Mark Levi

"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.