This publication offers a radical creation to the attention-grabbing international of section transitions in addition to many comparable subject matters, together with random walks, combinatorial difficulties, quantum box conception and S-matrix. basic recommendations of part transitions, similar to order parameters, spontaneous symmetry breaking, scaling alterations, conformal symmetry, and anomalous dimensions, have deeply replaced the trendy imaginative and prescient of many parts of physics, resulting in striking advancements in statistical mechanics, straightforward particle thought, condensed topic physics and string thought. This self-contained e-book presents a great advent to frontier subject matters of precisely solved versions in statistical mechanics and quantum box thought, renormalization crew, conformal versions, quantum integrable structures, duality, elastic S-matrix, thermodynamics Bethe ansatz and shape issue concept. The transparent dialogue of actual rules is observed through an in depth research of numerous branches of arithmetic, extraordinary for his or her attractiveness and wonder, reminiscent of limitless dimensional algebras, conformal mappings, crucial equations or modular services. along with complicated study issues, the ebook additionally covers many uncomplicated issues in statistical mechanics, quantum box idea and theoretical physics. every one argument is mentioned in nice aspect, being attentive to an total coherent knowing of actual phenomena. Mathematical historical past is equipped in vitamins on the finish of every bankruptcy, while applicable. The chapters also are by way of difficulties of other degrees of hassle. complicated undergraduate and graduate scholars will discover a wealthy and demanding resource for bettering their talents and for attaining a complete studying of the various elements of the subject.