The classical Rees construction (of common use in commutative algebra and Hodge theory) interpolates between filtrations, viewed as G(m)-equivariant vector bundles on the affine line, and their associated gradings. Various non-abelian versions have been proposed, where the multiplicative group G(m) is replaced by an arbitrary reductive group. Building on a construction due to P. O'Sullivan, we present a Galois correspondence between quasi-homogeneous spaces and certain monoidal categories, and apply it to monoidal categories of motives with concrete applications to algebraic cycles. In particular, we give a new proof and generalization of the Clozel-Deligne theorem about numerical equivalence on abelian varieties over finite fields.

In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more "motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the Andr & eacute;-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.

In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more “motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the André-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using p-adic Hodge theory. We show that an unconditional result lies behind this implication: the observability of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result. We also discuss another aspect of the Tate conjecture related to the transcendence of p-adic periods.

The relative algebraic monodromy of abelian logarithms (defined as the kernel of a map between algebraic monodromy groups attached to an abelian scheme with and without a section) was computed in : under natural assumptions, this vector group turns out to be maximal. The relative integral monodromy of abelian logarithms is defined similarly as a kernel of integral monodromy groups, without taking Zariski closures. We show that if the integral monodromy of the abelian scheme is a lattice in the algebraic monodromy (which is not always the case), then the relative integral monodromy of the abelian logarithm is also a lattice in the relative algebraic monodromy. The proof uses a Hodge-theoretic interpretation of sections of abelian schemes. We also consider relative integral monodromy groups in the more general context of normal functions.

Classical applications of Galois theory concern algebraic numbers and algebraic functions. Still, the night before his duel, Galois wrote that his last mathematical thoughts had been directed toward applying his "theory of ambiguity to transcendental functions and transcendental quantities". We outline some more or less recent ideas and results in this direction.

We have used a polarized spatially resolved microluminescence technique to investigate photocarrier charge and spin transport at 6 K in a GaAs nanowire (NW; n-type doping level approximate to 1017 cm-3). Because of the difference in expansion coefficients of the NW and of its SiO2 substrate, the NW is under strain, as revealed by the splitting between light- and heavy-hole emissions in the luminescence intensity spectrum. Light valence levels lie above the heavy valence ones, which is attributed as being caused by a tensile strain along both the axial and the lateral directions of the NW, equivalent to a compressive strain in the direction of light excitation. The symmetry group of the perturbed nanowire is then lowered to C2v. No spin polarization can be evidenced for the heavy valence levels. The electron spin polarization decays up to a distance of 5 mu m from the excitation spot, because of spin relaxation, and stays constant for larger distances because of the increased value of the drift velocity. Remarkably, the light-hole spin polarization exhibits damped spatial oscillations over as much as 5 mu m. Analysis of the effect of strain on valence states shows that these oscillations are caused by the spin-orbit interaction in the light valence level. It is found that, for the C2v point group, the corresponding Hamiltonian is linear in momentum. This spin-orbit interaction causes coherent oscillations rather than a spin relaxation process since transport essentially has a drift character in the internal electric field. The equivalent effective magnetic field induced by spin-orbit interaction and strain is, taking a light-hole g factor of 1, of the order of 60 mT.

GaAs(001) substrates nitrided with N-2 plasma at various temperatures were investigated after being exposed to air for 40 days. They were studied by means of parallel angle-resolved X-ray photoelectron spectroscopy, scanning electron microscopy, micro-photoluminescence and time-resolved photoluminescence (TRPL). Several nitrided GaAs Schottky diodes were manufactured to find the optimal nitridation conditions for high diode quality. An improvement on the ideality factor was achieved for a diode with 1.3 nm-thick GaN layer grown at room temperature and crystallized at 620 degrees C. The crystallization process was needed to enhance the air-exposed GaAs photoluminescence efficiency by a factor 15. TRPL measurements showed a spectacular increase in the decay time (x4), even for a sample exposed to air for 2 years. A high level of GaAs surface chemical protection was achieved. Indeed, neither the element arsenic As-0 nor Ga and As oxides states were detected at the GaN/GaAs interface for nitridation at high temperature (500 degrees C), yielding a 3.1 nm-thick GaN layer. However, for nitridation temperatures above 300 degrees C, pits with an inversed pyramidal shape and square base were formed at the surface, their size increased as nitridation temperature was raised. These pits acted as non-recombination centers which reduced the GaAs photoluminescence yield.

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discuss in detail. We then give a "catalogue raisonné" of examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology). We discuss a "weakly functorial" aspect of this result. "Splinters" are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. We also investigate their mysterious fpqc analogs, and prove that in prime characteristic, they are all regular. This leads us to the problem of descent of regularity by (non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma.

We have used a polarized microluminescence technique to investigate photocarrier charge and spin transport in n-type depleted GaAs nanowires (≈ 10 cm−3 doping level). At 6K, a longdistance tail appears in the luminescence spatial profile, indicative of charge and spin transport, only limited by the length of the NW. This tail is independent on excitation power and temperature. Using a self-consistent calculation based on the drift-diffusion and Poisson equations as well as on photocarrier statistics (Van Roosbroeck model), it is found that this tail is due to photocarrier drift in an internal electric field nearly two orders of magnitude larger than electric fields predicted by the usual ambipolar model. This large electric field appears because of two effects. Firstly, for transport in the spatial fluctuations of the conduction band minimum and valence band maximum, the electron mobility is activated by the internal electric field. This implies, in a counter intuitive way, that the spatial fluctuations favor long distance transport. Secondly, the range of carrier transport is further increased because of the finite NW length, an effect which plays a key role in one-dimensional systems.

