Forgetful maps between Deligne-Mostow ball quotients

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds

Sparse Selfreducible Sets and Nonuniform Lower Bounds

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in (Formula presented.), or even in (Formula presented.) that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that (Formula presented.) does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that (Formula presented.) does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of (Formula presented.) is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for (Formula presented.)

Partitioning multi-dimensional sets in a small number of ``uniform'' parts

In this paper we prove that every finite subset of ZxZ can be partitioned into a small number of subsets so that, in each part all vertical sections have aproximately the same size and all horyzontal sections have aproximately the same size. The generalization of this statement is used to give a combinatorial interpretation to every information inequality

Inverting Onto Functions and Polynomial Hierarchy

In this paper we construct an oracle under which the polynomial hierarchy is infinite but there are non-invertible polynomial time computable multivalued onto functions

High Entropy Random Selection Protocols

We study the two party problem of randomly selecting a common string among all the strings of length n. We want the protocol to have the property that the output distribution has high Shannon entropy or high min entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, Shannon entropy and simultaneously min entropy close to n/2. In the literature the randomness guarantee is usually expressed in terms of “resilience”. The notion of Shannon entropy is not directly comparable to that of resilience, but we establish a connection between the two that allows us to compare our protocols with the existing ones. We construct an explicit protocol that yields Shannon entropy n- O(1) and has O(log ∗n) rounds, improving over the protocol of Goldreich et al. (SIAM J Comput 27: 506–544, 1998) that also achieves this entropy but needs O(n) rounds. Both these protocols need O(n2) bits of communication. Next we reduce the number of rounds and the length of communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6 rounds, O(n) bits of communication and yields Shannon entropy n- O(log n) and min entropy n/ 2 - O(log n). Our protocol achieves the same Shannon entropy bound as, also non-explicit, protocol of Gradwohl et al. (in: Dwork (ed) Advances in Cryptology—CRYPTO ‘06, 409–426, Technical Report , 2006), however achieves much higher min entropy: n/ 2 - O(log n) versus O(log n). Finally we exhibit a very simple 3-round explicit “geometric” protocol with communication length O(n). We connect the security parameter of this protocol with the well studied Kakey

Randomised Individual Communication Complexity

In this paper we study the individual communication complexity of the following problem. Alice receives an input string x and Bob an input string y, and Alice has to output y. For deterministic protocols it has been shown in Buhrman et al. (2004), that C(y) many bits need to be exchanged even if the actual amount of information C(y|x) is much smaller than C(y). It turns out that for randomised protocols the situation is very diffe