuous K-theory Toggle abstract for Introduction to dualizable categories and continuous K-theory1

This talk will review the definition of dualizable categories, and explain (but not prove!) the characterization in terms of compactly exhaustible objects. We will use this to recall the dualizability of $Shv(X,\mathrm{Sp})$ and (time permitting) sketch a proof.
Anish Chedalavada (Johns Hopkins University): Introduction to rigidity and six functor formalism Toggle abstract for Introduction to rigidity and six functor formalism2

In this talk, we will review Verdier duality, construct the six operations, and define local rigidity and rigidity in connection with Verdier duality.
Liam Keenan (Brown University): Dualizable categories and trace methods — a sampler Toggle abstract for Dualizable categories and trace methods — a sampler3

Algebraic K-theory, topological Hochschild homology (THH), topological cyclic homology (TC), and topological restriction homology (TR), are all examples of localizing invariants. In this talk, I will sample some of the ways Efimov’s theory of dualizable categories and the category of localizing motives interact with these specific invariants.

In particular, I will discuss (1) the interaction of algebraic K-theory and infinite products of categories (originally studied by Kasprowski—Winges) and (2) how to express THH of any dualizable category completely algebraically. Time permitting, I will discuss some other applications and pose a question or two.
Marco Volpe (Universität Regensburg): K-theory of sheaves Toggle abstract for K-theory of sheaves4

Let X be a locally compact Hausdorff space, C a dualizable stable $\infty$ -category, and denote by $Shv(X, C)$ the $\infty$ category of C-valued sheaves on X. In this talk, we explain (in as much detail as time permits) how to compute the K-theory of Shv(X, C) in terms of the compactly supported cohomology of X with coefficients in the spectrum $K(C)$ .

The strategy we adopt, which was suggested by Dustin Clausen, consists in providing an axiomatic characterization of sheaf cohomology on compact Hausdorff spaces. If time permits, we will also outline applications to simple homotopy theory and functoriality of Becker-Gottlieb transfers.

AMTRaK I (September 2024, University of Pennsylvania): Equivariant and parametrized homotopy theory

AMTRaK I was held on September 20, 2024 at the University of Pennsylvania.

The first AMTRaK was themed around equivariant and parametrized homotopy theory and featured research talks by Andres Mejia and Lucy Yang, with pretalks by Albert Yang and Maxine Calle.

Schedule

Time	Event	Location
9:30-10	Welcome and coffee	DRL 4E17
10-11	Pretalk 1: Albert Yang	DRL A1
11-12	Pretalk 2: Maxine Calle	DRL A1
12-1	Lunch	—
1-2	Talk 1: Andres Mejia	DRL 2N36
2-3	Talk 2: Lucy Yang	DRL 2N36
3:15-4	Math Department tea	DRL 4E17
4-5:30	Q&A / Math jam	DRL 4E17
5:30 onward	Dinner and hanging out	—

Talks

Albert Yang (University of Pennsylvania): Introduction to Equivariant Homotopy Theory Toggle abstract for Introduction to Equivariant Homotopy Theory1

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.
Maxine Calle (University of Pennsylvania): Parametrized Equivariant Stable Homotopy Theory Toggle abstract for Parametrized Equivariant Stable Homotopy Theory2

In this talk, we will explain how the $\infty$ -categories of G-spaces and G-spectra upgrade to the parametrized $\infty$ -categorical setting. We will define genuine $C_{p}$ - $E_\infty$ -operads after Nardin-Shah and discuss how being a genuinely G-commutative monoid gives rise to norm maps.
Andres Mejia (University of Pennsylvania): G-commutative monoids and sheaves on G-manifolds Toggle abstract for G-commutative monoids and sheaves on G-manifolds3

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.
Lucy Yang (Columbia University): Normed E∞-rings in genuine equivariant C_{p}-spectra Toggle abstract for Normed E∞-rings in genuine equivariant C_{p}-spectra4

Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an $E_\infty$ -algebra. Our work concerns the $C_{p}$ - $E_\infty$ -algebras of Nardin–Shah with respect to a cyclic group $C_{p}$ 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 $E_\infty$ -algebras as a diagram category which we call normed algebras. Our main result provides a relatively straightforward criterion for identifying $C_{p}$ - $E_\infty$ -algebra structures. We visit some applications of our result to real motivic invariants.