From singularities to graphs

In this paper I analyze the problems which led to the introduction of graphs as tools for studying surface singularities. I explain how such graphs were initially only described using words, but that several questions made it necessary to draw them, leading to the elaboration of a special calculus with graphs. This is a non-technical paper intended to be readable both by mathematicians and philosophers or historians of mathematics.Comment: 23 pages, 27 figures. Expanded version of the talk given at the conference "Quand la forme devient substance : puissance des gestes, intuition diagrammatique et ph\'enom\'enologie de l'espace", which took place at Lyc\'ee Henri IV in Paris from 25 to 27 January 201

Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(\kappa,\tau)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.Comment: 19 page

On the proposed AdS dual of the critical O(N) sigma model for any dimension 2<d<4

We evaluate the 4-point function of the auxiliary field in the critical O(N) sigma model at O(1/N) and show that it describes the exchange of tensor currents of arbitrary even rank l>0. These are dual to tensor gauge fields of the same rank in the AdS theory, which supports the recent hypothesis of Klebanov and Polyakov. Their couplings to two auxiliary fields are also derived.Comment: 14 page

Spacetime Approach to Phase Transitions

In these notes, the application of Feynman's sum-over-paths approach to thermal phase transitions is discussed. The paradigm of such a spacetime approach to critical phenomena is provided by the high-temperature expansion of spin models. This expansion, known as the hopping expansion in the context of lattice field theory, yields a geometric description of the phase transition in these models, with the thermal critical exponents being determined by the fractal structure of the high-temperature graphs. The graphs percolate at the thermal critical point and can be studied using purely geometrical observables known from percolation theory. Besides the phase transition in spin models and in the closely related $\phi^4$ theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory.Comment: 59 pages, 18 figures. Write-up of Ising Lectures presented at the National Academy of Sciences, Lviv, Ukraine, 2004. 2nd version: corrected typo