Institut de Mathématiques de Jussieu-Paris Rive Gauche

We report on work to increase the number of well-measured Type Ia supernovae (SNe Ia) at high redshifts. Light curves, including high signal-to-noise Hubble Space Telescope data, and spectra of six SNe Ia that were discovered during 2001, are presented. Additionally, for the two SNe with z > 1, we present ground-based J-band photometry from Gemini and the Very Large Telescope. These are among the most distant SNe Ia for which ground-based near-IR observations have been obtained. We add these six SNe Ia together with other data sets that have recently become available in the literature to the Union compilation. We have made a number of refinements to the Union analysis chain, the most important ones being the refitting of all light curves with the SALT2 fitter and an improved handling of systematic errors. We call this new compilation, consisting of 557 SNe, the Union2 compilation. The flat concordance CDM model remains an excellent fit to the Union2 data with the best-fit constant equation-of-state parameter w = -0.997 +0.050 -0.054 (stat) +0.077 -0.082 (stat + sys together) for a flat universe, or w = -1.038 +0.056 -0.059 (stat) +0.093 -0.097 (stat + sys together) with curvature. We also present improved constraints on w(z). While no significant change in w with redshift is detected, there is still considerable room for evolution in w. The strength of the constraints depends strongly on redshift. In particular, at z 1, the existence and nature of dark energy are only weakly constrained by the data.

This paper proposes a general approach and a computationally convenient estimation procedure for the structural analysis of auction data. Considering first-price sealed-bid auction models within the independent private value paradigm, we show that the underlying distribution of bidders' private values is identified from observed bids and the number of actual bidders without any parametric assumptions. Using the theory of minimax, we establish the best rate of uniform convergence at which the latent density of private values can be estimated nonparametrically from available data. We then propose a two-step kernel-based estimator that converges at the optimal rate.

Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.

The multifractal formalism originally introduced for singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal functions (the wavelets actually play the role of “generalized boxes”). We report on a systematic comparison between this alternative method and the structure function approach which is commonly used in the context of fully developed turbulence. We comment on the intrinsic limitations of the structure functions which possess fundamental drawbacks and do not provide a full characterization of the singularities of a signal in many cases. We show that our method based on the wavelet transform modulus maxima decomposition works in most situations and is likely to be the ground of a unified multifractal description of singular distributions. Our theoretical considerations are both illustrated on pedagogical examples, e.g., generalized devil staircases and fractional Brownian motions, and supported by numerical simulations. Recent applications of the wavelet transform modulus maxima method to experimental turbulent velocity signals at inertial range scales are compared to previous measurements based on the structure function approach. A similar analysis is carried out for the locally averaged dissipation and the validity of the Kolmogorov’s refined similarity hypothesis is discussed. To conclude, we elaborate on a wavelet based technique which goes further than a simple statistical characterization of the scaling properties of fractal objects and provides a very promising tool for solving the inverse fractal problem, i.e., for uncovering their construction rule in terms of a discrete dynamical system.

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of “movable curves”, which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

We realize experimentally an atom-optics quantum-chaotic system, the quasiperiodic kicked rotor, which is equivalent to a 3D disordered system that allows us to demonstrate the Anderson metal-insulator transition. Sensitive measurements of the atomic wave function and the use of finite-size scaling techniques make it possible to extract both the critical parameters and the critical exponent of the transition, the latter being in good agreement with the value obtained in numerical simulations of the 3D Anderson model.

The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.

We prove new potential modularity theorems for n-dimensional essentially self-dual l -adic representations of the absolute Galois group of a totally real field. Most notably, in the ordinary case we prove quite a general result. Our results suffice to show that all the symmetric powers of any non-CM, holomorphic, cuspidal, elliptic modular newform of weight greater than one are potentially cuspidal automorphic. This in turns proves the Sato–Tate conjecture for such forms. (In passing we also note that the Sato–Tate conjecture can now be proved for any elliptic curve over a totally real field.)

A new classification of 15 relief patterns at the global scale combines a relief roughness indicator and the maximum altitude at a resolution of 30′ × 30′. Classical geographic terms have been retained but assigned to fixed relief roughness (RR = maximum minus minimum elevation per cell divided by half the cell length in meters/kilometer, or ‰) and altitude boundaries. Plains (33.2 Mkm2 of currently nonglaciated land surface) correspond to subhorizontal terrain (RR < 5‰). Lowlands (19.2 Mkm; 0–200 m) have a very low degree of roughness (5 <RR <20‰). Platforms and hills (30.5 Mkm2) correspond to the 200–500-m mean elevation class and have a greater degree of roughness (RR > 20‰). Plateaus (16.8 Mkm2), with mean elevations between 500 and 6000 m, have a medium degree of roughness (RR from 5 to 40‰). Mountains (33.3 Mkm2) are differentiated from hills by their higher mean elevation (>500 m) and from plateaus by their greater roughness (>20‰ then >40‰) in each elevation class. Accordingly, Tibet and the Altiplano are very high plateaus, not mountains. These quantitative definitions of relief patterns were divided into 15 classes, then clustered into 9 main types and mapped at the global scale at a resolution for which water runoff depth and population were previously determined. We also differentiated between exorheic areas (115.6 Mkm2 globally) and endorheic areas (17.36 Mkm2 globally) of potential runoff. Mountains thus account for 25% of the Earth's total land area, 32% of surface runoff, and 26% of the global population. The presence or vicinity of a rough and elevated landscape is less limiting to human settlement than water runoff.

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study citeBuanMarshReinekeReitenTodorov04 with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also citeCalderoChapotonSchiffler04) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel citeHappelPreiserRingel80, whose triangulated structure goes back to Auslander-Reiten's work citeAuslanderReiten87, citeReiten87, citeAuslanderReiten96.

The fractal scaling properties of DNA sequences are analyzed using the wavelet transform. Because the wavelet transform microscope can be made blind to the ``patchiness'' of genomic sequences, we demonstrate and quantify the existence of long-range correlations in genes containing introns and noncoding regions. Moreover, the fluctuations in the patchy landscapes of DNA walks are found to be homogeneous with Gaussian statistics.

This is the first of a series of papers about quantization in the context of derived algebraic geometry . In this first part, we introduce the notion of n - shifted symplectic structures ( n -symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n -stacks (see Toën and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toën in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n -symplectic structure, the derived mapping stack Map ( X , F ) is equipped with a canonical ( n − d )-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d . These two results imply the existence of many examples of derived moduli stacks equipped with n -symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π 1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234 , 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map ( E , T ∗ X ), for E an elliptic curve and T ∗ X is the total space of the cotangent bundle of a smooth scheme X . Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π 1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).

High-frequency trading is an algorithm-based computerized trading practice that allows firms to trade stocks in milliseconds. Over the last fifteen years, the use of statistical and econometric methods for analyzing high-frequency financial data has grown exponentially. This growth has been driven by the increasing availability of such data, the technological advancements that make high-frequency trading strategies possible, and the need of practitioners to analyze these data. This comprehensive book introduces readers to these emerging methods and tools of analysis. Yacine Ait-Sahalia and Jean Jacod cover the mathematical foundations of stochastic processes, describe the primary characteristics of high-frequency financial data, and present the asymptotic concepts that their analysis relies on. Ait-Sahalia and Jacod also deal with estimation of the volatility portion of the model, including methods that are robust to market microstructure noise, and address estimation and testing questions involving the jump part of the model. As they demonstrate, the practical importance and relevance of jumps in financial data are universally recognized, but only recently have econometric methods become available to rigorously analyze jump processes. Ait-Sahalia and Jacod approach high-frequency econometrics with a distinct focus on the financial side of matters while maintaining technical rigor, which makes this book invaluable to researchers and practitioners alike.

There are two ways to present this work; the most efficient is of course to start with the main syntactical definitions, and to end with semantics: this is the presentation that we follow in the body of the text: section 1, syntex; section 2, semantics. Another possibility is to follow the order of discovery of the concepts, which (as expected) starts with the semantics and ends with the syntex; we adopt this second way for our introduction, hoping that this orthogonal look at the same object will help to apprehend the concepts.

Abstract The growing need to store an increasing amount of renewable energy in a sustainable way has rekindled interest for sodium-ion battery technology, owing to the natural abundance of sodium. Presently, sodium-ion batteries based on Na 3 V 2 (PO 4 ) 2 F 3 /C are the subject of intense research focused on improving the energy density by harnessing the third sodium, which has so far been reported to be electrochemically inaccessible. Here, we are able to trigger the activity of the third sodium electrochemically via the formation of a disordered Na x V 2 (PO 4 ) 2 F 3 phase of tetragonal symmetry ( I 4 /mmm space group). This phase can reversibly uptake 3 sodium ions per formula unit over the 1 to 4.8 V voltage range, with the last one being re-inserted at 1.6 V vs Na + /Na 0 . We track the sodium-driven structural/charge compensation mechanism associated to the new phase and find that it remains disordered on cycling while its average vanadium oxidation state varies from 3 to 4.5. Full sodium-ion cells based on this phase as positive electrode and carbon as negative electrode show a 10–20% increase in the overall energy density.

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L1+ε-density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

Abstract We define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

We extend the methods of Wiles and of Taylor and Wiles from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> -adic lifts of certain automorphic mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> Galois representations of any dimension. We also make a conjecture about the structure of mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . Following Wiles' method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.

We define a generalized index of jump activity, propose estimators of that index for a discretely sampled process and derive the estimators’ properties. These estimators are applicable despite the presence of Brownian volatility in the process, which makes it more challenging to infer the characteristics of the small, infinite activity jumps. When the method is applied to high frequency stock returns, we find evidence of infinitely active jumps in the data and estimate their index of activity.

Soit -adique d’un groupe réductif connexe. On définit l’espace des fonctions de Schwartz-Harish-Chandra sur , à valeurs complexes, qui vérifient des conditions de croissance et de lissité. La formule de Plancherel exprime les valeurs d’une telle fonction en termes des opérateurs parcourt l’ensemble des classes de représentations lisses irréductibles et tempérées de . On démontre cette formule, ainsi que quelques résultats utiles d’analyse harmonique: l’existence du prolongement rationnel d’un opérateur d’entrelacement, la finitude (si est semi-simple) de l’ensemble des classes de représentations lisses irréductibles de carré intégrable de possédant un -type donné. Tous ces résultats sont dus à Harish-Chandra, qui les a démontrés dans un manuscrit non publié. Le présent article est une rédaction de ce manuscrit.AMS 2000 Mathematics subject classification: Primary 22E35; 22E50