Point simpliciality in Choquet theory on nonmetrizable compact spaces

Let H be a function space on a compact space K. The set of simpliciality of H is the set of all points of K for which there exists a unique maximal representing measure. Properties of this set were studied by M. Ba?ák in the paper Point simpliciality in Choquet representation theory, Illinois J. Math. 53 (2009) 289–302, mainly for K metrizable. We study properties of the set of simpliciality for K nonmetrizable.     

The topology of moduli spaces of group representations: The case of compact surface

Let G be a connected complex semisimple affine algebraic group, and let K be a maximal compact subgroup of G. Let X be a noncompact oriented surface. The main theorem of Florentino and Lawton (2009) [3] says that the moduli space of flat K-connections on X is a strong deformation retraction of the moduli space of flat G-connections on X. We prove that this statement fails whenever X is compact of genus at least two.     

Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces

As a consequence of the exposition of Dixmier type traces in the book of A. Connes (1994) [2], we were led to ask how general is this class of functionals within the space of all unitarily invariant functionals on the corresponding Marcinkiewicz ideal Mψ. In this paper we prove the surprising result that the set of all Dixmier traces on Mψ coincides with the set of all fully symmetric functionals on this space.     

Positive definite distributions and normed spaces

We answer a question of Alex Koldobsky. We show that for each −∞<p<2 and each n?3−p there is a normed space X of dimension n which embeds in Ls if and only if −n<s?p.     

LG/CY correspondence: The state space isomorphism

We prove the mirror duality conjecture for the mirror pairs constructed by Berglund, Hübsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi–Yau hypersurfaces inside a weighted projective space and the Fan–Jarvis–Ruan–Witten state space of the associated Landau–Ginzburg singularity theory.     

Projection bodies in complex vector spaces

The space of Minkowski valuations on an m-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is shown to satisfy geometric inequalities of the Brunn–Minkowski, Aleksandrov–Fenchel and Minkowski type.     

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the Hs norm of the solution when the time goes to infinity.     

A simple proof of the Grobman–Hartman theorem for nonuniformly hyperbolic flows

We give a simple and direct proof of the Grobman–Hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy. In particular, we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps. To the best of our knowledge, this is the first direct approach for nonuniform exponential dichotomies. We also show that the conjugacies are continuous in time and Hölder continuous in space. In addition, we describe the dependence of the conjugacies on the perturbation, and we obtain a reversibility result for the conjugacies of reversible differential equations. We emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial.     

Algebras, Projective Geometry, Mathematical Logic, and Constructing the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead

A. N. Whitehead (1861–1947) contributed notably to the foundations of pure and applied mathematics, especially from the late 1890s to the mid 1920s. An algebraist by mathematical tendency, he surveyed several algebras in his book Universal Algebra (1898). Then in the 1900s he joined Bertrand Russell in an attempt to ground many parts of mathematics in the newly developing mathematical logic. In this connection he published in 1906 a long paper on geometry, space and time, and matter. The main outcome of the collaboration was a three-volume work, Principia Mathematica (1910–1913): he was supposed to write a fourth volume on parts of geometries, but he abandoned it after much of it was done. By then his interests had switched to educational issues, and especially to space and time and relativity theory, where his earlier dependence upon logic was extended to an ontology of events and to a general notion of “process,” especially in human experience. These innovations led to somewhat revised conceptions of logic and of the philosophy of mathematics. © 2002 Elsevier Science (USA).A. N. Whitehead (1861–1947) contribuiu de forma marcante para os Fundamentos da Matemática Pura e Aplicada, especialmente entre o fim da década de 1890 e meados da década de 1920. Sendo um algebrista na sua vertente matemática, fez um levantamento de diversas álgebras no seu livro Universal Algebra (1898). Pouco depois de 1900 juntou-se a Bertrand Russell numa tentativa para basear várias partes da matemática sobre a lógica matemática, que se começava então a desenvolver. Nesse âmbito publicou em 1906 um longo artigo sobre geometria, espaço e tempo, e matéria. O principal resultado da colaboração foi um trabalho em três volumes, Principia Mathematica (1910–1913): estava previsto que Whitehead escrevesse um quarto volume sobre aspectos das geometrias, mas abandonou-o depois de uma boa parte já estar escrita. Por essa altura os seus interesses tinham-se voltado para questões educacionais; especialmente para o espaço e o tempo e para a teoria da relatividade, onde a sua anterior dependência da lógica se estendeu a uma ontologia de acontecimentos e a uma noção geral de “processo” especialmente na experiência humana. Estas inovações levaram a concepções um pouco revistas da lógica e da filosofia da matemática. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 00A30; 01A60; 03-03; 03A05.     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.     

BLO spaces associated with the Ornstein–Uhlenbeck operator

Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L^∞(γ) to BLOa(γ).     

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.     

Support theorem for the fundamental solution to the Schrödinger equation on certain compact symmetric spaces

In this paper we construct the fundamental solution to the Schödinger equation on a compact symmetric space with even root multiplicities using shift operators of Heckman and Opdam. Next, we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time. Moreover, we also show that generalized Gauss sums appear in the expression of the fundamental solution.     

Torus rational fibrations

We study rational fibrations where the fibre is an r-dimensional torus and the base is a formal space. We make use of the Eilenberg–Moore Spectral Sequence to prove the Toral Rank Conjecture in some cases.     

Algebraic K-theory and abstract homotopy theory

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer–Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra.     

Topological dynamical systems associated to II1-factors

If N⊂Rω is a separable II₁-factor, the space Hom(N,Rω) of unitary equivalence classes of unital ?-homomorphisms N→Rω is shown to have a surprisingly rich structure. If N is not hyperfinite, Hom(N,Rω) is an infinite-dimensional, complete, metrizable topological space with convex-like structure, and the outer automorphism group Out(N) acts on it by “affine” homeomorphisms. (If N≅R, then Hom(N,Rω) is just a point.) Property (T) is reflected in the extreme points – they?re discrete in this case. For certain free products N=Σ?R, every countable group acts nontrivially on Hom(N,Rω), and we show the extreme points are not discrete for these examples. Finally, we prove that the dynamical systems associated to free group factors are isomorphic.     

Local polynomial regression for circular predictors

We consider local smoothing of datasets where the design space is the d-dimensional (d≥1) torus and the response variable is real-valued. Our purpose is to extend least squares local polynomial fitting to this situation. We give both theoretical and empirical results.     

Compactness of Schrödinger semigroups with unbounded below potentials

By using the super Poincaré inequality of a Markov generator L₀ on L²(μ) over a σ-finite measure space (E,F,μ), the Schrödinger semigroup generated by L₀−V for a class of (unbounded below) potentials V is proved to be L²(μ)-compact provided μ(V?N)<∞ for all N>0. This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., Rd under the condition that V(x)→∞ as |x|→∞. Concrete examples are provided to illustrate the main result.     

Integral-type operators on continuous function spaces on the real line

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu^′′+βu^′+γu.Shape-preserving and regularity properties are also investigated.