A premature book review and a 60 year history with complex variables in 4 acts

“Visual Differential Geometry and Forms” (VDGF) by Tristan Needham is an incredible book.  Here is a premature review having only been through the first 82 pages of 464 pages of text.

Here’s why.

While mathematicians may try to tie down the visual Gulliver with Lilliputian strands of logic, there is always far more information in visual stimuli than logic can appreciate.  There is no such a thing as a pure visual percept (a la Bertrand Russell), as visual processing begins within the 10 layers of the retina and continues on from there.  Remember: half your brain is involved in processing visual information.  Which is a long winded way of saying that Needham’s visual approach to curvature and other visual constructs is an excellent idea.

Needham loves complex variables and geometry and his book is full of pictures (probably on 50% of the pages).

My history with complex variables goes back over 60 years and occurs in 4 acts.

 

Act I:  Complex variable course as an undergraduate. Time late 50s.  Instructor Raymond Smullyan a man who, while in this world, was definitely not of it.  He really wasn’t a bad instructor but he appeared to be thinking about something else most of the time.

 

Act II: Complex variable course at Rocky Mountain College, Billings Montana.  Time early 80s.  The instructor and MIT PhD was excellent.  Unfortunately I can’t remember his name.  I took complex variables again, because I’d been knocked out for probably 30 minutes the previous year and wanted to see if I could still think about the hard stuff.

 

Act III: 1999 The publication of Needham’s first book — Visual Complex Analysis.  Absolutely unique at the time, full of pictures with a glowing recommendation from Roger Penrose, Needham’s PhD advisor.  I read parts of it, but really didn’t appreciate it.

 

Act IV 2021 the publication of Needham’s second book, and the subject of this partial review.  Just what I wanted after studying differential geometry with a view to really understanding general relativity, so I could read a classmate’s book on the subject.  Just like VCA, and I got through 82 pages or so, before I realized I should go back and go through the relevant parts (several hundred pages) of VCA again, which is where I am now.  Euclid is all you need for the geometry of VCA, but any extra math you know won’t hurt.

 

I can’t recommend both strongly enough, particularly if you’ve been studying differential geometry and physics.  There really is a reason for saying “I see it” when you understand something.

 

Both books are engagingly and informally written, and I can’t recommend them enough (well at least the first 82 pages of VDGF).

 

High level mathematicians look like normal people

Have you ever had the pleasure of taking a course from someone who wrote the book? I did. I audited a course at Amherst from Prof. David Cox who was one of three authors of “Ideals, Varieties and Algorithms” It was uncanny to listen to him lecture (with any notes) as if he were reading from the book. It was also rather humbling to have a full professor correcting your homework. We had Dr. Cox for several hours each weak (all 11 or 12 of us). This is why Amherst is such an elite school. Ditto for Princeton back in the day, when Physics 103 was taught by John Wheeler 3 hours a week. Physics 103 wasn’t for the high powered among us who were going to be professional physicists (Heinz Pagels, Jim Hartle), it was for preMeds and engineers.

Dr. Cox had one very useful pedagogical device — everyone had to ask a question at the beginning of class, Cox being of the opinion that there is no such thing as a dumb question in math.

Well Dr. Cox and his co-authors (Little and O’Shea) got an award from the American Mathematical sociecty for their book. There’s an excerpt below. You should follow the link to the review to see what the three look like along with two other awardees. http://www.ams.org/publications/journals/notices/201604/rnoti-p417.pdf. Go to any midsize American city at lunchtime, and you’d be hard pressed to pick four of the five out of the crowd of middle aged men walking around. Well almost — one guy would be hard to pick out of the noonday crowd in Williamsburg Brooklyn or Tel Aviv. Four are extremely normal looking guys, not flamboyant or bizarre in any way. This is certainly true of the way Dr. Cox comports himself. The exception proving the rule however, is Raymond Smullyan who was my instructor in a complex variables course back in the day– quite an unusual and otherworldly individual — there’s now a book about him.

Here’s part of the citation. The link also contains bios of all.

“Even more impressive than its clarity of exposition is the impact it has had on mathematics. CLO, as it is fondly known, has not only introduced many to algebraic geometry, it has actually broadened how the subject could be taught and who could use it. One supporter of the nomination writes, “This book, more than any text in this field, has moved computational algebra and algebraic geometry into the mathematical mainstream. I, and others, have used it successfully as a text book for courses, an introductory text for summer programs, and a reference book.”
Another writer, who first met the book in an REU two years before it was published, says, “Without this grounding, I would have never survived my first graduate course in algebraic geometry.” This theme is echoed in many other accounts: “I first read CLO at the start of my second semester of graduate school…. Almost twenty years later I can still remember the relief after the first hour of reading. This was a math book you could actually read! It wasn’t just easy to read but the material also grabbed me.”
For those with a taste for statistics, we note that CLO has sold more than 20,000 copies, it has been cited more than 850 times in MathSciNet, and it has 5,000 citations recorded by Google Scholar. However, these numbers do not really tell the story. Ideals, Varieties, and Algorithms was chosen for the Leroy P. Steele Prize for Mathematical Exposition because it is a rare book that does it all. It is accessible to undergraduates. It has been a source of inspiration for thousands of students of all levels and backgrounds. Moreover, its presentation of the theory of Groebner bases has done more than any other book to popularize this topic, to show the powerful interaction of theory and computation in algebraic geometry, and to illustrate the utility of this theory as a tool in other sciences.”