Computational Arithmetic Dynamics. In progress.
Covers in details the major algorithms in Arithmetic Dynamics and the mathematics supporting them. This includes
Heights and Canonical Heights
Rational Preperiodic Points
Moduli Space Invariants and Normal Forms of conjugacy classes, such as minimal models
Automorphism Groups
Special Maps, such as postcritically finite maps.
An Experimental Introduction to Number Theory, published in 2018 by the American Mathematical Society in the series Pure and Applied Undergraduate Texts Volume 31.
ISBN: 978-1470430979
This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems.
The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration.
This course covers the main computational methods used in arithmetic dynamics. A working knowledge of abstract algebra and real analysis are assumed as well as basic number theory. Topics in computational algebra and algebraic geometry will be reviewed as needed. No prior experience with dynamical systems is assumed.
I often supervise undergraduate research projects and strongly recommend anyone consideration graduate study to complete a project. I have projects at a range of levels that range from purely programming projects to computational mathematics, to theoretic mathematics. If you are an undergraduate interested in a research project, send me an email that includes what math and/or programming courses you have taken so far and we can set up a time to meet.
I am able to supervise both MA and PhD thesis in mathematics. Please see my research page for more details on the kind of work that I do. If you are interested in working with me on your thesis (either MA or PhD) the first step is to set-up a time to meet and talk about your interests. Send me an email and we can coordinate a time.
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