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We consider the renormalization group equations of axion-like particle effective field theories and determine the corresponding anomalous dimensions at one loop via on-shell and unitarity-based methods. The calculation of the phase-space cut-integrals is carried out using different integration methods, among which the double-cut integration via Stokes' theorem proves to be technically simpler. A close comparison between the standard Feynman diagrammatic approach and the unitarity-based method enables us to explicitly verify the reduction of complexity in the latter case, along with a more direct and elegant way to establish a connection among anomalous dimensions of operators that are dual under the CP symmetry., Comment: 46 pages, 16 figures

We discuss a recently introduced strategy to study non-perturbatively thermal QCD up to temperatures of the order of the electro-weak scale, combining step scaling techniques and shifted boundary conditions. The former allow to renormalize the theory for a range of scales which spans several orders of magnitude with a moderate computational cost. Shifted boundary conditions remove the need for the zero temperature subtraction in the Equation of State. As a consequence, the simulated lattices do not have to accommodate two very different scales, the pion mass and the temperature, at the very same spacing. Effective field theory arguments guarantee that finite volume effects can be kept under control safely. With this strategy the first computation of the hadronic screening spectrum has been carried out over more than two orders of magnitude in the temperature, from $T\sim 1$ GeV up to $\sim 160$ GeV. This study is complemented with the first quantitative computation of the baryonic screening mass at next-to-leading order in the three-dimensional effective theory describing QCD at high temperatures. Both for the mesonic and the baryonic screening masses, the known leading behaviour in the coupling constant is found to be not sufficient to explain the non-perturbative data over the entire range of temperatures. These findings shed further light on the limited applicability of the perturbative approach at finite temperature, even at the electro-weak scale., Comment: arXiv admin note: substantial text overlap with arXiv:2410.12626

The thesis explores novel methods for Human Activity Recognition (HAR) using passive radar with a focus on non-intrusive Wi-Fi Channel State Information (CSI) data. Traditional HAR approaches often use invasive sensors like cameras or wearables, raising privacy issues. This study leverages the non-intrusive nature of CSI, using Spiking Neural Networks (SNN) to interpret signal variations caused by human movements. These networks, integrated with symbolic reasoning frameworks such as DeepProbLog, enhance the adaptability and interpretability of HAR systems. SNNs offer reduced power consumption, ideal for privacy-sensitive applications. Experimental results demonstrate SNN-based neurosymbolic models achieve high accuracy making them a promising alternative for HAR across various domains.

We present a recently introduced strategy to study non-perturbatively thermal QCD up to temperatures of the order of the electro-weak scale, combining step scaling techniques and shifted boundary conditions. The former allow to renormalize the theory for a range of scales which spans several orders of magnitude with a moderate computational cost. Shifted boundary conditions avoid the need for the zero temperature subtraction in the Equation of State. As a consequence, the simulated lattices do not have to accommodate two very different scales, the pion mass and the temperature. Effective field theory arguments guarantee that finite volume effects can be kept under control safely. As a first application of this strategy, we present the results of the computation of the hadronic screening spectrum in QCD with $N_f=3$ flavours of massless quarks for temperatures from $T\sim 1$ GeV up to $\sim 160$ GeV., Comment: 6 pages, 3 figures, Proceedings of the 42nd International Conference on High Energy Physics (ICHEP2024)

The toric fundamental group is the smallest extension of the \'etale fundamental group which can manage the monodromy of line bundles, in addition to the monodromy of finite \'etale covers. It is an extension of the \'etale fundamental group by a projective limit of tori. We prove the analogue of Grothendieck's section conjecture for the toric fundamental group over finite extensions of $\mathbb{Q}_{p}$., Comment: A. Tamagawa and Y. Hoshi pointed out to me a mistake in the proof of Proposition 15, so as of now the main results of the preprint are not proved. The general strategy might still work; I am working to try to fix the problem

We investigate the variational model for nematic elastomer proposed by Barchiesi and DeSimone with the director field defined on the deformed configuration under general growth conditions on the elastic density. This leads us to consider deformations in Orlicz-Sobolev spaces. Our work builds upon a previous paper by Henao and the Second Author, and extends their analysis to the quasistatic setting. The overall strategy parallels the one devised by the First author in the case of Sobolev deformations for a similar model in magnetoelasticity. We prove two existence results for energetic solutions in the rate-independent setting. The first result concerns quasistatic evolutions driven by time-dependent applied loads. For this problem, we establish suitable Poincar\'{e} and trace inequalities in modular form to recover the coercivity of the total energy. The second result ensures the existence of quasistatic evolution for both time-depend applied loads and boundary conditions under physical confinement. In its proof, we follow the approach advanced by Francfort and Mielke based on a multiplicative decomposition of the deformation gradient and we implement it for energies comprising terms defined on the deformed configuration. Both existence results rely on a compactness theorem for sequences of admissible states with uniformly bounded energy which yields the strong convergence of the composition of the nematic fields with the corresponding deformations. While proving it, we show the regular approximate differentiability of Orlicz-Sobolev maps with suitable integrability, thus generalizing a classical result for Sobolev maps due to Goffman and Ziemer.

