Dear Readers,
I’m Mark F. Kruelle, and I’m excited to introduce you to my book, An Introduction to the Galois Theory of Linear Differential Equations for Undergraduates. As a mathematician with a passion for education, I’ve dedicated the latter part of my career to making complex mathematical concepts accessible to bright young minds.
I am offering this book at no cost to you ( just click to get the book ) as a way of thanking the academic community for providing me with a world-class education at no cost to me.
After winning a softball throw contest in the sixth grade, I went on to win a full scholarship to Cornell to study engineering and classical applied mathematics, followed by a fellowship to Yale to study pure mathematics. After completing my studies at Yale, where I had the privilege of learning from renowned mathematicians such as the late Serge Lang, the late William Massey, the late Dan Mostow, the late George Seligman (my advisor), and Gregg Zuckerman, I spent years exploring interdisciplinary research in engineering, classical applied mathematics, and pure mathematics.
My interdisciplinary work is exemplified by an exact analytical solution I developed for the adjoint equations representing the missile miss distance problem 64 years after this problem was posited. An updated version of this paper, first copyrighted in 2016, will appear along with my forthcoming publications on this website.
Since I retired I have explored various areas of mathematics with a view towards writing mathematics books for bright young minds that fill new niches in mathematics. My journey has led me to write this book, which aims to introduce students who have studied classical Galois theory to the more advanced field of differential Galois theory.
This book is particularly close to my heart as it was inspired by my late friend and colleague, Dr. David O’Hanley. Dave’s enthusiasm for teaching Galois theory to high school students and his eagerness for the creation of this book provided the impetus for my authoring just such a book.
By writing this book, I’ve drawn from my experiences as both a student and an educator. I’ve tried to present the material in a way that’s accessible to students who have studied classical Galois theory. You’ll find that I’ve used analogies with classical Galois theory to introduce the concepts of differential Galois theory, along with worked examples and exercises to reinforce understanding.
Presenting differential Galois theory by means of analogies with classical Galois theory is necessary because proofs for differential Galois theory are beyond the reach of all but the most exceptional undergraduates. This is the primary reason that a book on differential Galois theory for undergraduates has not been written until now. The book covers topics such as Picard-Vessiot theory and Kovacic’s theorems, areas that are typically reserved for graduate-level study. My goal is to give you a taste of the frontiers of mathematics and inspire some of you to pursue further studies in this fascinating field.
On a personal note, I am a lifelong learner with diverse interests. When I’m not delving into mathematics, I enjoy writing poetry – in fact, I have a poetry book that is forthcoming. I’m also deeply appreciative of the spiritual guidance I’ve received throughout my life, which has shaped both my personal and professional journey. I hope this book challenges and excites prospective readers, opening new mathematical horizons for undergraduates. I look forward to learning your thoughts and experiences as you explore the world of differential Galois theory.
Wishing you all the best in your mathematical adventures,
Mark F. Kruelle
mk******@******le.edu
