Some favourite open problems:
  Disclaimer: This page, under construction, will be an eclectic collection of problems, which share the feature that I tried and failed to solve.
Problem number and authors PS
Problem 1: S. Mallat and O. Zeitouni, "Optimality of the Karhunen-Loeve basis in nonlinear reconstruction". Slides of an updated presentation for this problem can be found HERE
Problem 2: Conjecture: the (centered) Gaussian measure of the intersection of two convex, symmetric subsets of Euclidean space is greater or equal than the product of the measures of the two subsets. See for a review of the history of this problem and some partial results: Schechtman, G., Schlumprecht, Th., Zinn, J., On the Gaussian measure of the intersection. Ann. Probab. 26 (1998), 346--357. MATH REVIEWS LINK. G. Harge A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27 (1999), 1939--1951. MATH REVIEWS LINK --
Problem(s) 3: RWRE questions, conjectures: 0-1 laws and other problems. See for details the review "Random walks in random environments", Proceedings ICM 2002, vol III, pp. 117--127. For recent progress on related large deviations questions, see Varadhan's article MATH REVIEWS LINK. --