According to the classical uniformization theorem, every smooth Riemannian surface Z homeomorphic to the 2-sphere is conformally diffeomorphic to S2 (the unit sphere inR3 equipped with the Riemannian metric induced by the ambient Euclidean metric). The availability of a similar uniformization procedure for spheres with a “generalized conformal structure” is highly desirable, in particular in connection with Thurston’s hyperbolization conjecture. This was addressed by Cannon in his combinatorial Riemann mapping theorem [7]. He considers topological surfaces equipped with a sequence of “shinglings”—a combinatorial structure that leads to a notion of approximate conformal moduli of rings. He then finds conditions that imply the existence of coordinate systems on the surface that relate these combinatorial moduli to classical analytic moduli in the plane. In this paper we develop a uniformization theory for a different type of generalized conformal structure. We start with a metric space Z homeomorphic to S2 and ask for conditions under which Z can be mapped onto S2 by a quasisymmetric homeomorphism. The class of quasisymmetries is an appropriate analog of conformal1 mappings in a metric space context. Quasisymmetric homeomorphisms also arise in the theory of Gromov hyperbolic metric spaces—quasi-isometries between Gromov hyperbolic spaces induce quasisymmetric boundary homeomorphisms. Our setup has the advantage that we can exploit recent notions and methods from Analysis

/pdf/quasisymmetric-parametrizations-of-two-dimensional-metric-508fwkc6r3.pdf

Quasisymmetric parametrizations of two-dimensional metric spheres

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices This simple definition leads to a surprisingly rich theory featuring M\"obius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles We show how literally the same theory can be reinterpreted to addresses the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings

/pdf/discrete-conformal-maps-and-ideal-hyperbolic-polyhedra-51zetrh6km.pdf

Discrete conformal maps and ideal hyperbolic polyhedra

Maximal element theorems in product FC-spaces and generalized games

We study the critical properties of the ${\mathrm{QED}}_{3}$-Gross-Neveu model with $2N$ flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For $N=1$, this theory has recently been suggested to be dual to the SU(2) noncompact ${\mathrm{CP}}^{1}$ model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For $N=2$, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-$1/2$ systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the ${\mathrm{QED}}_{3}$-Gross-Neveu model for all values of $N$. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain ${\mathrm{QED}}_{3}$ (without scalar-field coupling) unstable at low values of $N$. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.

Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality

Fixed point results for multimaps in CAT(0) spaces

char ticky=0x22;char *first="main(){printf(";char *second="char ti cky=0x22;char *first=%c%s%c;char *second=%c%s%c;char *third=%c%s%c ;%s%c%s%c%s";char *third=",ticky,first,ticky,ticky,second,ticky,ti cky,third,ticky,first,ticky,second,ticky,third);}";main(){printf(" char ticky=0x22;char *first=%c%s%c;char *second=%c%s%c;char *third =%c%s%c;%s%c%s%c%s",ticky,first,ticky,ticky,second,ticky,ticky,thi rd,ticky,first,ticky,second,ticky,third);}

On Fixed Points

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