Volume 2012, Issue 667

We prove under a weak smoothness condition that two Riemannian manifolds are isomorphic if and only if there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.

We study semicomplete meromorphic vector fields on complex surfaces, that is, vector fields whose solutions are single-valued in restriction to the open set where the vector field is holomorphic. We show that, up to a birational transformation, a compact connected component of the curve of poles is either a rational or an elliptic curve of null self-intersection or it has the combinatorics of a singular fiber of an elliptic fibration. This result is then globalized by proving that, always up to a birational transformation, a semicomplete meromorphic vector field on a compact complex Kähler surface must satisfy at least one of the following conditions: to be globally holomorphic, to possess a non-trivial meromorphic first integral or to preserve a fibration. In particular, this extends the results established by Brunella for complete polynomial vector fields in the complex plane to the context of semicomplete ones.

Let X  ⊂ ℂ 3 be a surface with an isolated singularity at the origin, given by the equation Q ( x ,  y ,  z ) = 0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X  = ℂ 2 / G for G  <  SL 2 (ℂ) finite. Let Y  ≔  S n X be the n -th symmetric power of X . We compute the zeroth Poisson homology HP 0 (𝒪 Y ), as a graded vector space with respect to the weight grading, where 𝒪 Y is the ring of polynomial functions on Y . In the Kleinian case, this confirms a conjecture of Alev, that , where Weyl 2 n is the Weyl algebra on 2 n generators. That is, the Brylinski spectral sequence degenerates in degree zero in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, A γ , for all but countably many parameters γ in the elliptic curve. As a consequence, we deduce a bound on the number of irreducible finite-dimensional representations of all quantizations of Y . This includes the noncommutative spherical symplectic reflection algebras associated to G n  ⋊  S n .

For the ternary quadratic form Q ( x ) =  x 2  +  y 2  −  z 2 and a non-zero Pythagorean triple x 0  ∈ ℤ 3 lying on the cone Q ( x ) = 0, we consider an orbit 𝒪 =  x 0 Γ of a finitely generated subgroup Γ < SO Q (ℤ) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in 𝒪 whose hypotenuse, area, and product of side lengths have few prime factors, where “few” is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T , up to bounded constants.

We study relations between the k -Hessian energy , and the fractional Sobolev energy , where F k ( k  = 1, . . . ,  n ) is the k -Hessian operator on ℝ n , and u is a k -convex function vanishing at ∞. We prove that there is a constant C  > 0 such that C −1 E k [ u ] ≦ ℰ k [ u ] ≦  CE k [ u ], where the lower estimate is obtained under some additional assumptions on u . In general, we show that there exists a function u ̃ such that c −1  ≦ | u / u ̃| ≦  c , and C −1 E k [ u ̃] ≦ ℰ k [ u ] ≦  CE k [ u ̃], where c , C are positive constants depending only on k and n .

Given surjective homomorphisms R  →  T  ←  S of local rings, and ideals in R and S that are isomorphic to some T -module V , the connected sum R ⋕ T S is defined to be the ring obtained by factoring out the diagonal image of V in the fiber product R  × T S . When T is Cohen–Macaulay of dimension d and V is a canonical module of T , it is proved that if R and S are Gorenstein of dimension d , then so is R ⋕ T S . This result is used to study how closely an artinian ring can be approximated by a Gorenstein ring mapping onto it. When T is regular, it is shown that R ⋕ T S almost never is a complete intersection ring. The proof uses a presentation of the cohomology algebra as an amalgam of the algebras and over isomorphic polynomial subalgebras generated by one element of degree 2.

It is shown that, modulo the automorphisms which fix the canonical diagonal MASA point-wise, the group of those automorphisms of 𝒪 n which globally preserve both the diagonal and the core UHF-subalgebra is isomorphic, via restriction, with the group of those homeomorphisms of the full one-sided n -shift space which eventually commute along with their inverses with the shift transformation. The image of this group in the outer automorphism group of 𝒪 n can be embedded into the quotient of the automorphism group of the full two-sided n -shift by its center, generated by the shift. If n is prime then this embedding is an isomorphism.

We give a systematic construction of Hopf algebra structures on braided cofree coalgebras. The relevant underlying structures are braided algebras and braided coalgebras. We provide some interesting examples of these algebras and coalgebras related to quantum groups. We introduce quantum multi-brace algebras which are generalizations of both braided algebras and B ∞ -algebras, as the natural framework. This new subject enables one to quantize some important algebra structures in a uniform way. Particular interesting examples are quantum quasi-shuffle algebras.

We prove existence and uniform bounds for critical static Klein–Gordon–Maxwell–Proca systems in the case of 4-dimensional closed Riemannian manifolds.