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Dylan Wilson

Publications and source records attributed to Dylan Wilson.

At least 19 recordsLinked to original sources
arXiv2025

Optical Waveguide-based Spider Web Enables Resilient Impact Detection and Localization

Spiders use their webs as multifunctional tools that enable capturing and localizing prey and more general environmental sensing through vibrations. Inspired by their biological function, we present a spider web-inspired optical waveguide system for resilient impulse detection and localization. The structure consists of six clear thermoplastic polyurethane (TPU) waveguides arranged radially and interconnected by a spiral TPU thread, mimicking orb spider webs. Light transmission losses, induced by vibrations, are measured via coupled LEDs and photo-diodes, allowing real-time detection. We systematically characterize individual waveguides, analyzing key parameters such as tension, impulse position, and break angle to optimize vibrational response. The complete system is validated through controlled experiments, revealing a 5 ms propagation delay in vibration transfer between adjacent radii, enhancing localization capabilities. We demonstrate a robust impulse detection and localization algorithm leveraging time delay analysis, achieving reliable event identification even in cases of sensor failure. This study highlights the potential of bioinspired optical waveguide structures for adaptive sensing, with applications in soft robotics, structural monitoring, and environmental sensing.

eess.SP↗
arXiv2024

Adjusted Overfitting Regression

In this paper, I will introduce a new form of regression, that can adjust overfitting and underfitting through, "distance-based regression." Overfitting often results in finding false patterns causing inaccurate results, so by having a new approach that minimizes overfitting, more accurate predictions can be derived. Then I will proceed with a test of my regression form and show additional ways to optimize the regression. Finally, I will apply my new technique to a specific data set to demonstrate its practical value.

cs.LG↗
arXiv2024

LLM Tree Search

This project aims to investigate a novel sequence generation method inspired by the AlphaGo paradigm, adapting it for use with large language models (LLMs). The proposed approach involves creating search trees of different possible completions and evaluating these completions based on model confidence. By considering various paths in the search tree and scoring them according to the model's confidence in each completion, we can generate diverse and high-quality sequences. This research explores the implementation of this paradigm by using confidence as a proxy for response quality akin to beam search \citep{vijayakumar2016diverse}. The primary goal of this paper is to outline the paradigm and demonstrate its potential, rather than focusing on achieving perfect results. The paper will outline the reasons why we believe this paradigm has the potential to improve LLMs in the following manners: 1) increase output quality, 2) decrease errors, 3) eliminate or reduce the compound error problems, 4) generate diverse and creative completions, 5) allow for iterative problem-solving, and 6) self-training. We expect this approach to yield a set of diverse and coherent sequences, offering insights into balancing exploration and exploitation in sequence generation. Potential applications include creative text generation tasks, such as storytelling and content creation, as well as other natural language processing domains, like machine translation and automated summarization. The goal is that the model will be far more effective as it will be able to consider many possible variations allowing it to find the ideal completion. This research aims to contribute to the understanding of effective search strategies in sequence generation and their impact on generating high-quality, varied textual outputs.

cs.CL↗
arXiv2024

Syntomic cohomology of Morava K-theory

We compute the MU-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n+1})$, of all $\mathbb{E}_1$-MU-algebra forms of connective Morava K-theory k(n). As qualitative consequences, we deduce the Lichtenbaum--Quillen conjecture, telescope conjecture, and redshift conjecture for the algebraic K-theories of all $\mathbb{E}_{1}$-$\mathbb{S}$-algebra forms of $(2p^n-2)$-periodic Morava K-theory. Notably, the motivic spectral sequence computing $\pi_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$.

math.KT↗
arXiv2023

Examples of disk algebras

We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum. Consequently, topological Hochschild homology relative to these rings inherits a circle action.

math.AT↗
arXiv2022

A motivic filtration on the topological cyclic homology of commutative ring spectra

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

math.KT↗
arXiv2021

On the $C_p$-equivariant dual Steenrod algebra

We compute the $C_p$-equivariant dual Steenrod algebras associated to the constant Mackey functors $\underline{\mathbb{F}}_p$ and $\underline{\mathbb{Z}}_{(p)}$, as $\underline{\mathbb{Z}}_{(p)}$-modules. The $C_p$-spectrum $\underline{\mathbb{F}}_p \otimes \underline{\mathbb{F}}_p$ is not a direct sum of $RO(C_p)$-graded suspensions of $\underline{\mathbb{F}}_p$ when $p$ is odd, in contrast with the classical and $C_2$-equivariant dual Steenrod algebras.

math.AT↗
arXiv2020

Redshift and multiplication for truncated Brown-Peterson spectra

We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm{K}(\mathrm{BP}\langle n \rangle)_{(p)} \to \mathrm{L}_{n+1}^{f} \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.

