On calcule les restrictions aux points fixes de deux bases de la cohomologie équivariante des tours de Bott, et on précise la structure multiplicative de ces algèbres de cohomologie. En s’aidant du travail de Duan, on en déduit une méthode de calcul des constantes de structure de la cohomologie équivariante des variétés de drapeaux (calcul de Schubert équivariant).

‘Equivariant cohomology of Bott towers and equivariant Schubert calculus’. We calculate the restrictions to fixed points of two bases of the equivariant cohomology of Bott towers, and we describe the multiplicative structure of these cohomology algebras. We build on the work of Duan to give a method to calculate the structure constants of the equivariant cohomology of the flag varieties (equivariant Schubert calculus).

We examine 2-complexes $X$ with the property that for any compact connected $Y$, and immersion $Y\rightarrow X$, either $\unicode[STIX]{x1D712}(Y)\leqslant 0$ or $\unicode[STIX]{x1D70B}_{1}Y=1$. The mapping torus of an endomorphism of a free group has this property. Every irreducible 3-manifold with boundary has a spine with this property. We show that the fundamental group of any 2-complex with this property is locally indicable. We outline evidence supporting the conjecture that this property implies coherence. We connect the property to asphericity. Finally, we prove coherence for 2-complexes with a stricter form of this property. As a corollary, every one-relator group with torsion is coherent.

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of $\mathit{SO}_{2n+1}$ of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

We show that a twisted Hilbert space with an unconditional basis is isomorphic to a Hilbert space.

AMS 2000 Mathematics subject classification: Primary 46B10. Secondary 46C15

We use coefficient systems on the affine Bruhat–Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

Consider a $C^{2}$ family of mixing $C^{4}$ piecewise expanding unimodal maps $t\in [a,b]\mapsto f_{t}$, with a critical point $c$, that is transversal to the topological classes of such maps. Given a Lipchitz observable $\unicode[STIX]{x1D719}$ consider the function

$$\begin{eqnarray}{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)=\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t},\end{eqnarray}$$

$\unicode[STIX]{x1D707}_{t}$

$f_{t}$

$\unicode[STIX]{x1D70E}_{t}>0$

$t\in [a,b]$

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{t}^{2}=\unicode[STIX]{x1D70E}_{t}^{2}(\unicode[STIX]{x1D719})=\lim _{n\rightarrow \infty }\int \left(\frac{\mathop{\sum }_{j=0}^{n-1}\left(\unicode[STIX]{x1D719}\circ f_{t}^{j}-\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t}\right)}{\sqrt{n}}\right)^{2}\,d\unicode[STIX]{x1D707}_{t}.\end{eqnarray}$$

We show that

$$\begin{eqnarray}m\left\{t\in [a,b]:t+h\in [a,b]\text{ and }\frac{1}{\unicode[STIX]{x1D6F9}(t)\sqrt{-\log |h|}}\left(\frac{{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t+h)-{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)}{h}\right)\leqslant y\right\}\end{eqnarray}$$

$$\begin{eqnarray}\frac{1}{\sqrt{2\unicode[STIX]{x1D70B}}}\int _{-\infty }^{y}e^{-\frac{s^{2}}{2}}\,ds,\end{eqnarray}$$

$\unicode[STIX]{x1D6F9}(t)$

$m$

$[a,b]$

$m([a,b])=1$

${\mathcal{R}}_{\unicode[STIX]{x1D719}}$

$[a,b]$

We consider valued fields with a value-preserving automorphism and improve on model-theoretic results by Bélair, Macintyre and Scanlon on these objects by dropping assumptions on the residue difference field. In the equicharacteristic 0 case we describe the induced structure on the value group and the residue difference field.

Let $\mathcal{L}^{S}\left (s,\pi ,{\mathfrak{st}}\right )$ be a partial $\mathcal{L}$-function of degree $7$ of a cuspidal automorphic representation $\pi $ of the exceptional group $G_2$. In this paper we construct a Rankin–Selberg integral for representations having a certain Fourier coefficient.

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of . In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

Using the $\ell $-invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of $p$-adic $L$-functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

in this article we compare different conditions on abelian schemes with real multiplication which occur in the integral models of the hilbert–blumenthal shimura variety considered by rapoport, deligne, pappas and kottwitz. we show that the models studied by deligne/pappas and kottwitz are isomorphic over $\mathrm{spec}\mathbb{z}_{(p)}$. we also examine the associated local models and prove that they are equal.

Following Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T}_d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound.

It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

We deal with the incompressible Navier–Stokes equations with vortex patches as initial data. Such data describe an initial configuration for which the vorticity is discontinuous across a hypersurface. We give an asymptotic expansion of the solutions in the vanishing viscosity limit which exhibits an internal layer where the fluid vorticity has a sharp variation. This layer moves with the flow of the Euler equations.

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

The notion of gluing of abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied further by Polishchuk. We observe that this notion is a particular case of a general categorical construction.

We then apply this general notion to the study of the ring of global differential operators $\mathcal{D}$ on the basic affine space $G/U$ (here $G$ is a semi-simple simply connected algebraic group over $\mathbb{C}$ and $U\subset G$ is a maximal unipotent subgroup).

We show that the category of $\mathcal{D}$-modules is glued from $|W|$ copies of the category of $D$-modules on $G/U$ where $W$ is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that the algebra $\mathcal{D}$ is Noetherian, and get some information on its homological properties.

AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20