| dc.description.abstract | This book came out of an attempt to explain to a class of motivated students at the
University of Illinois at Chicago what sorts of problems I thought about in my
research. In the course, we had just talked about the integral solutions to the
Pythagorean Equation and it seemed only natural to use the Pythagorean Equation
as the context to motivate the answer. Basically, I motivated my own research, the
study of rational points of bounded height on algebraic varieties, by posing the
following question: What can you say about the number of right triangles with
integral sides whose hypotenuses are bounded by a large number X? How does this
number depend on X? In attempting to give a truly elementary explanation of the
solution, I ended up having to introduce a fair bit of number theory, the Gauss circle
problem, the Möbius function, partial summation, and other topics. These topics
formed the material in Chapter 13 of the present text.
Mathematicians never develop theories in the abstract. Despite the impression
given by textbooks, mathematics is a messy subject, driven by concrete problems
that are unruly. Theories never present themselves in little bite-size packages with
bowties on top. Theories are the afterthought. In most textbooks, theories are
presented in beautiful well-defined forms, and there is in most cases no motivation
to justify the development of the theory in the particular way and what example or
application that is given is to a large extent artificial and just “too perfect.” Perhaps
students are more aware of this fact than what professional mathematicians tend to
give them credit for—and in fact, in the case of the class I was teaching, even
though the material of Chapter 13 was fairly technical, my students responded quite
well to the lectures and followed the technical details enthusiastically. Apparently, a
bit of motivation helps. | en_US |
