In this talk we will present a near-optimal compression scheme for bounded-round randomized 2-party communication protocols. Previously, such a scheme was only known for protocols where the inputs to the parties are independent. The results yield a new optimal direct sum theorem for bounded-round communication. They also reveal a tight connection between the Information Cost of a problem and its amortized Communication Complexity. Joint work with Anup Rao.
We give a simple combinatorial proof of the Chernoff-Hoeffding concentration
bound for sums of independent Boolean random variables. Unlike the standard
proofs, our proof does not rely on the method of higher moments, but rather uses
an intuitive counting argument. In addition, this new proof is constructive in the
following sense: if the given random variables fail the concentration bound, then
we can efficiently find a subset of the variables that are statistically dependent.
As easy corollaries, we also get concentration bounds for [0, 1]-valued random
Semidefinite programming bounds are widely used in combinatorial optimization, quantum computing and complexity theory. The first semidefinite programming bound to gain fame is the so-called theta number developed by Lov\'asz to compute the Shannon capacity of the five-cycle graph. The semidefinite relaxation for the maximal cut problem has led to the near ubiquitous use of semidefinite programming in designing approximation algorithms.
We shall discuss new pseudorandom generators for regular read-once branching programs of small width. A branching program is regular if the in-degree of every vertex in it is (either 0 or) 2. For every width d and length n, the pseudorandom generator uses a seed of length O((log d + log log n + log(1/p)) log n) to produce n bits that cannot be distinguished from a uniformly random string by any regular width d length n read-once branching program, except with probability p > 0
In this survey I will present several extremal problems, and some solutions, concerning convex lattice polytopes.
A typical example is to determine the smallest area that a convex lattice polygon can have if it has exactly n vertices.
The mathematical problems arising from modern celestial mechanics, which originated with Isaac Newton’s Principia in 1687, have led to many mathematical theories. Poincaré (1854-1912) discovered that a system of several celestial bodies moving under Newton’s gravitational law shows chaotic dynamics. Earlier, Euler (1707–83) and Lagrange (1736–1813) found instances of stable motion; a spacecraft in the gravitational fields of the sun, earth, and the moon provides an interesting system of this kind. Helmut Hofer, Professor in the School of Mathematics, explains how these observations have led to the development of a geometry based on area rather than distance.
Collateralized Default Obligations (CDOs) and related financial derivatives have been at the center of the last financial crisis and subject of ongoing regulatory overhaul.
In this talk, I shall describe an ongoing project to develop a complexity theory for cryptographic (multi-party computations. Different kinds of cryptographic computations involve different constraints on how information is accessed. Our goal is to qualitatively -- and if possible, quantitatively -- characterize the "cryptographic complexity" (defined using appropriate notions of reductions) of these different modes of accessing information. Also, we explore the relationship between such cryptographic complexity and computational intractability.
For four years now, we have been conducting "medium-scale" experiments in how human subjects behave in strategic and economic settings mediated by an underlying network structure. We have explored a wide range of networks inspired by generative models from the literature, and a diverse set of collective strategic problems, including biased voting, graph coloring, consensus, and networked trading. These experiments have yielded a wealth of both specific findings and emerging general themes about how populations of human subjects interact in strategic networks.