Jackson Morris

Jackson Morris

Mathematics PhD

Research

My research interests are in homotopy theory, motivated from the chromatic perspective, and the tools I use are often computational. Here are some things that I have been interested in recently.

I have written user’s guides to some of my papers that you can check out here.

Preprints/Publications

#TitleAuthorsLinksStatus
7.Motivic K-theory cooperations over the rationals
We compute the rings of cooperations for algebraic and hermitian K-theory over the $p$-adics and rationals, as well as for integral motivic cohomology. We further produce spectrum-level splittings for $\mathrm{kgl}\otimes$ and $\mathrm{H}\mathbb{Z}\otimes\mathrm{H}\mathbb{Z}$. As consequences of our work, we recompute the algebraic K-theory, hermitian K-theory, and motivic cohomology of $\mathbb{Q}_p$, of $\mathbb{Q}_2$, and of $\mathbb{Q}$.
Jackson Morris, Sarah Petersen, Liz Tatum In preparation
6.Periodicity in stable motivic homotopy theory
Over the last 50 years, many of the insights into stable homotopy theory have been made from the chromatic perspective. This persepctive organizes the category of spectra into more tractible localizations governed by the moduli stack of formal groups. Over the last 20 years, the field of motivic homotopy theory has developed into a far reaching subject bearing similarity with classical stable homotopy theory. In this survey article, we examine periodicity in motivic homotopy theory. We pay particular attention to the properties of classical spectra which analogize nicely to motivic spectra, new phenomena which is unique to the motivic setting, and computational tools which give a tactile way to understand the complexity of motivic homotopy theory.
Jackson Morris In preparation
5.Immersions of $\mathrm{C}_2$-projective spaces via $\mathrm{K}\mathbb{R}$-theory
We compute the Atiyah Real K-theory of $\mathrm{C}_2$-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces due to Bhattacharya–Waugh–Zeng–Zou, together with a variant of the localized slice spectral sequence introduced by Meier–Shi–Zeng. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.
Manyi Guo, Jackson Morris, Alex Waugh, Albert YangarXivSubmitted
4.Splittings of truncated motivic Brown-Peterson cooperation algebras
We construct spectrum-level splittings of $\text{BPGL}\langle 1\rangle\wedge \text{BPGL}\langle 1\rangle$ at all primes $p$, where $\text{BPGL}\langle 1 \rangle$ is the first truncated motivic Brown-Peterson spectrum. Classically, $\text{BP}\langle 1 \rangle \wedge \text{BP}\langle 1\rangle$ was first described by Kane and Mahowald in terms of Brown-Gitler spectra. This splitting was subsequently reinterpreted by Lellman and Davis-Gitler-Mahowald in terms of Adams covers. In this paper, we give motivic lifts of these splittings in terms of Adams covers, over the base fields $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{F}_q$, where $\text{char}(\mathbb{F}_q)\neq p$. As an application, we compute the $E_1$-page of the $\text{BPGL}\langle 1 \rangle$-based Adams spectral sequence as a module over $\text{BPGL}\langle 1 \rangle$, both in homotopy and in terms of motivic spectra. We also record analogous splittings for $\text{BPGL}\langle 0 \rangle \wedge \text{BPGL} \langle 0 \rangle$.
Jackson Morris, Sarah Petersen, Liz TatumarXivSubmitted
3.Rings of cooperations for hermitian K-theory over finite fields
We compute the ring of cooperations $\pi_{**}(\text{kq} \otimes \text{kq})$ for the very effective Hermitian K-theory over all finite fields $\mathbb{F}_q$ where $\text{char}(\text{F}_q) \neq 2$. To do this, we use the motivic Adams spectral sequence and show that all differentials are determined by the integral motivic cohomology of $\mathbb{F}_q$. As an application, we compute the $E_1$-page of the kq-resolution.
Jackson MorrisarXiv, User’s GuideSubmitted
2.On the ring of cooperations for real hermitian K-theory
Let $\text{kq}$ denote the very effective cover of the motivic hermitian K-theory spectrum. We analyze the ring of cooperations $\pi_{**}(\text{kq} \otimes \text{kq})$ in the stable motivic homotopy category $\text{SH}(\mathbb{R})$, giving a full description in terms of Brown-Gitler comodules. To do this, we decompose the $\text{E}_2$-page of the motivic Adams spectral sequence and show that it must collapse. The description of the $\text{E}_2$-page is accomplished by a series of algebraic Atiyah-Hirzebruch spectral sequences which converge to the summands of the $\text{E}_2$-page. Along the way, we prove a splitting result for the very effective symplectic K-theory $\text{ksp}$ over any base field of characteristic not two.
Jackson MorrisarXiv, User’s GuideAccepted to Annals of K-theory
1.Toric double determinantal varieties
We examine Li’s double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double determinantal varieties are smooth. We use this framework to give a straighforward formula for their dimension. Finally, we use the smallest nontrivial toric double determinantal variety to provide some empirical evidence concerning an open problem in local algebra.
Alexander Blose, Patricia Klein, Owen McGrath, Jackson MorrisarXivCommunications in Algebra (2021)
coolchart
The ring of cooperations $\pi_{**}(\text{kq} \otimes \text{kq})$ over $\mathbb{F}_3$.

Selected Recent Talks