We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

Let $(T,M)$ be a complete local (Noetherian) ring such that $\dim T\geq 2$ and $|T|=|T/M|$ and let $\{p_i\} _{i \in \mathcal I}$ be a collection of elements of T indexed by a set $\mathcal I$ so that $|\mathcal I | < |T|$. For each $i \in \mathcal{I}$, let $C_i:=\{Q_{i1},\dots,Q_{in_i}\}$ be a set of nonmaximal prime ideals containing $p_i$ such that the $Q_{ij}$ are incomparable and $p_i\in Q_{jk}$ if and only if $i=j$. We provide necessary and sufficient conditions so that T is the ${\bf m}$-adic completion of a local unique factorization domain $(A, {\bf m})$, and for each $i \in \mathcal I$, there exists a unit $t_i$ of T so that $p_it_i \in A$ and $C_i$ is the set of prime ideals Q of T that are maximal with respect to the condition that $Q \cap A = p_it_iA$. We then use this result to construct a (nonexcellent) unique factorization domain containing many ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A most of whose formal fibers are geometrically regular.

No abstract.

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases. In the last part of the paper, we describe a noncommutative deformation functor for quasi-coherent sheaves of modules on a ringed space $(X,\mathcal{A})$. We show that for any good $\mathcal{A}$-affine open cover $\mathsf{U}$ of X, the forgetful functor $\mathsf{QCoh}\mathcal{A} \to \mathsf{PreSh}(\mathsf{U}, \mathcal{A})$ induces an isomorphism of noncommutative deformation functors. \emph{Applications.} We consider noncommutative deformations of quasi-coherent $\mathcal{A}$-modules on X when $(X, \mathcal{A}) = (X, \mathcal{O}_X)$ is a scheme or $(X, \mathcal{A}) = (X, \mathcal{D})$ is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of X closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left $\sh D_X$-module $\mathcal{D}_X$ when X is an elliptic curve as an example.

If an algebraic torus T acts on a complex projective algebraic variety X, then the affine scheme $\operatorname{Spec} H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H₂^T(X;\mathbb C).$ In many situations the ordinary cohomology ring of X can be described in terms of this arrangement.

Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C.~L.\ Jiang.

We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

The space of Monge–Ampère functions, introduced by J.~H.~G.~Fu, is a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D² u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge–Ampère functions. We also prove that if a Monge–Ampère function u on a bounded set $\Omega\subset\R²$ satisfies the equation $\operatorname{Det} D² u=0$ in a particular weak sense, then the graph of u is a developable surface, and moreover u enjoys somewhat better regularity properties than an arbitrary Monge–Ampère function of 2 variables.

Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X={X_j}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= div_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.

An x-pseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.

It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration $C_{14}$. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations. Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective non-affine point-transitive Steiner triple system S.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras $\LL_q$. The center of $\LL_q$ now is generally infinite dimensional. In this paper, $\Z$-graded Verma modules $\widetilde{V}(\varphi)$ over $\LL_q$ and their corresponding irreducible highest weight modules $V(\varphi)$ are defined for some linear functions $\varphi$. Necessary and sufficient conditions for $V(\varphi)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\widetilde{V}(\varphi)$ to be irreducible are obtained.

We study p-indivisibility of the central values $L(1,E_d)$ of quadratic twists $E_d$ of a semi-stable elliptic curve E of conductor N. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families $\FF_S$, indexed by subsets S of the primes dividing N. Let $\delta_S= \gcd_{d\in \mathcal{F}_S} L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ denotes the algebraic part of the central L-value, $L(1,E_d)$. Our main theorem relates the p-adic valuations of $\delta_S$ as S varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation between Waldspurger packets on $\widetilde{\operatorname{SL}₂}$ and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of p-indivisibility of L-values of quadratic twists.

We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.

Let K denote a compact real symmetric subset of $\mC$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on K that are analytic in the interior $K^\circ$ of K, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)²$ which are reducible. In addition, for an arbitrary compact K in $\mathbb C$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A(K)$ is 1. Finally, we also characterize all compact real symmetric sets K such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank 1.

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

In this paper we study the supersingular locus of the reduction modulo p of the Shimura variety for GU(1,s) in the case of an inert prime p. Using Dieudonné theory we define a stratification of the corresponding moduli space of p-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat-Tits building of a unitary group. In the case of GU(1,2), we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

We use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety $\Flag_G$ generated by degree 2. We use this result to show that the vertices of the moment map image of $\Flag_G$ lie on a paraboloid.