at the

American Institute of Mathematics, San Jose, California

organized by

Aleksandar Nikolov and Kunal Talwar

Hereditary discrepancy is a combinatorial quantity associated with collections of sets that has deep connections to uniformity of distributions and a number of applications to theoretical computer science. Recent work by Nikolov and Talwar showed that hereditary discrepancy is equivalent, up to logarithmic factors, to a classical factorization norm from Banach space theory. This led to easier proofs of classical results, and a much better understanding of the discrepancy of some natural collections of sets, most prominently sets induced by axis-aligned boxes in high dimension and by homogeneous arithmetic progressions.

The goal of this workshop is to refine the connections between functional analysis and convex geometry, on one hand, and combinatorial discrepancy, on the other, and to explore further the implications of such techniques. Some of our goals for the workshop are to:

- make the connection between the gamma-2 factorization norm and hereditary discrepancy fully constructive, for example by developing a constructive proof of Banaszczyk's bound for the Komlos problem;
- show an analogue of the equivalence between the gamma-2 factorization norm and hereditary discrepancy for other notions of combinatorial discrepancy, in particular for the problem of balancing vectors with respect to an arbitrary norm;
- find a geometric quantity, e.g. a different factorization constant, that approximates hereditary discrepancy more tightly;
- exploit the power of factorization norms for bounding combinatorial discrepancy, together with classical transference theorems, to give new constructions of pointsets distributed uniformly relative to an arbitrary measure.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

An algorithm for Komlós conjecture matching Banaszczyk's bound

by Nikhil Bansal, Daniel Dadush, and Shashwat Garg, *57th Annual IEEE Symposium on Foundations of Computer Science–FOCS 2016, 788–799, IEEE Computer Soc., Los Alamitos, CA, 2016. * MR3631042

Algorithmic discrepancy beyond partial coloring

by Nikhil Bansal and Shashwat Garg

Towards a constructive version of Banaszczyk's vector balancing theorem

by Daniel Dadush, Shashwat Garg, Shachar Lovett, and Aleksandar Nikolov, *Approximation, randomization, and combinatorial optimization. Algorithms and techniques, Art. No. 28, 12 pp., LIPIcs. Leibniz Int. Proc. Inform., 60, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016 * MR3566770