  Beginning with the projectively invariant method for linear programming,
interior point methods have led to powerful algorithms for many difficult
computing problems, in combinatorial optimization, logic, number theory and
non-convex optimization. Algorithms for convex optimization benefitted from
many pre-established ideas from classical mathematics, but non-convex problems
require new concepts. Lecture series I am presenting at the conference on
Foundations of Computational Mathematics, 2014, outlines some of these
concepts{computational models based on the concept of the continuum, algorithms
invariant w.r.t. projective, bi-rational, and bi-holomorphic transformations on
co-ordinate representation, extended proof systems for more efficient
certificates of optimality, extensions of Grassmanns extension theory,
efficient evaluation methods for the effect of exponential number of
constraints, theory of connected sets based on graded connectivity, theory of
curved spaces adapted to the problem data, and concept of relatively algebraic
sets in curved space. Since this conference does not have a proceedings, the
purpose of this article is to provide the material being presented at the
conference in more widely accessible form.