On the NASA 2020 rover mission to Jezero crater, the remote determination of the texture, mineralogy and chemistry of rocks is essential to quickly and thoroughly characterize an area and to optimize the selection of samples for return to Earth. As part of the Perseverance payload, SuperCam is a suite of five techniques that provide critical and complementary observations via Laser-Induced Breakdown Spectroscopy (LIBS), Time-Resolved Raman and Luminescence (TRR/L), visible and near-infrared spectroscopy (VISIR), high-resolution color imaging (RMI), and acoustic recording (MIC). SuperCam operates at remote distances, primarily 2–7 m, while providing data at sub-mm to mm scales. We report on SuperCam’s science objectives in the context of the Mars 2020 mission goals and ways the different techniques can address these questions. The instrument is made up of three separate subsystems: the Mast Unit is designed and built in France; the Body Unit is provided by the United States; the calibration target holder is contributed by Spain, and the targets themselves by the entire science team. This publication focuses on the design, development, and tests of the Mast Unit; companion papers describe the other units. The goal of this work is to provide an understanding of the technical choices made, the constraints that were imposed, and ultimately the validated performance of the flight model as it leaves Earth, and it will serve as the foundation for Mars operations and future processing of the data.

We have investigated the photocarrier charge transport in n-type metallic GaAs nanowires ( 10^17 cm^-3 doping level), grown by hydride vapor phase epitaxy (HVPE) on Si(111) substrates. Analysis of the luminescence intensity spatial profiles for selected energies in the spectrum allows us to determine the spatial distribution of photoelectrons, minority photoholes and electrons of the Fermi sea as a function of distance from the light excitation spot. This analysis shows that charge can be transported over record distances larger than 25 (micro)m at 6K, in spite of the expected localization of minority holes in the potential fluctuations generated by statistical fluctuations of the donor concentration. It is shown that transport is little affected by the fluctuations because of the build up of large internal electric fields which strongly increase the hole and electron mobilities and therefore enable transport. Comparison of the spatial profiles of the emissions due to hot electrons and to the Fermi sea gives evidence for at least three spatial zones, including a zone of excess intrinsic electrons near the excitation spot and a zone of depletion of these electrons at a distance larger than 2-10 (micro)m depending on excitation power. The internal outward electric field increases the kinetic energy of photoholes in the fluctuations so that after a given distance to the excitation spot, there occurs ballistic transport over the fluctuations.

We prove that for $1$-motives defined over an algebraically closed subfield of $\mathbf{C}$, viewed as Nori motives, the motivic Galois group coincides with the Mumford–Tate group. In particular, the Hodge realization of the Tannakian category of Nori motives generated by $1$-motives is fully faithful. This result extends an earlier result by the author, according to which Hodge cycles on abelian varieties are motivated (a weak form of the Hodge conjecture).

We describe the Griffiths group of the product of a curve C and a surface S as a quotient of the Albanese kernel of S over the function field of C. When C is a hyperplane section of S varying in a Lefschetz pencil, we prove the nonvanishing in Griff(C x S) of a modification of the graph of the embedding C hooked right arrow S for infinitely many members of the pencil, provided the ground field k is of characteristic 0, the geometric genus of S is > 0, and k is large or S is "of motivated abelian type".

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of homological conjectures in commutative algebra [H1][HH2]. Namely, for any local homomorphism $ R\to R'$ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay $R$-algebra and some Cohen-Macaulay $R'$-algebra. When $R$ contains a field, this is already known [[3.9]{HH2}]. When $R$ is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz's refined treatment [D] of weak functoriality of Cohen-Macaulay algebras in characteristic $p$; in fact, developing a tilting argument due to K. Shimomoto, we combine the perfectoid techniques of [A1][A2] with Dietz's result.

These notes give a quick overview of recent work by the speaker and of some subsequent works introducing perfectoid geometry into homological commutative algebra and singularity theory. The emphasis is on big Cohen-Macaulay algebras and applications. The progresses take place primarily in mixed characteristic, but sometimes provide a bridge between characteristic p and characteristic 0.

We show evidence that tunnel junctions (TJs) in GaN grown by metal-organic vapor phase epitaxy are dominated by defect level-assisted tunneling. This is in contrast with the common belief that highly doped layers (>10(20)cm(-3)) are required to narrow the TJ space charge region and promote the band-to-band tunneling. Our conclusion stems from the study and the review of the major doping limitations of carefully optimized p(++)and n(++)layers. The secondary ions mass spectroscopy profiles of GaN based TJ LEDs show a strong oxygen concentration located close to the p(++)/n(++)interface, typical for three dimensional growth. In addition, considering the doping limitation asymmetry and Mg carry-over, our simulations indicate a depletion region of more than 10 nm which is buried in a rough and defective n(++)layer. However, decent electrical characteristics of the studied TJ based LEDs are obtained, with a low penalty voltage of 1.1 V and a specific differential resistance of about 10(-2)omega.cm(2)at 20 mA. This indicates that a common TJ could be greatly optimized by using a moderate doping (similar to 10(19)cm(-3)) while intentionally introducing local defects within the TJ.