1. "Analysis of a Strategy for Sequencing the Human Genome" Dick Karp University of Washington The rapid increase in human genome sequencing effort and the emergence of several alternative strategies for large-scale sequencing raise the need for a thorough comnparison of such strategies. This talk surveys the various strategies and provides a mathematical analysis of one of them: the BAC-end strategy of Venter, Smith and Hood.
2. "Algorithms in real algebraic geometry" Ricky Pollack Courant Institute, New York University For the first decade or so of its existence, computational geometry was concerned primarily with a linear world: arrangements of points, lines, line segments and their higher dimensional analogues. In the past decade, motivated by questions in robotics, motion planning, computer vision, and learning theory, as well as by theoretical studies of arrangements in general, computational geometry has investigated arrangements of sets which cannot be described by linear equations and inequalities but are instead arrangements of ``semi-algebraic sets''. These are the solution sets of polynomial equalities and inequalities over the reals. The study of geometric, topological, algebraic and algorithmic issues about them constitutes a significant portion of real algebraic geometry. We begin the talk with an introduction to the subject, presenting the ways in which it is both similar to and different from the linear world and introducing many of the problems in which we are primarily interested. This is followed by a discussion of earlier results and the techniques that have made increasingly efficient algorithms for these problems possible. We report on recent developments by the speaker together with S. Basu and M. F. Roy, which have produced improved algorithms for the existential theory of the reals, quantifier elimination over real closed fields, the computation of roadmaps for semi-algebraic sets and the computation of a semi-algebraic description of the connected components of a semi-algebraic set. We conclude by presenting some very recent results of S. Basu on algorithms to compute the Euler characteristic of a semi-algeberaic set and his bounds on the complexity of a single cell in an arrangement of n surface patches in d-space.
3. Jennifer Chayes, Microsoft Finite-Size Scaling in Percolation and Satisfiability Phase transitions are abrupt changes in the global behavior of a system in response to small changes in an external parameter. They are familiar phenomena in physical systems. But such phenomena also occur in many probabilistic models and even in some problems in theoretical computer science. Technically, phase transitions occur only in infinite systems. Finite-size scaling is the study of how the transition behavior emerges from the behavior of large, finite near the critical value of the external parameter. In particular, finite-size scaling characterizes the sharpness of the transition as a function of the size of the system. In this talk, I will focus on finite-size scaling in two systems: percolation and the 2-satisfiability problem.
4. "Transforming Men Into Mice" Pavel Pevzner University of Southern California Many people (including myself) believe that transformations of humans into mice happen only in fairy tales. However, despite some differences in appearance and habits, men and mice are genetically very similar. In the pioneering paper, Nadeau and Taylor, 1984 estimated that surprisingly few genomic rearrangements (approximately 180) happened since the divergence of human and mouse 80 million years ago. However, their analysis is non-constructive and no rearrangement scenario for human-mouse evolution has been suggested yet. The problem is complicated by the fact that rearrangements in multi-chromosomal genomes include inversions, translocations, fusions and fissions of chromosomes, a rather complex set of operations. We prove a duality theorem which expresses the genomic distance in terms of easily computable parameters reflecting different combinatorial properties of sets of strings. This theorem leads to a polynomial-time algorithm for computing most parsimonious rearrangement scenarios. Based on this result and the comparative physical mapping data we have constructed a scenario of human-mouse evolution with 131 reversals/translocations/fusions/fissions. A combination of the genome rearrangement algorithm with the recently proposed experimental techniques suggests a new "rearrangement-based" approach to the problem of reconstructing mammalian evolution. This is a joint work with Sridhar Hannenhalli (SmithKline Beecham Pharmaceuticals).
5. "Verification of Open Systems" Moshe Y. Vardi Rice University In computer system design, we distinguish between closed and open systems. A closed system is a system whose behavior is completely determined by the state of the system. An open system is a system that interacts with its environment and whose behavior depends on this interaction. The ability of temporal logics to describe an ongoing interaction of a reactive program with its environment makes them particularly appropriate for the specification of open systems. Nevertheless, model-checking algorithms used for the verification of closed systems are not appropriate for the verification of open systems. Correct verification of open systems should check the system with respect to arbitrary environments and should take into account uncertainty regarding the environment. This is not the case with current model-checking algorithms and tools. Module checking is an algorithmic method that checks, given an open system (modeled as a finite structure) and a desired requirement (specified by a temporal-logic formula), whether the open system satisfies the requirement with respect to all environments. In this paper we describe and examine module checking problem, and study its computational complexity. Our results show that module checking is computationally harder than model checking. This work was done jointly with Orna Kupferman during the DIMACS Special Year on Logic and Algorithms.
6. "Sampling by Markov Chains: How to Speed It Up?" Laszlo Lovasz Yale University Random walk techniques have many applications to the problem of sampling, i.e., to the problem of generating a random element from a given distribution. Applications of such methods include simulation in statistical mechanics, combinatorial enumeration, integration, volume computation, generation of contingency tables, and optimization over convex bodies. The crucial issue in this applications is the "mixing time" of the random walk, i.e., the speed of convergence to the stationary distributions. Focusing on the application to sampling from a convex body, we discuss basic techniques an new developments in the analysis and improvement of sampling algorithms. In particular, we show how a "log-Cheeger inequality" improves mixing time bounds (joint work with Ravi Kannan), and how "lifting", a general construction suggested by the work of Diaconis, Holmes and Neal, leads to dramatic improvement in the mixing time (not just in the bound!) in some cases (joint work with Fang Chen and Igor Pak).