SU(N) Meander Determinants

We propose a generalization of meanders, i.e., configurations of non-selfintersecting loops crossing a line through a given number of points, to SU(N). This uses the reformulation of meanders as pairs of reduced elements of the Temperley-Lieb algebra, a SU(2)-related quotient of the Hecke algebra, with a natural generalization to SU(N). We also derive explicit formulas for SU(N) meander determinants, defined as the Gram determinants of the corresponding bases of the Hecke algebra.Comment: TeX using harvmac.tex and epsf.tex, 60 pages (l-mode), 5 figure

Ricci flow of warped Berger metrics on R4\mathbb{R}^{4}

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension $n > 3$ of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead $g_{0}$ has no necks, its curvature decays and the Hopf fibers are not collapsed, then the solution is immortal. Finally, we prove that if the flow is Type-I, then there exist minimal 3-spheres for times close to the maximal time.Comment: 40 pages, final version. Accepted in Calc. Va

Combinatorial point for higher spin loop models

Integrable loop models associated with higher representations (spin k/2) of U_q(sl(2)) are investigated at the point q=-e^{i\pi/(k+2)}. The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries.Comment: latest version adds some reference

Folding Transitions of the Square-Diagonal Lattice

We address the problem of "phantom" folding of the tethered membrane modelled by the two-dimensional square lattice, with bonds on the edges and diagonals of each face. Introducing bending rigidities $K_1$ and $K_2$ for respectively long and short bonds, we derive the complete phase diagram of the model, using transfer matrix calculations. The latter displays two transition curves, one corresponding to a first order (ferromagnetic) folding transition, and the other to a continuous (anti-ferromagnetic) unfolding transition.Comment: TeX using harvmac.tex and epsf.tex, 22 pages (l mode), 17 figure

Integrable Combinatorics

We review various combinatorial problems with underlying classical or quantum integrable structures. (Plenary talk given at the International Congress of Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1

Simulating lattice field theories on multiple thimbles

Simulating thimble regularization of lattice field theory can be tricky when more than one thimble is to be taken into account. A couple of years ago we proposed a solution for this problem. More recently this solution proved to be effective in the case of 0+1 dimensional QCD. A few lessons we can learnt, including the role of symmetries and general hints on algorithmic solutions.Comment: 8 pages, 2 figures; Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spai

Digital technologies for virtual recomposition : the case study of Serpotta stuccoes

The matter that lies beneath the smooth and shining surface of stuccoes of the Serpotta family, who used to work in Sicily from 1670 to 1730, has been thoroughly studied in previous papers, disclosing the deep, even if empirical, knowledge of materials science that guided the artists in creating their master- works. In this work the attention is focused on the solid perspective and on the scenographic sculpture by Giacomo Serpotta, who is acknowledged as the leading exponent of the School. The study deals with some particular works of the artist, the so-called "teatrini" (Toy Theater), made by him for the San Lorenzo Oratory in Palermo. On the basis of archive documents and previous analogical photogrammetric plotting, integrated with digital solutions and methodologies of computer- based technologies, the study investigates and interprets the geometric-formal genesis of the examined works of art, until the prototyping of the whole scenic apparatus.peer-reviewe