jonalfcam.github.io

Published

• Devissage and Localization for the Grothendieck Spectrum of Varieties (joint w/ Inna Zakharevich). 2022.
  • Published in Advances
  • arxiv version
• Iterated Traces in Bicategories and Lefschetz Theorems (joint w/ Kate Ponto). 2022.
  • Published in AGT
  • arxiv version
  • We show an agreement between two different “iterated traces”, generalizing work of Ben-Zvi and Nadler. This can be used to prove a wide array of index-type theorems, as well as recover the modular invariance of categorical 2-characters.
• Topological Hochschild Homology and Higher Characteristics (joint w/ Kate Ponto). 2019.
  • Published in AGT
  • arxiv version
  • We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows, topological Hochschild homology, and Morita invariance in bicategorical generality.
• Derived Zeta Functions (joint w/ Jesse Wolfson and Inna Zakharevich). 2019.
  • Published in Advances
  • arxiv version
• The K-Theory Spectrum of Varieties. 2019.
  • Published in Transactions
  • arxiv version
  • The Grothendeick ring of varieties is a fundamental object of study in algebraic geometry. Using her formalism of assemblers, Inna Zhakarevich defined a spectrum whose $pi_0$ is this Grothendieck ring. By modifying techniques of Waldhausen, I give another definition of this spectrum, as well as producing maps out of it to other spectra of interest. These maps can be considered as various forms of “derived” motivic measures.
• A Guide for Computing Stable Homotopy Groups (joint w/ Agnes Beaudry). 2018.
  • Published in Topology and Quantum Field Theory in Interaction
  • arxiv version
  • This paper complements “Homotopy Theoretic Classification of Symmetry Protected Phases” (see below). We provide background on computations in homotopy theory, and compute a number of low-dimensional homotopy groups of cobordism spectra.

Accepted

• Riemann-Roch theorems in monoidal 2-categories (joint w/ Kate Ponto)
  • To appear in Quarterly Journal of Mathematics
  • arxiv

Submitted

• Hilbert’s Third Problem and a Conjecture of Goncharov (joint w/ Inna Zakharevich)
  • arxiv
• Spectral Waldhausen categories, the S∙-construction, and the Dennis trace (joint w/ John Lind, Cary Malkiewich, Kate Ponto, Inna Zakharevich)
  • arxiv
• K-theory of endomorphisms, the TR-trace, and zeta functions (joint w/ John Lind, Cary Malkiewich, Kate Ponto, Inna Zakharevich)
  • arxiv

In Preparation

• K-Theory with Squares (joint w/ Mona Merling and Inna Zakharevich)
• Translation Scissors Congruence (joint w/ Inna Zakharevich)
• Unnamed Projected on Witt Vectors

Preprints / Never prints

• Homotopy Theoretic Classification of Symmetry Protected Phases. 2017.
  • arxiv version
  • Though this contains some novel computations, this will likely never be published.
• Facets of the Witt Vectors.
  • arxiv
  • An exposition of the Witt vectors contains a (as far as I know), novel norm map.
• Topological Hochschild Homology and Koszul Duality -arxiv