Analyse complexe

Résumé

L'analyse complexe est un domaine des mathématiques traitant des fonctions à valeurs complexes (ou, plus généralement, à valeurs dans un C-espace vectoriel) et qui sont dérivables par rapport à une ou plusieurs variables complexes. Les fonctions dérivables sur un ouvert du plan complexe sont appelées holomorphes et satisfont de nombreuses propriétés plus fortes que celles vérifiées par les fonctions dérivables en analyse réelle. Entre autres, toute fonction holomorphe est analytique et vérifie le principe du maximum. Le principe des zéros isolés permet de définir le corps des fonctions méromorphes comme ensemble des quotients de fonctions entières, c'est-à-dire de fonctions holomorphes définies sur tout le plan complexe. Parmi ces fonctions méromorphes, les fonctions homographiques forment un groupe qui agit sur la sphère de Riemann, constituée du plan complexe muni d'un point à l'infini. Le prolongement analytique mène à la définition des surfaces de Riemann, qui permettent de rame

Source officielle

https://fr.wikipedia.org/wiki/Analyse_complexe

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This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

Le cours étudie les concepts fondamentaux de l'analyse complexe et de l'analyse de Laplace en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applications, notably we will prove rationality of the Weil Zeta function.

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Ising Model and Field Theory: Lattice Local Fields and Massive Scaling Limit

Sung Chul Park

This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions for the building blocks of the continuum theory (spin and energy density) have been rigorously established [HoSm13, CHI15]. We study how the field-theoretic description of these random fields extends beyond the critical regime of the model. Concretely, the thesis consists of two parts: The first part studies the behaviour of lattice local fields in the critical Ising model. A lattice local field is a function of a finite number of spins at microscopic distances from a given point. We study one-point functions of these fields (in particular, their asymptotics under scaling limit and conformal invariance). Our analysis, based on discrete complex analysis methods, results in explicit computations which are of interest in applications (e.g. [HKV17]). The second part considers the behaviour of the massive spin field. In the subcritical massive scaling limit regime first considered by Wu, McCoy, Tracy, and Barouch [WMTB76], we show that the correlations of the massive spin field in a bounded domain have a scaling limit. Furthermore, to this end we generalise the notions and methods of discrete complex analysis in the critical case to the massive regime, and give a new derivation of the formula for the two-point correlation in the full plane in terms of a Painlevé III transcendent.

EPFL2019

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Discrete Holomorphicity and Ising Model Operator Formalism

Clément Hongler, Ali Zahabi

We explore the connection between the transfer matrix formalism and discrete complex analysis approach to the two dimensional Ising model. We construct a discrete analytic continuation matrix, analyze its spectrum and establish a direct connection with the critical Ising transfer matrix. We show that the lattice fermion operators of the transfer matrix formalism satisfy, as operators, discrete holomorphicity, and we show that their correlation functions are Ising parafermionic observables. We extend these correspondences also to outside the critical point. We show that critical Ising correlations can be computed with operators on discrete Cauchy data spaces, which encode the geometry and operator insertions in a manner analogous to the quantum states in the transfer matrix formalism.

2014

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Conformal Invariance of Spin Pattern Probabilities in the Planar Ising Model

Clément Hongler, Sung Chul Park

We study the 2-dimensional Ising model at critical temperature on a smooth simply-connected graph Ω.We rigorously prove the conformal invariance of arbitrary spin-pattern probabilities centered at a point a and derive formulas to compute the probabilities as functions of the conformal map from Ω to the unit disk. Our methods extend those of [Hon10] and [CHI13] which proved conformal invariance of energy densities and spin correlations for points fixed far apart from each other. We use discrete complex analysis techniques and construct a discrete multipoint fermionic observable that takes values related to pattern probabilities in the planar Ising model. Refined analysis of the convergence of the discrete observable to a continuous, conformally covariant function completes the result.

2015

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