For a list of my publications, visit my profiles on Google Scholar and dblp.
Here you find a selection of some of my recent papers.
I study families of depth measures defined by natural sets of axioms. I show that any such depth measure is a constant factor approximation of Tukey depth. I further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy the most restrictive set of axioms, which includes Tukey depth. Along the way, I introduce and study a new depth measure called enclosing depth, which I believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.
We develop a query-efficient algorithm to find an approximate fixpoint of a map that is contracting under an lp-norm. Previously, such algorithms were only known for the l-infinity metric. To prove our results, we introduce the notion of lp-halfspaces and generalize the classical centerpoint theorem from discrete geometry. The proof of this generalized centerpoint theorem uses topological methods, namely Brouwer's fixpoint theorem.
Schaefer's dichotomy theorem states that a boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of six given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, together with Simon Weber we analyze boolean CSPs in terms of their topological complexity, instead of their computational complexity. We attach a natural topological space to the set of solutions of a boolean CSP and introduce the notion of projection-universality. We prove that a boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer's dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality.