arXiv subjects

David Chan

Publications and source records attributed to David Chan.

At least 19 recordsLinked to original sources

arXiv2026

Multiplicatively cohomological Tambara functors

We study a nice class of Tambara functors, called multiplicatively cohomological Tambara functors. The first result is a proof of a conjecture of Balchin, Quigley, and Spitz showing that every multiplicatively cohomological Tambara functor is also additively cohomological. Building on this, we prove Nakayama's lemma for multiplicatively cohomological Tambara functors. As an application, we identify a condition under which finitely generated projective modules over local Tambara functors are always free. This applies, in particular, to the constant $C_{p^n}$-Tambara functor $\underline{\mathbb{Z}}_p$.

math.AT↗

arXiv2026

Computations in Equivariant Topological Hochschild Homology

One of the most effective approaches to computations in algebraic $K$-theory is trace methods, which compare algebraic $K$-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschild homology ($\mathrm{ETHH}$) which receives a trace map from Merling's genuine equivariant algebraic $K$-theory. In this paper, we perform foundational computations of $\mathrm{ETHH}$ that can serve as input for future computations of $\mathrm{ETHH}$ and equivariant topological cyclic homology. Namely, we compute $\mathrm{ETHH}(H\underline{\mathbb{F}}_p)$ for odd primes, showcasing the complexity of B\"okstedt periodicity in this setting. Furthermore, we give computations of $\mathrm{ETHH}$ for the equivariant complex cobordism spectra $MU_G$ and $MU_{\mathbb{R}}$.

math.AT↗

arXiv2026

Topological symmetric and braid homologies

We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $\Delta \mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies.

math.AT↗

arXiv2026

Radicals and Nilpotents in Equivariant Algebra

Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{Spec}(T)$, analogous to the Zariski spectrum of a commutative ring. We establish that this topological space is spectral. This result follows from an analysis of the notion of nilpotence in Tamabra functors. We prove that the nilradical of a Tambara functor $T$ (the intersection of all of its prime ideals) is computed levelwise, i.e. consists precisely of the nilpotent elements in $T$. In contrast to ordinary commutative algebra, the nilpotents of $T$ are not the same as the elements $x$ such that $T[1/x] = 0$; we therefore also give a classification of these elements. As a corollary, we observe that the set of these elements in $\pi_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.

math.AT↗

arXiv2025

An Adaptive Graphical Lasso Approach to Modeling Symptom Networks of Common Mental Disorders in Eritrean Refugee Population

Despite the significant public health burden of common mental disorders (CMDs) among refugee populations, their underlying symptom structures remain underexplored. This study uses Gaussian graphical modeling to examine the symptom network of post-traumatic stress disorder (PTSD), depression, anxiety, and somatic distress among Eritrean refugees in the Greater Washington, DC area. Given the small sample size (n) and high-dimensional symptom space (p), we propose a novel extension of the standard graphical LASSO by incorporating adaptive penalization, which improves sparsity selection and network estimation stability under n < p conditions. To evaluate the reliability of the network, we apply bootstrap resampling and use centrality measures to identify the most influential symptoms. Our analysis identifies six distinct symptom clusters, with somatic-anxiety symptoms forming the most interconnected group. Notably, symptoms such as nausea and reliving past experiences emerge as central symptoms linking PTSD, anxiety, depression, and somatic distress. Additionally, we identify symptoms like feeling fearful, sleep problems, and loss of interest in activities as key symptoms, either being closely positioned to many others or acting as important bridges that help maintain the overall network connectivity, thereby highlighting their potential importance as possible intervention targets.

stat.AP↗

arXiv2025

A comparison of definitions of equivariant trees

We show that various categories of trees can be modeled by Grothendieck constructions on categories of trees with a fixed set of leaves. We prove this result for the dendroidal category $\Omega$, the category $\Omega^G$ of trees with a $G$-action for a finite group $G$, and finally for the category of genuine equivariant trees $\Omega_G$ that has played an important role in recent work on genuine equivariant operads.

