Events for the week of November 09 2009

Abstract. http://www.math.ucla.edu/~ploh/ipamseminar/1110yury-person.pdf

Abstract. http://www.math.ucla.edu/~ploh/ipamseminar/1112-kevin-woods.pdf

Abstract. Two fundamental computationally hard problems in number theory are the discrete logarithm problem and factoring a large integer. To solve the discrete logarithm problem on a cyclic group of size $n$, J.M. Pollard provided two heuristic algorithms in 1979 which suggest a running time of $O(\sqrt{n})$, which are still some of the best we have to date for general cyclic groups. In recent works (with Montenegro and Kim-Montenegro-Peres), we confirm both heuristic behaviors with rigorous analysis. Time permitting, I will also mention a result on the stopping time of the square dependence problem (of Pomerance from 1994) arising in the analysis of quadratic sieve and other factoring algorithms. In joint work with Croot, Granville and Pemantle, we determine such a stopping time up to a small multiplicative constant, leaving open the question of a sharp threshold.

Abstract. The geographic profiling problem is the problem of constructing an estimate for the location of the home base of a serial offender from the locations of the offender's crime sites. In this talk, we shall present a mathematical survey of some of the algorithms that have been used to solve the geographic profiling problem. We will then present a new mathematical framework for the geographic profiling problem based on Bayesian methods that is able to incorporate both geographic features that influence the choice of a crime site as well as geographic features that affect the location of the offender's anchor point. We have implemented the resulting algorithm in software, and it is now being evaluated for effectiveness by a number of police agencies across the country. We will conclude with a discussion of open questions.