Slope and geometry in variational mathematics

Various notions of the "slope" of a real-valued function pervade optimization, and variational mathematics more broadly. In the semi-algebraic setting - an appealing model for concrete variational problems - the slope is particularly well-behaved. This talk sketches a variety of surprising applications, illustrating the unifying power of slope. Highlights include error bounds for level sets, the existence and regularity of steepest descent curves in complete metric spaces (following Ambrosio et al.), transversality and the convergence of von Neumann's alternating projection algorithm, and the geometry underlying nonlinear programming active-set algorithms. This talk will be self-contained, requiring no familiarity with variational analysis, optimization theory, or semi-algebraic geometry. Joint work with A. Daniilidis (Barcelona), A.D. Ioffe (Technion), M. Larsson (Lausanne), A.S. Lewis (Cornell).