This page contains the Reading selections of UES members. Click on the list below to jump directly to the person’s recommendations:
1. Keeping with my anti-establishment leaning mathematics, here is a paper entitled Down with Determinants! by Sheldon Axler. Axler claims that “determinants are difficult, non-intuitive, and often defined without motivation.” Hence the paper strips determinants from the major ideas of linear algebra, beginning by redefining eigenvalues without using determinants and then proving their existence for any linear operator on a finite dimensional complex vector space. The paper follows this with determinant-free proofs of major theorems. Finally, he redefines the determinant in terms of eigenvalues and then shows that his definitions are equivalent to standard definitions.
This paper preceded his linear algebra textbook “Linear Algebra Done Right” which, like the paper, develops linear algebra without determinants—they’re relegated to the last chapter, and we all know teachers never get to the last chapters of any textbook.
This paper is a good reminder that mathematics is forever evolving. We can view every idea from many perspectives and often finding a new perspective is the insight required to advance the science. Hence this paper gives students a glimpse at how mathematics is practiced.
Anyone who has taken MAS4105 should be able to understand this paper, with suitable review.
2. Richard Hamming started an entire line of research in information theory with his 1950 paper on error-detecting and error-correcting codes. These codes make sure that the strings of bits sent over your modem don’t have errors, and hence make sure that information is sent reliably.
In 1986, while working at Bell Labs, he gave a speech entitled “You and Your Research.” He argues that solutions to tough problems don’t appear magically, they are the result of a lot of hard work thinking about those problems. Hence, “If you do not work on an important problem, it’s unlikely you’ll do important work.” This is some of the best research advice I’ve ever heard.
3. Any book from the new Springer Undergraduate Mathematics Series. These are, on average, well written textbooks at an accessible level. They are designed for self-study, hence they have complete solutions to all exercises. Also, they are all usually under $40. There are some very interesting titles, so check them out.
- American Caesar by William Manchester
- When Character Was King by Peggy Noonan
- Philip of Spain by Henry Kamen
- The Mystery of Capital by Hernando de Soto
- Principles of Economics by Gregory Mankiw
- Macroeconomics, 6th Ed., by Gregory Mankiw
- Prime Obsession by John Derbyshire
- Godel, Escher, Bach by Douglas Hofstadter
- Fermat’s Enigma by Simon Singh (only since I met Oktay Sinanoglu by it)
- Matt: I also recommend this book. Many writers have chronicled Fermat’s Last Theorem and this book is among the best. Furthermore, even a slow reader like myself quickly read this book.
- Statecraft by Margaret Thatcher
- Colossus by Niall Ferguson
- The Uses of the Past by Herbert Muller
- Rural Studio
- The Fountainhead by Ayn Rand
- Invariances by Robert Nozick
- The Republic by Plato
- The Federalist Papers by Hamilton, et al
- Of Grammatology by Jacques Derrida
Star Trek (ones in bold are mandatory reading)
- Federation by J&G Reeves-Stevens
- Probe by Margaret Wander Bonanno
- Spock’s World by Diane Duane
- Final Frontier by Diane Carey
- Best Destiny by Diane Carey
- Sarek by A.C. Crispin
- Ashes of Eden trilogy by William Shatner
- Spectre trilogy by William Shatner
- A Flag Full of Stars by Brad Ferguson
- TNG - Immortal Coil
- TNG - Q-Squared
- DS9 - Millennium trilogy by J&G Reeves-Stevens
- DS9 - A Stitch in Time by Andrew Robinson
- DS9 - Mission Gamma tetralogy
- Voyager - Mosaic