math.AT↗

arXiv2025

POrTAL: Plan-Orchestrated Tree Assembly for Lookahead

When tasking robots in partially observable environments, these robots must efficiently and robustly plan to achieve task goals under uncertainty. Although many probabilistic planning algorithms exist for this purpose, these algorithms can be inefficient if executed with the robot's limited computational resources, or may produce policies that take more steps than expected to achieve the goal. We therefore created a new, lightweight, probabilistic planning algorithm, Plan-Orchestrated Tree Assembly for Lookahead (POrTAL), that combines the strengths of two baseline planning algorithms, FF-Replan and POMCP. We demonstrate that POrTAL is an anytime algorithm that generally outperforms these baselines in terms of the final executed plan length given bounded computation time, especially for problems with only moderate levels of uncertainty.

cs.RO↗

arXiv2025

Realizing compatible pairs of transfer systems by combinatorial $N_\infty$-operads

We investigate how the notions of pairings of operads of May and compatible pairs of indexing systems of Blumberg--Hill relate via the correspondence between indexing systems and $N_{\infty}$-operads. We show that a pairing of operads induces a pairing on the associated indexing systems. Conversely, we show that in many cases, compatible pairs of indexing systems can be realized by a pairing of $N_{\infty}$-operads.

math.AT↗

arXiv2025

$C_p$-Mackey functors in Macaulay2

We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.

math.AT↗

arXiv2025

The algebraic $K$-theory of Green functors

In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group. From the spectral sequence we deduce a complete calculation of the algebraic $K$-theory of the constant $C_2$-Green functor associated to the field with two elements, and a calculation of the $p$-completion of the algebraic $K$-theory of the constant $G$-Green functor associated to the integers when $G$ is a cyclic $p$-group. Additionally, we introduce the notion of a Green meadow to abstract the Green functor structure underlying clarified Tambara fields, and show, under mild conditions, that every finitely generated projective module over a $G$-Green meadow is free when $G$ is a cyclic $p$-group. This gives a computation of $K_0$ for such Green functors.

math.KT↗

arXiv2025

A genuine $G$-spectrum for the cut-and-paste $K$-theory of $G$-manifolds

Recent work has applied scissors congruence $K$-theory to study classical cut-and-paste ($SK$) invariants of manifolds. This paper proves the conjecture that the squares $K$-theory of equivariant $SK$-manifolds arises as the fixed points of a genuine $G$-spectrum. Our method utilizes the framework of spectral Mackey functors as models for genuine $G$-spectra, and our main technical result is a general procedure for constructing spectral Mackey functors using squares $K$-theory.

math.KT↗

arXiv2025

Segal K-theory factors through Waldhausen categories

We show that Segal's K-theory of symmetric monoidal categorizes can be factored through Waldhausen categories. In particular, given a symmetric monoidal category $C$, we produce a Waldhausen category $\Gamma(C)$ whose K-theory is weakly equivalent to the Segal K-theory of $C$. As a consequence, we show that every connective spectrum may be obtained via Waldhausen K-theory.

math.KT↗

arXiv2025

The spectrum of the Burnside Tambara functor

We compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_8$, the alternating group $A_4$, and the general linear group $GL_3(F_2)$.

math.AT↗

arXiv2025

Trace methods for equivariant algebraic K-theory

In the past decades, one of the most fruitful approaches to the study of algebraic $K$-theory has been trace methods, which construct and study trace maps from algebraic $K$-theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic $K$-theory have emerged, but thus far few tools are available for the study and computation of these theories. In this paper, we lay the foundations for a trace methods approach to equivariant algebraic $K$-theory. For $G$ a finite group, we construct a Dennis trace map from equivariant algebraic $K$-theory to a $G$-equivariant version of topological Hochschild homology; for $G$ the trivial group this recovers the ordinary Dennis trace map. We show that upon taking fixed points, this recovers the trace map of Adamyk--Gerhardt--Hess--Klang--Kong, and gives a trace map from the fixed points of coarse equivariant $A$-theory to the free loop space. We also establish important properties of equivariant topological Hochschild homology, such as Morita invariance, and explain why it can be considered as a multiplicative norm.

