arXiv subjects

Nicolas Ricka

Publications and source records attributed to Nicolas Ricka.

12 recordsLinked to original sources

arXiv2026

Quantitative Evaluation of the Severity of Posttraumatic Stress Disorder through Transfer Learning from Specific Phobia Data

Posttraumatic stress disorder (PTSD) is a prevalent and debilitating mental health condition with significant personal and societal impacts. Current clinical assessments of PTSD often rely on subjective evaluations, which can be time-consuming, costly, and prone to human bias. This study proposes a machine learning (ML) approach based on multivariate kernel density estimation (MKDE) technique for the objective evaluation of PTSD severity. We collected heart rate (HR) and galvanic skin response (GSR) signals as well as PTSD Checklist - Military Version (PCL-M) labels from 21 participants during an immersive simulation. A fear-response model was trained on a public arachnophobia dataset, and predictive features of PTSD were extracted from the fear-response curves estimated on the military dataset. The model achieved an accuracy of 86\% in classifying PTSD status, effectively distinguishing participants with and without PTSD (PCL-M threshold of 36). The average mean absolute error (MAE) of the models is 5.6, and it estimated a clinical PTSD severity scale with a mean absolute percentage error of 17\%. Our algorithm demonstrates promising potential for enhancing estimation of PTSD severity and followup by offering an objective and low-effort evaluation approach using physiology. These findings suggest clinical utility in both screening and follow-up settings.

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arXiv2019

Conjugation Spaces are Cohomologically Pure

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces.

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arXiv2018

C-motivic modular forms

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.

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arXiv2017

Motivic modular forms from equivariant stable homotopy theory

In this paper, we produce a cellular motivic spectrum of motivic modular forms over $\R$ and $\C$, answering positively to a conjecture of Dan Isaksen. This spectrum is constructed to have the appropriate cohomology, as a module over the relevant motivic Steenrod algebra. We first produce a $\G$-equivariant version of this spectrum, and then use a machinery to construct a motivic spectrum from an equivariant one. We believe that this machinery will be of independent interest.

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arXiv2017

The stable Picard group of $\mathcal{A}(2)$

Using a form of descent in the stable category of $\mathcal{A}(2)$-modules, we show that there are no exotic elements in the stable Picard group of $\mathcal{A}(2)$, \textit{i.e.} that the stable Picard group of $\mathcal{A}(2)$ is free on $2$ generators.

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arXiv2016

Local study of stable module categories via tensor triangulated geometry

We investigate the particular properties of the stable category of modules over a finite dimensional cocommutative graded connected Hopf algebra $A$, via tensor-triangulated geometry. This study requires some mild conditions on the Hopf algebra $A$ under consideration (satisfied for example by all finite sub-Hopf-algebras of the modulo $2$ Steenrod algebra). In particular, we study some particular covers of its spectrum of prime ideals $\mathrm{Spc}(A)$, which are related to Margolis' Work. We then exploit the existence of Margolis' Postnikov towers in this situation to show that the localization at an open subset $U$ of $\mathrm{Spc}(A)$, for various $U$, assembles in an $\infty$-stack. Finally, we turn to applications in the study of Picard groups of Hopf algebras and localizations in the stable categories of modules.

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arXiv2016

The Picard group of motivic A(1)

We show that the Picard group $Pic(A(1))$ of the stable category of modules over $\mathbb{C}$-motivic $A(1)$ is isomorphic to $\mathbb{Z}^4$. By comparison, the Picard group of classical $A(1)$ is $\mathbb{Z}^2 \oplus \mathbb{Z}/2$. One extra copy of $\mathbb{Z}$ arises from the motivic bigrading. The joker is a well-known exotic element of order $2$ in the Picard group of classical $A(1)$. The $\mathbb{C}$-motivic joker has infinite order.

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arXiv2016

The stable Picard group of Hopf algebras via descent, and an application

Let $A$ be a cocommutative finite dimensional Hopf algebra over the field with two elements, satisfying some mild hypothesis. We set up a descent spectral sequence which computes the Picard group of the stable category of modules over $A$. The starting point is the observation that the stable category of $A$-modules can be reconstructed, as an $\infty$-category, as the totalization of a cosimplicial $\infty$-category whose layers are related to the stable categories of modules over the quasi-elementary sub-Hopf-algebras of $A$. This leads to a spectral sequence computing the Picard group which, in some cases, is completely understood. This also leads to a spectral sequence answering a lifting problem in the category of $A$-modules. We then show how to apply this machinery to compute Picard groups and solve lifting problems in the case of $\mathcal{A}(1)$-modules, where $\mathcal{A}(1)$ is the subalgebra of the Steenrod algebra generated by the two first Steenrod squares.

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arXiv2014

Equivariant Anderson duality and Mackey functor duality

We show that the $\mathbb{Z}/2$-equivariant Morava K-theories with reality (as defined by Hu) are self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in Morava K-theory with reality. As a particular example, we recover the self-duality of the spectrum $KO$. The study of $\mathbb{Z}/2$-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of $RO(\mathbb{Z}/2)$-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.

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arXiv2014

Subalgebras of the Z/2-equivariant Steenrod algebra

The aim of this paper is to study sub-algebras of the $\mathbb{Z}/2$-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor $\mathbb{F}_2$) which come from quotient Hopf algebroids of the $\mathbb{Z}/2$-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical $\mathcal{A}(n)$ and $\mathcal{E}(n)$ and show that the Steenrod algebra is free as a module over these.

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arXiv2014

Height h detection and connective real k-theory of elementary abelian 2-groups

In this paper, we determine the connective K-cohomology with reality of elementary abelian $2$-groups as a module over $\mathbb{Z}[v_1,a]$, where $v_1$ is the equivariant Bott class and $a$ the Euler class of the sign representation. This gives in particular a new approach to the computation of the connective real K-theory of such groups. The originality here is to make all computations in the $\mathbb{Z}/2$-equivariant stable category, considering only $\mathbb{Z}/2$-equivariant cohomology theories, and to use relative homological algebra over certain subalgebras of the equivariant Steenrod algebra to perform explicit computations.

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arXiv2009

On the Tate spectrum of tmf at the prime 2

Computations involving the root invariant prompted Mahowald and Shick to develop the slogan: "the root invariant of v_n periodic homotopy is v_n torsion." While neither a proof, nor a precise statement, of this slogan appears in the literature, numerous authors have offered computational evidence in support of its fundamental idea. The root invariant is closely related to Mahowald's inverse limit description of the Tate spectrum, and computations have shown the Tate spectrum of v_n periodic cohomology theories to be v_n torsion. The purpose of this paper is to split the Tate spectrum of tmf as a wedge of suspensions of kO, providing yet another example in support of the slogan to the existing literature.

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