On the Number of Type Change Loci of a Generalized Complex Structure

In this note, we describe a procedure to construct generalized complex structures whose type change locus has arbitrarily many path components on products of the circle with a connected sum of closed 3-manifolds. As an application, we use the procedure to exhibit such geometric structures on a myriad of simply connected 4-manifolds among many others.     

Coverings,correspondences, and noncommutative geometry

We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover cobordisms. We consider dynamical systems obtained from associated convolution algebras endowed with time evolutions defined in terms of the underlying geometries. We describe the relevance of our construction to the problem of spectral correspondences in noncommutative geometry.     

Contact Spheres and Hyperkähler Geometry

A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles.This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in hyperkähler geometry. The classification of taut contact spheres on closed 3-manifolds includes the known classification of 3-Sasakian 3-manifolds, but the local Riemannian geometry of contact spheres is much richer. We construct two examples of taut contact spheres on open subsets of \({\mathbb{R}^3}\) with nontrivial local geometry; one from the Helmholtz equation on the 2-sphere, and one from the Gibbons-Hawking ansatz. We address the Bernstein problem whether such examples can give rise to complete metrics.     

Interaction dynamics of bright solitons in linearly coupled asymmetric systems

In this research work, composite media based on metamaterials including random distribution of spherical nanoparticles in a polymeric foam host are suggested to achieve negative effective refractive index in the visible spectrum. For this purpose structures including single, two and three layer spherical particles are investigated. Based on simulation results, media including single layer spheres (metallic and dielectric particles) and two layer nanospheres (core–shell particles consist of metallic core and dielectric shell) based on superposition of nanoparticles with different sizes and fill fractions are proposed for desired result. In this work, to obtain optimized band with negative RI media, superposition of three layer nanoparticles and doped semiconductor are designed.

     

Internal symmetry groups of quantum geons

In canonical quantum gravity asymptotically trivial diffeomorphisms not deformable to the identity can act nontrivially on the quantum state space. We show that for many 3-manifolds, the inequivalent diffeomorphisms comprise coverings in SU(2) of crystallographic groups. When the diffeomorphism R_2π associated with 2π-rotation is nontrivial, state vectors can have half-integral angular momentum; we list all 3-manifolds with R_2π trivial.     

Deformations of 2k-Einstein structures

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.     

3-Manifolds for relativists

In canonical quantum gravity certain topological properties of 3-manifolds are of interest. This article gives an account of those properties which have so far received sufficient attention, especially those concerning the diffeomorphism groups of 3-manifolds. We give a summary of these properties and list some old and new results concerning them. The appendix contains a discussion of the group of large diffeomorphisms of thel-handle 3-manifold.     

分形结构对随机取向烟尘团簇粒子光散射特性的影响

利用蒙特卡罗方法对不同分形维数和分形前向因子的随机取向烟尘团簇粒子的分形结构进行了仿真,采用离散偶极子近似（DDA）方法对随机取向烟尘团簇粒子的缪勒矩阵元进行了数值计算,并与球形粒子模型进行了比较,深入探讨了烟尘团簇粒子的分形维数和分形前因子对其散射特性的影响。研究表明,等效球形粒子的光散射特性与随机取向烟尘团簇粒子的光散射特性存在很大差别,并且此差别随着团簇粒子的分形维数以及分形前向因子的增大而减小;分形维数对表征团簇粒子散射特性的缪勒矩阵元的影响在一定散射角范围内均比较明显,分形前向因子对团簇粒子的缪勒矩阵元角分布的影响与分形维数的影响类似,不过其影响相对分形维数较弱。     

Collapsing and monopole classes of 3-manifolds

By applying the L²$L^2$-estimate of the scalar curvature of a Riemannian 3-manifold with a Seiberg–Witten monopole class to a collapsing sequence of metrics, we obtain conditions to be a monopole class on certain 3-manifolds. This also gives a relation between a maximizing sequence of the Yamabe constants and the collapsing on a 3-manifold with a non-torsion monopole class.     

On the nonexistence of Einstein metrics on 4-manifolds

By using the gluing formulae of the Seiberg–Witten invariant, we show the nonexistence of Einstein metrics on manifolds obtained from a 4-manifold with a nontrivial Seiberg–Witten invariant by performing sufficiently many connected sums or appropriate surgeries along circles or homologically trivial 2-spheres with closed oriented 4-manifolds with negative-definite intersection form.     

Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space ofT 2

We describe a cut-and-paste method for computing Chern-Simons invariant of flatG-connections on 3-manifolds decomposed along tori, especially forG=SU(2) andSL(2,C). We use this method to make computations ofSU(2) Chern-Simons invariants of graph manifolds which generalize Fintushel and Stern's computations for Seifert-fibered spaces. We also use this technique to give a simple derivation of a formula of Yoshida relating the flatSL(2,C) Chern-Simons invariant of the holonomy representation to the volume and the metric Chern-Simons invariant for cusped hyperbolic 3-manifolds.     

A TQFT from Quantum Teichmüller Theory

By using quantum Teichmüller theory, we construct a one parameter family of TQFTs on the categroid of admissible leveled shaped 3-manifolds.     

Exponential decrease of thermodynamic functions of photon gas in cubic and spherical cavities with low adiabatic invariants

The Planck radiation spectrum of ideal cubic and spherical cavities with low adiabatic invariants, γ = TV ^1/3, is discrete and strongly dependent on the cavity geometry and temperature. This behavior is a consequence of the random distribution of the state weights in the cubic cavity and of the random overlapping of successive multiplet components, in the case of a spherical cavity. The total energy density of cavities with low adiabatic invariant, γ (obtained by summing up the exact contributions of the eigenvalues and their weights) no longer obeys the Stefan-Boltzmann law. The new law includes a corrective factor depending on γ, which imposes an exponential decrease of the total energy density to zero, when γ → 0. A similar behavior is demonstrated for specific heat and for all other thermodynamic functions of photon gas in cubic and spherical cavities with low adiabatic invariants. This special quantum regime, defined by the limits of principal quantum numbers or by adiabatic invariants, is shown to be similar for cubic and spherical cavities.     

Walker 4-manifolds with proper almost complex structures

It is shown that a Walker 4-manifold, endowed with a canonical neutral metric depending on three arbitrary functions, admits a specific almost complex structure (called proper) and an associated opposite almost complex structure. We study when these two almost complex structures are integrable and when the corresponding Kähler forms are symplectic. The conditions for the canonical neutral metric to be Kähler imply that the three arbitrary functions in the metric are all harmonic with respect to two coordinate variables, and we obtain a useful method of constructing indefinite Kähler 4-manifolds. Petean’s example of a nonflat indefinite Kähler–Einstein 4-manifold is a special case of this construction.     

Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory

We study nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary. We establish the basic regularity and compactness properties (assuming L^p-bounds on the curvature for p>2) as well as the Fredholm theory in a compact model case. The motivation for studying this boundary value problem lies in the construction of an instanton Floer homology for 3-manifolds with boundary. The present paper is part of a program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres.Acknowledgement I would like to thank Dietmar Salamon for his constant help and encouragement in pursuing this project. Part of this research was supported by the Swiss National Science Foundation.     

Singularities of renormalization group flows

We discuss singularity formation in certain renormalization group flows. Special cases are the Ricci Yang–Mills and B$B$-field flows. We point out some results suggesting that topological hypotheses can make RG flows much less singular than Ricci flow. In particular we show that for rotationally symmetric initial data on S²×S¹$S^2\times S^1$ one gets long time existence and convergence of RYM flow, in stark contrast to the case for Ricci flow [S. Angenent, D. Knopf, An example of neckpinching for Ricci flow on Sⁿ⁺¹$S^{n+1}$, Math. Res. Lett. 11 (4) (2004) 493–518]. Other results are given which allow one to rule out many singularity models under strictly topological hypotheses. A conjectural picture of singularity formation for RG flow on 3-manifolds is given.     

Gauge Theories Labelled by Three-Manifolds

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional ${\mathcal{N} = 2}$ gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that ${S^{3}_{b}}$ partition functions of two mirror 3d ${\mathcal{N} = 2}$ gauge theories are equal. Three-dimensional ${\mathcal{N} = 2}$ field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional ${\mathcal{N} = 2}$ SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.     

Pair correlations in a ferromagnetic colloid

We discuss some properties of the equation of state and of the static correlations for spherical ferromagnetic grains, at thermal equilibrium, in a passive fluid. At high external fieldsH, low concentrations, and not too high temperatures, the grains tend to formchains along the directions ofH. In zero field (but otherwise identical conditions), some chains are still present but oriented at random and in competition with closed rings and clusters. Various experimental methods which could give information on these chain structures are listed.