|Mark Levi received his Ph. D. from the Courant Institute under the direction of Jürgen Moser.
His research interests include dynamical systems with applications
to physical problems. One of his hobbies is creating and collecting physical devices that he
uses to illustrate mathematical phenomena. His book “Mathematical Mechanic” was included in
the “Top 10 books in science” by Amazon editors in 2009, and was one of Outstanding Academic
Titles, 2009, by CHOICE magazine. His recent book is titled “Why cats land on their feet and 76 other
physical paradoxes and puzzles”. His book “Classical mechanics, calculus of variations
and optimal control -- an intuitive introduction” will appear in March 2014.
"Mathematics by physical reasoning:" Physics often provides mathematics not only with a problem, but sometimes also with the idea of a solution. Some calculus problems can be solved by a physical argument more quickly and easily than by the "standard" approach used in college courses. When they work, these solutions can be strikingly short and simple in comparison with the standard ones. Quite a few theorems which may seem somewhat mysterious become completely obvious when interpreted physically; the trick is to find a suitable interpretation. This is the case for some "elementary" theorems (the Pythagorean Theorem, Pappus' theorems, some trig identities (e.g., cos(x+y)=...,) Euler's formula V-E+F=2, and more), Cauchy-Schwarz inequality, and for some less elementary ones: Green's theorem, the Riemann Mapping Theorem, the Gauss--Bonnet theorem, Noether's theorem on conserved quantities, Poincar integral invariance, uniformization theorem, Moser’s theorem on densities, and more. Furthermore, the fundamental ideas of Hamiltonian dynamics are drastically clarified by using certain static analogies: seemingly unmotivated definitions become completely natural, and some fundamental theorems (whose textbook proofs rely on computation) become utterly obvious and with little mathematical background. I will describe a selection from the above topics, taking into account the audience's preferences.