Speaking about his stay at Princeton during 1955-57, Bott provides a vivid picture of the mathematics and mathematical personalities he encountered there: Atiyah, Serre, Kodaira, Hirzebruch, among others. Two mathematical paradigms of this century form the focus of his talk. Bott shows how the first of these, the maximal principle, is "quantified" by Morse theory. The second topic is the Brouwer Fixed Point Theorem, which Bott connects to the ideas of Lefschetz and to sheaf theory. (Providence, RI, 1988)