Kyle Ormsby
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View article: Enumerating submonoids of finite commutative monoids
Enumerating submonoids of finite commutative monoids Open
Given a finite commutative monoid $M$, we show that submonoids of $M\times [n]$ - where $[n] = \{0,1,\ldots,n\}$ is equipped with the max operation $\vee$ - may be enumerated via the transfer matrix method. When $M$ is also idempotent, we …
View article: A motivic analogue of the $K(1)$-local sphere spectrum
A motivic analogue of the $K(1)$-local sphere spectrum Open
We identify the motivic KGL/2 -local sphere as the fiber of \psi^{3}-1 on (2,\eta) -completed Hermitian K -theory, over any base scheme containing 1/2 . This is a motivic analogue of the classical resolution of the K(1) -local sphere, and …
View article: The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$
The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$ Open
We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct $N_{\infty} $ operads for dihedral groups $D_{p^n}$ , $p \gt 2$ prime, and cyclic gro…
View article: Hochschild homology of mod-𝑝 motivic cohomology over algebraically closed fields
Hochschild homology of mod-𝑝 motivic cohomology over algebraically closed fields Open
We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod- motivic Steenrod a…
View article: Homotopical Combinatorics
Homotopical Combinatorics Open
Communities summer conference Homotopical Combinatorics, one of four topical research conferences offered this year that are focused on collaborative research and professional development for early-career mathematicians.Additional informat…
View article: Topological Network Analysis of Beta‐Amyloid and Tau in Alzheimer’s Disease Using PET Imaging Data
Topological Network Analysis of Beta‐Amyloid and Tau in Alzheimer’s Disease Using PET Imaging Data Open
Background Alzheimer’s disease (AD) is characterized by the presence of beta‐amyloid (Aβ) and tau in the brain. Understanding the interplay between these proteins and their spatial distribution may provide insights into disease progression…
View article: Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games
Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games Open
We prove that Hill's characteristic function $χ$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $χ$ have unique maxima which are exactly the saturated transfer systems. In order to a…
View article: A motivic analogue of the K(1)-local sphere spectrum
A motivic analogue of the K(1)-local sphere spectrum Open
We identify the motivic $KGL/2$-local sphere as the fiber of $ψ^3-1$ on $(2,η)$-completed Hermitian $K$-theory, over any base scheme containing $1/2$. This is a motivic analogue of the classical resolution of the $K(1)$-local sphere, and e…
View article: The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$
The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$ Open
We provide a general recursive method for constructing transfer systems on finite lattices. Using this we calculate the number of homotopically distinct $N_\infty$ operads for dihedral groups $D_{p^n}$, $p > 2$ prime, and cyclic groups $C_…
View article: Lifting $N_\infty$ operads from conjugacy data
Lifting $N_\infty$ operads from conjugacy data Open
We isolate a class of groups -- called lossless groups -- for which homotopy classes of $G$-$N_\infty$ operads are in bijection with certain restricted transfer systems on the poset of conjugacy classes $\operatorname{Sub}(G)/G$.
View article: Composition closed premodel structures and the Kreweras lattice
Composition closed premodel structures and the Kreweras lattice Open
We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization system…
View article: Hochschild homology of mod-$p$ motivic cohomology over algebraically closed fields
Hochschild homology of mod-$p$ motivic cohomology over algebraically closed fields Open
We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating fu…
View article: Self-duality of the lattice of transfer systems via weak factorization systems
Self-duality of the lattice of transfer systems via weak factorization systems Open
For a finite group $G$, $G$-transfer systems are combinatorial objects which encode the homotopy category of $G$-$N_\infty$ operads, whose algebras in $G$-spectra are $E_\infty$ $G$-spectra with a specified collection of multiplicative nor…
View article: Saturated and linear isometric transfer systems for cyclic groups of order $p^mq^n$
Saturated and linear isometric transfer systems for cyclic groups of order $p^mq^n$ Open
Transfer systems are combinatorial objects which classify $N_\infty$ operads up to homotopy. By results of A. Blumberg and M. Hill, every transfer system associated to a linear isometries operad is also saturated (closed under a particular…
View article: Model structures on finite total orders
Model structures on finite total orders Open
We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial …
View article: Self-duality of the lattice of transfer systems via weak factorization\n systems
Self-duality of the lattice of transfer systems via weak factorization\n systems Open
For a finite group $G$, $G$-transfer systems are combinatorial objects which\nencode the homotopy category of $G$-$N_\\infty$ operads, whose algebras in\n$G$-spectra are $E_\\infty$ $G$-spectra with a specified collection of\nmultiplicativ…
View article: Biased permutative equivariant categories
Biased permutative equivariant categories Open
For a finite group G, we introduce the complete suboperad Q G of the categorical G-Barratt-Eccles operad P G .We prove that P G is not finitely generated, but Q G is finitely generated and is a genuine E ∞ G-operad (i.e., it is N ∞ and inc…
View article: The homotopy groups of the η-periodic motivicsphere spectrum
The homotopy groups of the η-periodic motivicsphere spectrum Open
We compute the homotopy groups of the {\eta}-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that …
View article: Biased permutative equivariant categories
Biased permutative equivariant categories Open
For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it…
View article: 1-smooth pro-p groups and Bloch-Kato pro-p groups
1-smooth pro-p groups and Bloch-Kato pro-p groups Open
Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a homomorphism of pro-$p$ groups $G\to1+p\mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-$p$ Galois groups of f…
View article: On the ring of cooperations for 2‐primary connective topological modular forms
On the ring of cooperations for 2‐primary connective topological modular forms Open
We analyze the ring tmf_*tmf of cooperations for the connective spectrum of\ntopological modular forms (at the prime 2) through a variety of perspectives:\n(1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a\ndecompositi…
View article: Vanishing in stable motivic homotopy sheaves
Vanishing in stable motivic homotopy sheaves Open
We determine systematic vanishing regions for the bigraded homotopy sheaves of the motivic sphere spectrum over a field of characteristic different from two.
View article: VANISHING IN STABLE MOTIVIC HOMOTOPY SHEAVES
VANISHING IN STABLE MOTIVIC HOMOTOPY SHEAVES Open
We determine systematic vanishing regions for the bigraded homotopy sheaves of the motivic sphere spectrum over a field of characteristic different from two.
View article: The stable Galois correspondence for real closed fields
The stable Galois correspondence for real closed fields Open
In previous work, the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if $L/k$ is a finite Galois extension of fields with Galois group $G$, there is a functor $c_{L/k}^*$ f…
View article: On the homotopy of Q(3) and Q(5) at the prime 2
On the homotopy of Q(3) and Q(5) at the prime 2 Open
We study modular approximations [math] , [math] , of the [math] –local sphere at the prime [math] that arise from [math] –power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with [math] and record H…