Ivar Ekeland

[1][7][8] These books were first written in French and then translated into English and other languages, where they received praise for their mathematical accuracy as well as their value as literature and as entertainment.

[3] When the novel was adapted for the film Jurassic Park by Steven Spielberg, Ekeland and Gleick were consulted by the actor Jeff Goldblum as he prepared to play the mathematician specializing in chaos theory.

[12] Ekeland's variational principle can be used when the lower level set of a minimization problem is not compact, so that the Bolzano–Weierstrass theorem can not be applied.

[9] Ivar Ekeland is an expert on variational analysis, which studies mathematical optimization of spaces of functions.

In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimization methods on problems that were known to be non-convex.

[17][15][18] Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions.

Picture of the Julia set
Ivar Ekeland has written popular books about chaos theory and about fractals , [ 1 ] [ 2 ] such as the Julia set (animated) . Ekeland's exposition provided mathematical inspiration to Michael Crichton 's discussion of chaos in Jurassic Park . [ 3 ]
Picture of Jeff Goldblum
Actor Jeff Goldblum consulted Ekeland while preparing to play a mathematician specializing in chaos theory in Spielberg's Jurassic Park . [ 6 ]
The Shapley–Folkman lemma depicted by a diagram with two panes, one on the left and the other on the right. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. Comparing the left array and the right pane, one confirms that the right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.
Ivar Ekeland applied the Shapley–Folkman lemma to explain Claude Lemarechal's success with Lagrangian relaxation on non-convex minimization problems. This lemma concerns the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowski sum of the four non-convex sets ( right ) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown in red).
Picture of the Feigenbaum bifurcation of the iterated logistic-function
The Feigenbaum bifurcation of the iterated logistic function system was described as an example of chaos theory in Ekeland's Mathematics and the unexpected . [ 1 ]