arXiv subjects

Marco Armenta

Publications and source records attributed to Marco Armenta.

9 recordsLinked to original sources

arXiv2026

Protected corners and a trichotomy for Han's conjecture

Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. In the tau-Hochschild framework of Cibils, Lanzilotta, Marcos and Solotar, it splits into persistence (Gap A) and survival (Gap B), and a Gap-A failure is already a counterexample. We prove a protected corner theorem bounding Ext at a surviving vertex, settling the Liu-Morin extension conjecture beyond the monomial and special biserial cases. We then establish a trichotomy for Gap-A failures on three strongly connected vertices: the all-infinite case is impossible, and the two-infinite case is completely classified as a mutual dumbbell.

math.RT↗

arXiv2026

$\tau$-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin--Tsygan calculus

We relate two recent enrichments of the Hochschild theory of a finite-dimensional algebra $\Lm$: the $\tau$-Hochschild (co)homology of Cibils, Lanzilotta, Marcos and Solotar, built from Iyama's higher Auslander--Reiten translates of the regular bimodule, and the Coxeter automorphism $\sigma_\Lm$ of the Tamarkin--Tsygan calculus. We show that the Nakayama functor of the enveloping algebra transforms Happel's minimal resolution into a complex representing $\D\Lm\Ltimes_\Lm \D\Lm$, the square of the Serre bimodule whose shift generates $\sigma_\Lm$, and that the $\tau$-translates $\tau_n\Lm$ are precisely the cycle bimodules of this complex. This produces extensions $0\to \B_n\to \tau_n\Lm\to \Tor_n^\Lm(\D\Lm,\D\Lm)\to 0$ whose outer term is dual to $\Ext^n_{\Lme}(\Lm,\Lme)$ and whose inner term is a strictly Morita-theoretic residue of the minimal model. In top degree $d=\gldim\Lm$ the residue vanishes and $\tau_d\Lm$ is the dual of the degree-one component of the $(d+1)$-preprojective algebra of Iyama--Oppermann; for $\Lm=\kk Q$ hereditary, $\tau_{\Lme}\Lm\cong \D\Pi(Q)_1$ and $\HH^1_\tau(\kk Q)$ is the degree-one part of the zeroth Hochschild homology of the preprojective algebra. For self-injective algebras, the derived part vanishes identically, which explains structurally the growth of $\tau$-cohomology for the Buchweitz--Green--Madsen--Solberg algebras. Taking Euler characteristics in the Cibils--Lanzilotta--Marcos--Solotar dimension formulas recovers Happel's trace formula $\sum_i(-1)^i\dim\HH^i(\Lm)=-\tr\sigma_\Lm$. We prove that the two refinements are transversal, propose the combined Morita invariant, exhibit derived-equivalent algebras of finite global dimension whose $\tau$-translates have identical dimension but opposite composition, and pose the problem of derived invariance of $\tau$-Hochschild theory over the smooth locus.

math.RT↗

arXiv2026

The Coxeter transformation as an automorphism of the Tamarkin--Tsygan calculus

Let $A$ be a finite-dimensional algebra over a field $\kk$. We show that the Auslander--Reiten bimodule $\ar_A:=\D A[-1]$ is central in the derived Picard group of $A$ and that, when $\gldim A<\infty$, it induces through the derived-invariance functor $\HB$ of \cite{Armenta19,ArmentaKeller17,ArmentaKeller19} a canonical automorphism $\sigma_A$ of the Tamarkin--Tsygan calculus of $A$; the pair $(\HB(A),\sigma_A)$ is invariant under derived equivalence. We then compute both components of $\sigma_A$. On the Hochschild homology of an elementary algebra, which is concentrated in degree zero, the matrix of $\sigma_A$ in the basis of idempotent traces is $-C_A^{-1}C_A^{\mathrm{T}}$, so its characteristic polynomial is the Coxeter polynomial; the enriched calculus strictly refines both the calculus and the Coxeter polynomial, as the path algebras of quivers of types $\mathbb{A}_4$ and $\mathbb{D}_4$ show, although it is not a complete derived invariant, as the smallest cospectral pair of trees shows. On Hochschild cohomology we prove that $\sigma_A$ is the identity: the left and right actions of $\HH^\bullet(A)$ on the bimodule $\D A$ coincide for every finite-dimensional $A$. This yields a short conceptual proof that the Nakayama automorphism of a Frobenius algebra acts trivially on Hochschild cohomology, recovering a recent theorem of Su\'arez-\'Alvarez. Finally we extend the construction to smooth and proper differential graded algebras, hence to perfect derived categories of smooth projective varieties; the enrichment degenerates precisely on Calabi--Yau categories, and on $\PP^n$ it is governed by the Coxeter polynomial $(x+(-1)^n)^{n+1}$ of the Beilinson algebra. Happel's trace formula and de la Pe\~na's cyclotomicity theorem for fractionally Calabi--Yau algebras become statements internal to the enriched calculus.

