Quasiconformal surgery in holomorphic dynamics(Cambridge Studies in Advanced Mathematics 141)ByBodil Branner and Núria Fagella

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

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Mathematical Knowledge and the Interplay of Practices

10.23943/princeton/9780691167510.001.0001 ◽

2015 ◽

Cited By ~ 12

Author(s):

José Ferreirós

Keyword(s):

Mathematical Knowledge ◽

A Priori ◽

Philosophy Of Mathematics ◽

Advanced Mathematics ◽

New Approach ◽

Agent Based ◽

Epistemology Of Mathematics ◽

Actual Experience ◽

Elementary Math ◽

Mathematical Tradition

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

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Advanced Mathematics for Optimizing Missile Seeker Signal Processing

10.21236/ada398610 ◽

2001 ◽

Author(s):

Harry A. Schmitt

Keyword(s):

Signal Processing ◽

Advanced Mathematics

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Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽

10.3390/e23020237 ◽

2021 ◽

Vol 23 (2) ◽

pp. 237

Author(s):

Rostislav Grigorchuk ◽

Supun Samarakoon

Keyword(s):

Operator Algebras ◽

Probabilistic Approach ◽

Schur Complement ◽

Automata Theory ◽

Random Schrödinger Operators ◽

Rational Maps ◽

Holomorphic Dynamics ◽

Von Neumann ◽

Intermediate Growth ◽

Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

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