We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure.

Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G\left( \lambda M \right)=V\left( \lambda M \right)+O\left( {{\lambda }^{d-1-\varepsilon \left( d \right)}} \right)$ for some positive $\varepsilon (d)$. Here we give for general convex bodies the weaker estimate

$$|G\left( \lambda M \right)-V\left( \lambda M \right)|\,\le \,\frac{1}{2}{{S}_{{{Z}^{d}}}}\left( M \right){{\lambda }^{d-1}}+o\left( {{\lambda }^{d-1}} \right)$$

where ${{S}_{{{Z}^{d}}}}\left( M \right)$ denotes the lattice surface area of $M$. The term ${{S}_{{{Z}^{d}}}}\left( M \right)$ is optimal for all convex bodies and $o\left( {{\lambda }^{d-1}} \right)$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$.

Further we deal with families $\left\{ {{P}_{\lambda }} \right\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have

$$|G\left( {{P}_{\lambda }} \right)-V\left( {{P}_{\lambda }} \right)|\,\le \,dV\left( {{P}_{\lambda }},\,K;1 \right)+o\left( S\left( {{P}_{\lambda }} \right) \right)$$

where the convex body $K$ satisfies some simple condition, $V\left( {{P}_{\lambda }},K;1 \right)$ is some mixed volume and $S\left( {{P}_{\lambda }} \right)$ is the surface area of ${{P}_{\lambda }}$.

In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime ${{\mathbb{L}}^{n+1}}$ . We also present an isoperimetric inequality in the Minkowski spacetime ${{\mathbb{L}}^{n+1}}$ as a consequence of this Brunn-Minkowski type theorem.

We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, i.e., to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra.

We consider a class of fractal subsets of ${{\mathbb{R}}^{d}}$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

The cohomology algebra mod $p$ of the complex projective Stiefel manifolds is determined for all primes $p$. When $p=2$ we also determine the action of the Steenrod algebra and apply this to the problem of existence of trivial subbundles of multiples of the canonical line bundle over a lens space with 2-torsion, obtaining optimal results in many cases.

In this article we characterize the univalent harmonic mappings from the exterior of the unit disk, $\Delta $, onto a simply connected domain $\Omega $ containing infinity and which are solutions of the system of elliptic partial differential equations $\overline{{{f}_{{\bar{z}}}}\left( Z \right)}=a\left( z \right){{f}_{z}}\left( z \right)$ where the second dilatation function $a\left( z \right)$ is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in ${{\text{C}}^{\text{2}}}$ is studied in details.

The trace formula contains terms on the spectral side that are constructed from unramified automorphic $L$-functions. We shall establish an identify that relates these terms with corresponding terms attached to endoscopic groups of $G$. In the process, we shall show that the $L$-functions of $G$ that come from automorphic representations of endoscopic groups have meromorphic continuation.

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a ${{C}^{*}}$ -coaction $\delta :D\to D\otimes {{C}^{*}}\left( G/N \right)$ of a quotient group $G/N$ of a discrete group $G$ to a ${{C}^{*}}$-coaction $\text{Ind}\,\delta \text{:}\,\text{Ind}\,D\to \text{Ind}\,D\otimes {{C}^{*}}\left( G \right)\,\text{of }G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\text{Ind}\,D{{\times }_{\text{Ind}\,\delta }}\,G\,\text{and }D{{\times }_{\delta }}\,G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

A field $F$ is said to be tractable when a condition described below on the simultaneous representation of quaternion algebras holds over $F$. It is shown that a global field $F$ is tractable iff $F$ has at most one dyadic place. Several other examples of tractable and nontractable fields are given.

This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G\left( E \right)$, with periods by a subgroup $G\left( F \right)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E=F\oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy.

The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces $H\left( E \right)/H\left( F \right)$. This offers a sharpening of the theory of liftings, where periods play a key role.

The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.

Finally, the fundamental lemma is verified for $\text{SL}\left( 2 \right)$.

We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Gamma$-automorphic Laplacian below a given bound when $\Gamma$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Gamma$ in the family is assumed to contain a principal congruence subgroup whose index in $\Gamma$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Gamma$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice $\Gamma$.

We develop a method of separable reduction for Fréchet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fréchet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

This article studies algebras $R$ over a simple artinian ring $A$, presented by a quiver and relations and graded by a semigroup $\Sigma $. Suitable semigroups often arise from a presentation of $R$. Throughout, the algebras need not be finite dimensional. The graded ${{K}_{0}}$, along with the $\Sigma $-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.

A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert $\Sigma $-series in the associated path incidence ring.

The rationality of the $\Sigma $-Euler characteristic, the Hilbert $\Sigma $-series and the Poincaré-Betti $\Sigma $-series is studied when $\Sigma $ is torsion-free commutative and $A$ is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

Let $E$ be an elliptic curve over $\mathbb{Q}$ , and let $r$ be an integer. According to the Lang-Trotter conjecture, the number of primes $p$ such that ${{a}_{p}}\left( E \right)=r$ is either finite, or is asymptotic to ${{C}_{E,r}}\sqrt{x}/\log x$ where ${{C}_{E,r}}$ is a non-zero constant. A typical example of the former is the case of rational $\ell $-torsion, where ${{a}_{p}}\left( E \right)=r$ is impossible if $r\equiv 1\,\left( \bmod \,\ell \right)$. We prove in this paper that, when $E$ has a rational $\ell $-isogeny and $\ell \ne 11$, the number of primes $p$ such that ${{a}_{p}}\left( E \right)\equiv r\,\left( \bmod \,\ell \right)$ is finite (for some $r$ modulo $\ell $) if and only if $E$ has rational $\ell $-torsion over the cyclotomic field $\mathbb{Q}\left( {{\zeta }_{\ell }} \right)$ . The case $\ell =11$ is special, and is also treated in the paper. We also classify all those occurences.

A reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one-dimensional group generated by all reflections with fixed axis and center.

In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori.

In general, the tensor product, $A\otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a capped tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for every lattice $B$ with zero; we shall call such lattices amenable.

The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition $\left( \text{T} \right)$.

We prove that a finite lattice $A$ is amenable iff it is sharply transferable as a join-semilattice.

For a general lattice $A$ with zero, we obtain the result: $A$ is amenable iff $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice.

This yields, for example, that a finite lattice $A$ is amenable iff $A\otimes \text{F}\left( 3 \right)$ is a lattice iff $A$ satisfies $\left( \text{T} \right)$, with respect to join. In particular, ${{M}_{3}}\,\otimes \,\text{F}\left( 3 \right)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system $\left( cf.\,[27],\,[28],\,[13] \right)$.