We propose a model for quasistatic growth of cavities and cracks in two-dimensional nonlinear elasticity. Cavities and cracks are modeled as discrete and compact subsets of a planar domain, respectively, and deformations are defined only outside of cracks. The model accounts for the irreversibility of both processes of cavitation and fracture and it allows for the coalescence of cavities into cracks. Our main result shows the existence of quasistatic evolutions in the case of a finite number of cavities, under an a priori bound on the number of connected components of the cracks.

We give a positive answer to a question of J. Doyle and J. Silverman about fields of definition of dynamical systems on $\mathbb{P}^{n}$. We prove that, for fixed $n$, there exists a constant $C_{n}$ such that every dynamical system $\mathbb{P}^{n}\to\mathbb{P}^{n}$ is defined over an extension of degree $\le C_{n}$ of the field of moduli. More generally, the same bound works for any kind of "algebraic structure" defined over $\mathbb{P}^{n}$, such as embedded curves, hypersurfaces, algebraic cycles. As a consequence we prove that, if $x\in X(k)$ is a rational point of an $n$-dimensional variety with quotient singularities, there exists a field extension $k'/k$ of degree $\le C_{n-1}$ such that $x$ lifts to a $k'$-rational point of any resolution of singularities.

J. Silverman proved that a dynamical system on $\mathbb{P}^{1}$ descends to the field of moduli if it is polynomial or it has even degree, but for non-polynomial ones of odd degree the picture is less clear. We give a complete characterization of which dynamical systems over $\mathbb{P}^{1}$ descend to the field of moduli., Comment: There is a mistake in the proof of the main theorem. In the middle of page 5, we state that \phi has equal orders of vanishing in 0 and \infty. However, the argument only proves the weaker statement that the ramification degrees are equal. This is not sufficient to conclude

We investigate the existence of minimizers of variational models with Eulerian-Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do so, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples., Comment: 63 pages, 8 figures

We present first non-perturbative results for the renormalization constants of the QCD energy-momentum tensor, based on the framework of thermal QCD with shifted and twisted (for quarks only) boundary conditions in the compact direction. We also show preliminary results for the entropy density obtained with the very same numerical strategy. This opens the way to the determination of the QCD Equation of State up to very high temperatures., Comment: 8 pages, contribution to the 40th International Symposium on Lattice Field Theory

We elaborate on the application of on-shell and unitarity-based methods for evaluating renormalization group coefficients, and generalize this framework to account for the mixing of operators with different dimensions and leading mass effects. We derive a master formula for anomalous dimensions stemming from the general structure of operator mixings, up to two-loop order, and show how the Higgs low-energy theorem can be exploited to include leading mass effects. A few applications on the renormalization properties of popular effective field theories showcase the strength of the proposed approach, which drastically reduces the complexity of standard loop calculations. Our results provide a powerful tool to interpret experimental measurements of low-energy observables, such as flavor violating processes or electric and magnetic dipole moments, as induced by new physics emerging above the electroweak scale., Comment: 6 pages, 5 figures. Text improved, appendix and figure added. Published version

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve $C$ over an algebraically closed field $K$ of characteristic $0$ with field of moduli $k_{C}\subset K$ is defined by a polynomial with coefficients in $k'$, where $k'/k_{C}$ is an extension with $[k':k_{C}]\le 3$ and $[k':k_{C}]\mid \operatorname{deg} C$., Comment: Comments are welcome! In particular, I would be very grateful for historical information regarding the statement of Theorem 1

We prove that every hyperbolic curve with a faithful action of a non-cyclic $p$-group (with a few exceptions if $p=2$) has a twisted form of index $1$ which satisfies Grothendieck's section conjecture. Furthermore, we prove that for every hyperbolic curve $S$ over a field $k$ finitely generated over $\mathbb{Q}$ there exists a finite extension $K/k$ and a finite \'etale cover $C\to S_{K}$ such that $C$ satisfies the conjecture.