math.AT↗
arXiv2020

Odd primary analogs of Real orientations

We define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of $\mu_p$-orientation analogous to a $C_2$-equivariant Real orientation. The definition hinges on a $C_p$-space $\mathbb{CP}^{\infty}_{\mu_p}$, which we prove to be homologically even in a sense generalizing recent $C_2$-equivariant work on conjugation spaces. We prove that the height $p-1$ Morava $E$-theory is $\mu_p$-oriented and that $\mathrm{tmf}(2)$ is $\mu_3$-oriented. We explain how a single equivariant map $v_1^{\mu_p}:S^{2\rho} \to \Sigma^{\infty} \mathbb{CP}^{\infty}_{\mu_p}$ completely generates the homotopy of $E_{p-1}$ and $\mathrm{tmf}(2)$, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.

math.AT↗
arXiv2020

Equivariant nonabelian Poincar\'e duality and equivariant factorization homology of Thom spectra

In this paper, we study genuine equivariant factorization homology and its interaction with equivariant Thom spectra, which we construct using the language of parametrized higher category theory. We describe the genuine equivariant factorization homology of Thom spectra, and use this description to compute several examples of interest. A key ingredient for our computations is an equivariant nonabelian Poincar\'e duality theorem, in which we prove that factorization homology with coefficients in a $G$-space is given by a mapping space. We compute the Real topological Hochschild homology ($THR$) of the Real bordism spectrum $MU_\mathbb{R}$ and of the equivariant Eilenberg--MacLane spectra $H\underline{\mathbb{F}}_2$ and $H\underline{\mathbb{Z}}_{(2)}$, as well as factorization homology of the sphere $S^{2\sigma}$ with coefficients in these Eilenberg--MacLane spectra. In Appendix B, Jeremy Hahn and Dylan Wilson compute $THR(H\underline{\mathbb{Z}})$.

math.AT↗
arXiv2019

Real topological Hochschild homology and the Segal conjecture

We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of $\mathbb{F}_2$. This determines the $\mathrm{E}_2$-page of the descent spectral sequence for the map $\mathrm{N}\mathbb{F}_2 \to \mathbb{F}_2$, where $\mathrm{N}\mathbb{F}_2$ is the $C_2$-equivariant Hill--Hopkins--Ravenel norm of $\mathbb{F}_2$. The $\mathrm{E}_2$-page represents a new upper bound on the $RO(C_2)$-graded homotopy of $\mathrm{N}\mathbb{F}_2$, from which the Segal conjecture is an immediate corollary.

math.AT↗
arXiv2019

$C_2$-equivariant Homology Operations: Results and Formulas

In this note we state corrected and expanded versions of our previous results on power operations for $C_2$-equivariant Bredon homology with coefficients in the constant Mackey functor on $\mathbb{F}_2$. In particular, we give a version of the Adem relations. The proofs rely on certain results in equivariant higher algebra which we will supply in a longer version of this paper.

math.AT↗
arXiv2019

Mod 2 power operations revisited

In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of $\mathbb{E}_{\infty}$-ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.

math.AT↗
arXiv2018

Quotients of even rings

We prove that if $R$ is an $\mathbb{E}_2$-ring with homotopy concentrated in even degrees, and $\{x_j\}$ is any sequence of elements in $\pi_{2*}(R)$, then $R/(x_1,x_2,\cdots)$ admits the structure of an $\mathbb{E}_1$-$R$-algebra. This removes an assumption, common in the literature, that $\{x_j\}$ be a regular sequence.

math.AT↗
arXiv2018

Eilenberg-MacLane spectra as equivariant Thom spectra

We prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $\mathrm{H}\mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.

math.AT↗
arXiv2017

On categories of slices

In this paper we give an algebraic description of the category of $n$-slices for an arbitrary group $G$, in the sense of Hill-Hopkins-Ravenel. Specifically, given a finite group $G$ and an integer $n$, we construct an explicit $G$-spectrum $W$ (called an isotropic slice $n$-sphere) with the following properties: (i) the $n$-slice of a $G$-spectrum $X$ is equivalent to the data of a certain quotient of the Mackey functor $\underline{[W,X]}$ as a module over the endomorphism Green functor $\underline{[W,W]}$; (ii) the category of $n$-slices is equivalent to the full subcategory of right modules over $\underline{[W,W]}$ for which certain restriction maps are injective. We use this theorem to recover the known results on categories of slices to date, and exhibit the utility of our description in several new examples. We go further and show that the Green functors $\underline{[W,W]}$ for certain slice $n$-spheres have a special property (they are "geometrically split") which reduces the amount of data necessary to specify a $\underline{[W,W]}$-module. This step is purely algebraic and may be of independent interest.

math.AT↗
arXiv2016

Power operations for $\text{H}\underline{\mathbb{F}}_2$ and a cellular construction of $\text{BP}\mathbf{R}$

We study some power operations for ordinary $C_2$-equivariant homology with coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$. In addition to a few foundational results, we calculate the action of these power operations on a $C_2$-equivariant dual Steenrod algebra. As an application, we give a cellular construction of the $C_2$ equivariant Brown-Peterson spectrum $\text{BP}\mathbf{R}$ and deduce its slice tower.

math.AT↗