math.AT↗

arXiv2024

On the Tambara Affine Line

Tambara functors are the analogue of commutative rings in equivariant algebra. Nakaoka defined ideals in Tambara functors, leading to the definition of the Nakaoka spectrum of prime ideals in a Tambara functor. In this work, we continue the study of the Nakoaka spectra of Tambara functors. We describe, in terms of the Zariski spectra of ordinary commutative rings, the Nakaoka spectra of many Tambara functors. In particular: we identify the Nakaoka spectrum of the fixed point Tambara functor of any $G$-ring with the GIT quotient of its classical Zariski spectrum; we describe the Nakaoka spectrum of the complex representation ring Tambara functor over a cyclic group of prime order $p$; we describe the affine line (the Nakaoka spectra of free Tambara functors on one generator) over a cyclic group of prime order $p$ in terms of the Zariski spectra of $\mathbb{Z}[x]$, $\mathbb{Z}[x,y]$, and the ring of cyclic polynomials $\mathbb{Z}[x_0,\ldots,x_{p-1}]^{C_p}$. To obtain these results, we introduce a "ghost construction" which produces an integral extension of any $C_p$-Tambara functor, the Nakaoka spectrum of which is describable. To relate the Nakaoka spectrum of a Tambara functor to that of its ghost, we prove several new results in equivariant commutative algebra, including a weak form of the Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors. These results also allow us to compute the Krull dimensions of many Tambara functors.

math.AT↗

arXiv2024

The Galois-equivariant $K$-theory of finite fields

We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied $RO(G)$-graded coefficient groups of the equivariant Eilenberg--MacLane spectrum $H\underline{\mathbb Z}$. Our comparison between the equivariant $K$-theory spectrum and $H\underline{\mathbb Z}$ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where $G$ has prime order, we provide an explicit presentation of the equivariant $K$-groups.

math.KT↗

arXiv2024

Multi-Stage Multi-Modal Pre-Training for Automatic Speech Recognition

Recent advances in machine learning have demonstrated that multi-modal pre-training can improve automatic speech recognition (ASR) performance compared to randomly initialized models, even when models are fine-tuned on uni-modal tasks. Existing multi-modal pre-training methods for the ASR task have primarily focused on single-stage pre-training where a single unsupervised task is used for pre-training followed by fine-tuning on the downstream task. In this work, we introduce a novel method combining multi-modal and multi-task unsupervised pre-training with a translation-based supervised mid-training approach. We empirically demonstrate that such a multi-stage approach leads to relative word error rate (WER) improvements of up to 38.45% over baselines on both Librispeech and SUPERB. Additionally, we share several important findings for choosing pre-training methods and datasets.

cs.CL↗

arXiv2024

conv_einsum: A Framework for Representation and Fast Evaluation of Multilinear Operations in Convolutional Tensorial Neural Networks

Modern ConvNets continue to achieve state-of-the-art results over a vast array of vision and image classification tasks, but at the cost of increasing parameters. One strategy for compactifying a network without sacrificing much expressive power is to reshape it into a tensorial neural network (TNN), which is a higher-order tensorization of its layers, followed by a factorization, such as a CP-decomposition, which strips a weight down to its critical basis components. Passes through TNNs can be represented as sequences of multilinear operations (MLOs), where the evaluation path can greatly affect the number of floating point operations (FLOPs) incurred. While functions such as the popular einsum can evaluate simple MLOs such as contractions, existing implementations cannot process multi-way convolutions, resulting in scant assessments of how optimal evaluation paths through tensorized convolutional layers can improve training speed. In this paper, we develop a unifying framework for representing tensorial convolution layers as einsum-like strings and a meta-algorithm conv_einsum which is able to evaluate these strings in a FLOPs-minimizing manner. Comprehensive experiments, using our open-source implementation, over a wide range of models, tensor decompositions, and diverse tasks, demonstrate that conv_einsum significantly increases both computational and memory-efficiency of convolutional TNNs.

cs.LG↗