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arXiv2024

Hidden Activations Are Not Enough: A General Approach to Neural Network Predictions

We introduce a novel mathematical framework for analyzing neural networks using tools from quiver representation theory. This framework enables us to quantify the similarity between a new data sample and the training data, as perceived by the neural network. By leveraging the induced quiver representation of a data sample, we capture more information than traditional hidden layer outputs. This quiver representation abstracts away the complexity of the computations of the forward pass into a single matrix, allowing us to employ simple geometric and statistical arguments in a matrix space to study neural network predictions. Our mathematical results are architecture-agnostic and task-agnostic, making them broadly applicable. As proof of concept experiments, we apply our results for the MNIST and FashionMNIST datasets on the problem of detecting adversarial examples on different MLP architectures and several adversarial attack methods. Our experiments can be reproduced with our \href{https://github.com/MarcoArmenta/Hidden-Activations-are-not-Enough}{publicly available repository}.

cs.LG↗

arXiv2024

Fermionic Machine Learning

We introduce fermionic machine learning (FermiML), a machine learning framework based on fermionic quantum computation. FermiML models are expressed in terms of parameterized matchgate circuits, a restricted class of quantum circuits that map exactly to systems of free Majorana fermions. The FermiML framework allows for building fermionic counterparts of any quantum machine learning (QML) model based on parameterized quantum circuits, including models that produce highly entangled quantum states. Importantly, matchgate circuits are efficiently simulable classically, thus rendering FermiML a flexible framework for utility benchmarks of QML methods on large real-world datasets. We initiate the exploration of FermiML by benchmarking it against unrestricted PQCs in the context of classification with random quantum kernels. Through experiments on standard datasets (Digits and Wisconsin Breast Cancer), we demonstrate that FermiML kernels are on-par with unrestricted PQC kernels in classification tasks using support-vector machines. Furthermore, we find that FermiML kernels outperform their unrestricted candidates on multi-class classification, including on datasets with several tens of relevant features. We thus show how FermiML enables us to explore regimes previously inaccessible to QML methods.

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arXiv2021

Double framed moduli spaces of quiver representations

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation function, called ReLU, to a moduli space using the symplectic reduction approach to quiver moduli.

math.RT↗

arXiv2021

Neural Teleportation

In this paper, we explore a process called neural teleportation, a mathematical consequence of applying quiver representation theory to neural networks. Neural teleportation "teleports" a network to a new position in the weight space and preserves its function. This phenomenon comes directly from the definitions of representation theory applied to neural networks and it turns out to be a very simple operation that has remarkable properties. We shed light on surprising and counter-intuitive consequences neural teleportation has on the loss landscape. In particular, we show that teleportation can be used to explore loss level curves, that it changes the local loss landscape, sharpens global minima and boosts back-propagated gradients at any moment during the learning process. Our results can be reproduced with the code available here: https://github.com/vitalab/neuralteleportation

cs.LG↗

arXiv2019

On the cap product in Hochschild theory

In this paper, we give an axiomatic characterization of the cap product in the Hochschild theory of associative unital algebras which are projective over a commutative unital ring. We also give an interpretation of the cap product with coefficients in the algebra via chain maps. We illustrate these results by computing the cap product for truncated polynomial algebras $k[x]/(x^N)$ and for polynomial algebras, where it is identified with the contraction of differential forms by polyvector fields.

math.KT↗

arXiv2018

Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra

We prove that Hochschild cohomology with coefficients in $A^*=\Hom_k(A,k)$ under conditions on the algebra structure of $A^*$ is a Batalin-Vilkovisky algebra. We also show that for symmetric and Frobenius algebras, this recovers the known BV-structures in Hochschild cohomology with coefficients in $A$ but admits an easy-to-describe BV-operator. Finally, we show that for monomial algebras $A = kQ/\langle T \rangle$, the Hochschild cohomology with coefficients in $A^*$ is always a Batalin-Vilkovisky algebra.

math.KT↗