On tubular neighborhoods of piecewise linear and topological manifolds.

Analogues of the smooth tubular neighborhood theorem are developed for the topological and piecewise linear categories.

A new construction of Calabi–Yau manifolds: Generalized CICYs

We present a generalization of the complete intersection in products of projective space (CICY) construction of Calabi–Yau manifolds. CICY three-folds and four-folds have been studied extensively in the physics literature. Their utility stems from the fact that they can be simply described in terms of a ‘configuration matrix’, a matrix of integers from which many of the details of the geometries can be easily extracted. The generalization we present is to allow negative integers in the configuration matrices which were previously taken to have positive semi-definite entries. This broadening of the complete intersection construction leads to a larger class of Calabi–Yau manifolds than that considered in previous work, which nevertheless enjoys much of the same degree of calculational control. These new Calabi–Yau manifolds are complete intersections in (not necessarily Fano) ambient spaces with an effective anticanonical class. We find examples with topology distinct from any that has appeared in the literature to date. The new manifolds thus obtained have many interesting features. For example, they can have smaller Hodge numbers than ordinary CICYs and lead to many examples with elliptic and K3-fibration structures relevant to F-theory and string dualities.

Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface

We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles SrM of a Riemannian manifold (M,g) of dim ≥ 3, assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I^G and symplectic structure ω^G on the manifold TM , generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds TM and SrM.

Riemannian questions with a fundamental differential system

Introduzimos o leitor ao estudo de um sistema diferencial exterior fundamental, descoberto anteriormente pelo autor, que se pode sempre associar a qualquer dada variedade riemanniana M de dimensão n+1. Depois de recordarmos a geometria do fibrado de esferas tangente SM--->M com a métrica de Sasaki, apresentamos o sistema de formas diferencias de grau n que complementa a conhecida estrutura de contacto de SM. A partir daí vemos como o sistema diferencial se aplica ao estudo de problemas métricos em hipersuperfícies de M, bem como a outros que são próprios de SM, e as diversas questões que se podem colocar neste novo contexto.

Automorphisms of irreducible holomorphic symplectic manifolds and related problems

We study automorphisms and the mapping class group of irreducible holomorphic symplectic (IHS) manifolds. We produce two examples of manifolds of K3[2] type with a symplectic action of the alternating group A7. Our examples are realized as double EPW-sextics, the large cardinality of the group allows us to prove the irrationality of the associated families of Gushel-Mukai threefolds. We describe the group of automorphisms of double EPW-cubes. We give an answer to the Nielsen realization problem for IHS manifolds in analogy to the case of K3 surfaces, determining when a finite group of mapping classes fixes an Einstein (or Kähler-Einstein) metric. We describe, for some deformation classes, the mapping class group and its representation in second cohomology. We classify non-symplectic involutions of manifolds of OG10 type determining the possible invariant and coinvariant lattices. We study non-symplectic involutions on LSV manifolds that are geometrically induced from non-symplectic involutions on cubic fourfolds.

Quantum Riemannian geometry on graphs for gauge field theory

In questo lavoro estendiamo concetti classici della geometria Riemanniana al fine di risolvere le equazioni di Maxwell sul gruppo delle permutazioni $S_3$. Cominciamo dando la strutture algebriche di base e la definizione di calcolo differenziale quantico con le principali proprietà. Generalizziamo poi concetti della geometria Riemanniana, quali la metrica e l'algebra esterna, al caso quantico. Tutto ciò viene poi applicato ai grafi dando la forma esplicita del calcolo differenziale quantico su $\mathbb{K}(V)$, della metrica e Laplaciano del secondo ordine e infine dell'algebra esterna. A questo punto, riscriviamo le equazioni di Maxwell in forma geometrica compatta usando gli operatori e concetti della geometria differenziale su varietà che abbiamo generalizzato in precedenza. In questo modo, considerando l'elettromagnetismo come teoria di gauge, possiamo risolvere le equazioni di Maxwell su gruppi finiti oltre che su varietà differenziabili. In particolare, noi le risolviamo su $S_3$.

Photophysical properties of a photocytotoxic fluorinated chlorin conjugated to four beta-cyclodextrins

A meso-tetrakis(pentafluorophenyl)-chlorin with the reduced pyrrole ring linked to an isoxazolidine ring (FC) has been conjugated to four beta-cyclodextrins (CDFC). The CDFC exhibits excellent water solubility and is a potent photosensitizer towards proliferating NCTC 2544 human keratinocytes. The study by conventional steady state absorption and fluorescence spectroscopies and by time-resolved femto- and nanosecond laser flash spectroscopies suggests that in ethanol and pH 7 buffer the beta-cyclodextrins embed the highly hydrophobic tetrakis(pentafluorophenyl)-chlorin macrocycle and strongly interact with the chlorin rings in the singlet and triplet manifolds. In these solvents, femtosecond spectroscopy suggests that the conjugate undergoes a rapid relaxation in the upper excited singlet states induced by photochemical and/or conformation change(s) at a rate of about 5 ps(-1) to fluorescent states whose lifetime is similar to 8 ns. This interaction is destroyed upon addition of Triton X100 to buffer. Both FC and CDFC strongly fluoresce (Phi(F) similar to 0.5) in micelles. Similar behavior is observed at the triplet level. In ethanol and water, the initial transient triplet state absorbance decays within 1-3 mu s yielding a longer lived triplet with spectral properties indistinguishable from that of original difference absorbance spectra. The determination of the molar absorbance in the 440-460 nm region (similar to 35 000 M(-1) cm(-1)) leads to an estimate of similar to 0.2 for the triplet formation quantum yield of FC in toluene and of FC and CDFC in Triton X100 micelles. Quenching of the CDFC triplets by dioxygen in buffer produces (1)O(2) in a good yield consistent with the effective photocytotoxicity of the chlorin-cyclodextrins conjugate towards cultured NCTC 2544 human keratinocytes. By contrast, FC which aggregates in buffer produces little if any (1)O(2).

Transport properties in nontwist area-preserving maps

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3247349]

Heat flow for Yang-Mills-Higgs fields, part I

The Yang-Mills-Higgs field generalizes the Yang-Mills field. The authors establish the local existence and uniqueness of the weak solution to the heat flow for the Yang-Mills-Higgs field in a vector bundle over a compact Riemannian 4-manifold, and show that the weak solution is gauge-equivalent to a smooth solution and there are at most finite singularities at the maximum existing time.

Bishadowing and hyperbolicity

Shadowing of a dynamical system is often used to justify the validity of computer simulations of the system, and in numerical calculations an inverse form of the shadowing concept is also of some interest. In this paper we characterize the notion of shadowing in terms of stability, and express the notion of hyperbolicity using the concept of inverse shadowing.