It is a classical fact going back to F. Klein that an elliptic curve $E$ over $\bar{\mathbb{Q}}$ is defined by a homogeneous polynomial in $3$ variables with coefficients in $\mathbb{Q}(j_{E})$, where $j_{E}$ is the $j$-invariant of $E$, and $\mathbb{Q}(j_{E})$ is the field of moduli of $E$. The general definition of field of moduli goes back to T. Matsusaka and G. Shimura. With few exceptions, it coincides with the intersection of the fields where the curve is defined. We prove that every smooth plane curve of degree prime with $6$ is defined by a homogeneous polynomial with coefficients in the field of moduli. Furthermore, we show that most plane curves in arbitrary degree, and more generally most algebraic cycles in $\mathbb{P}^{2}$ with finite automorphism group, descend to a Brauer-Severi surface over the field of moduli.

If $X$ is a variety with an algebraic structure $\xi$, such as a marked point, a divisor, a group structure and so forth, then it is possible to study whether the pair $(X,\xi)$ is defined over the field of moduli. There exists a precise definition of ``algebraic structures'' which covers essentially all of the obvious concrete examples. We prove several formal results about algebraic structures. There are immediate applications to the study of fields of moduli of curves and finite sets in $\mathbb{P}^{2}$, but the results are completely general. First, we give a way of classifying algebraic structures on $X$ up to equivalence. Secondly we show that, under some assumptions, every algebraic structure on $X$ is equivalent to the structure given by some $0$-cycle. Fix $G$ a finite group of automorphisms of $X$. Third, we give a cohomological criterion for checking the existence of algebraic structures not defined over the field of moduli with group of automorphisms equal to $G$. Fourth, we identify geometric conditions regarding the action of $G$ on $X$ which ensure that every algebraic structure with group of automorphisms equal to $G$ is defined over the field of moduli.

For every $n\ge 6$, we give an example of a finite subset of $\mathbb{P}^{2}$ of degree $n$ which does not descend to any Brauer-Severi surface over the field of moduli. Conversely, for every $n\le 5$ we prove that a finite subset of degree $n$ always descends to a $0$-cycle on $\mathbb{P}^{2}$ over the field of moduli.

We present our progresses in the use of the non-perturbative renormalization framework based on considering QCD at finite temperature with shifted and twisted (for quarks only) boundary conditions in the compact direction. We report our final results in the application of this method for the non-perturbative renormalization of the flavor-singlet local vector current. We then discuss the more challenging case of the renormalization of the energy-momentum tensor, and show preliminary results on the relevant one-point functions for the computation of the renormalization constants of its non-singlet components., Comment: 9 pages, 2 figures, contribution to the 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Bonn, Germany

Let $k$ be a field, $X$ a variety with tame quotient singularities and $\tilde{X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde{X}$ by the Lang-Nishimura theorem, but if $x$ is singular this might be false. For certain types of singularities the rational point is guaranteed to lift, though; these are called singularities of type $\mathrm{R}$. This concept has applications in the study of the fields of moduli of varieties and yields an enhanced version of the Lang-Nishimura theorem where the smoothness assumption is relaxed. We classify completely the tame quotient singularities of type $\mathrm{R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\mathrm{R}$, and most cyclic singularities are of type $\mathrm{R}$ too.

Let $k$ be a field with algebraic closure $\bar{k}$ and $D \subset \mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree $n \ge 3$, write $k(D)$ for the field of moduli of $D$. A. Marinatto proved that when $n$ is odd, or $n = 4$, $D$ descends to a divisor on $\mathbb{P}^{1}_{k(D)}$. We analyze completely the problem of when $D$ descends to a divisor on a smooth, projective curve of genus $0$ on $k(D)$, possibly with no rational points. In particular, we study the remaining cases $n \ge 6$ even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins about the field of moduli of hyperelliptic curves.

Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety $X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of $\overline{k}$ of elements fixed by field automorphisms $\gamma\in\operatorname{Gal}(\overline{k}/k)$ such that the twist $X_{\gamma}$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset\overline{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. Furthermore, D\`ebes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname{Aut}(X,p)$ is finite, \'etale and of degree prime to $d!$ is defined over its field of moduli.

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing $\mathbb{Q}_{p}$-points to $\mathbb{F}_{p}$-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a $\mathbb{Q}_{p}$-point will specialize to an $\mathbb{F}_{p^{n}}$-point for some $n$. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang-Nishimura theorem holds for tame stacks.