Richard Montgomery, University of California, Santa Cruz will deliver a talk entitled, "An Octahedral Gem Hidden in Newton's Three Body Problem." The lecture will take place on July 25, 2012 at the Fields Institute, as part of the conference on "Geometry, Symmetry, Dynamics, and Control: The Legacy of Jerry Marsden."
Richard Montgomery received undergraduate degrees in both mathematics and physics from Sonoma State in Northern California. He completed his PhD under Jerry Marsden at Berkeley in 1986, after which he held a Moore Instructorship at MIT for two years, followed by two years of postdoctoral studies at University of California, Berkeley.
Montgomery's research fields are geometric mechanics, celestial mechanics, control theory and differential geometry and is perhaps best known for his rediscovery - with Alain Chenciner - of Cris Moore's figure eight solution to the three-body problem, which led to numerous new 'choreography' solutions. He also established the existence of the first-known abnormal minimizer in sub-Riemannian geometry, and is known for investigations using gauge-theoretic ideas of how a falling cat lands on its feet. He has written one book on sub-Riemannian geometry.
The PIMS Marsden Memorial Lecture Series is dedicated to the memory of Jerrold E Marsden (1942-2010), a world-renowned Canadian applied mathematician. Marsden was the Carl F Braun Professor of Control and Dynamical Systems at Caltech, and prior to that he was at the University of California, Berkeley, for many years. He did extensive research in the areas of geometric mechanics, dynamical systems and control theory. He was one of the original founders in the early 1970s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today.
The inaugural Marsden Memorial Lecture was given by Alan Weinstein (University of California, Berkeley) in July of 2011 at ICIAM in Vancouver.
Number theory is concerned with Diophantine equations and their solutions, encoded in discrete structures involving integers, rational numbers or algebraic quantities. Topology studies the properties of shapes that are unchanged under continuous or smooth deformations, a technique of choice being the construction of appropriate homological invariants. It turns out--perhaps surprisingly to the uninitiated--that these invariants can be endowed with sufficient structure to capture a tremendous amount of arithmetic information. The powerful interplay between arithmetic and topological ideas underlies the most important breakthroughs in the study of Diophantine equations, such as Faltings’ proof of the Mordell Conjecture and Wiles’ proof of Fermat’s Last Theorem. It is also at the heart of more recent and still very fragmentary attempts to construct algebraic points on elliptic curves when their existence is predicted by the Birch and Swinnerton-Dyer conjecture. This lecture will attempt to give a non-technical sampler of some of the rich, fascinating interactions between arithmetic questions and topological insights.
Predictive policing seeks to anticipate the times and locations of crimes to better allocate law enforcement resources to combat these crimes. The key to predictive policing is modeling that
combines available data to forecast or estimate the areas most threatened by crimes at different times. We have developed models that integrate geographic, demographic, and social media information from a specific area of interest to produce the needed predictions. In this presentation, I describe our approach to this predictive modeling, which combines spatial-temporal generalized additive models (STGAM) with a new approach to text mining. We use the STGAM to predict the probability of criminal activity at a given location and time within the area of interest. Our new approach to text mining combines Latent Dirichlet Allocation (LDA) with Latent Semantic Indexing (LSI) to identify and use key topics in social media relevant to criminal activity. We use social media since these data provide a rich, event-based context for criminal incidents. I present our application of this approach to actual criminal incidents in Charlottesville, Virginia. Our results indicate that this combined modeling approach outperforms models that only use geographic and demographic data.
In this paper we study the dynamics of a population where the individuals can either be contributors (tax payers) or no contributors (tax evaders or cheaters). We introduce a 2D cellular automaton on which the probability of transition from one of the above states to the other is the sum of the local effect and of the global field effect. The model also includes the policy that allocates a fraction of the budget to fight tax evasion. This scheme allowed us to simulate the cases in which inhomogeneous strategies in contrasting tax evasion is applied in a region and the case in which cooperative policies are adopted by neighbor societies.
This talk focuses on the application of point process methods to crime and security data. We will discuss semi- and non- parametric models, as well as their estimation using Expectation-Maximization algorithms. We conclude the talk with some results from a randomized controlled trial in Los Angeles where police patrols are determined each day using a semi-parametric self-exciting point process.
In the first part of the talk, we consider the Short et.al. model of crime. This model exhibits hot-spots of crime -- localized areas of high criminal activity. In a certain asymptotic limit, we use singular perturbation theory to construct the profile of these hot-spots and then study their stability.
In the second part of the talk, we extend the original model to incorporate biased Levi Flights for the criminal's motion. Such motion is considered to be more realistic than the biased diffusion that was originally proposed. This generalization leads to fractional Laplacians. We then investigate the effect of introducing the Levi Flights on the formation of hot-spots using linear stability and full numerics.
Joint works with Jonah Breslau, Tum Chaturapruek, Daniel Yazdi, Scott McCalla, Michael Ward and Juncheng Wei.
Security is a critical concern around the world, whether it's the challenge of protecting ports, airports and other critical national infrastructure, or protecting wildlife and forests, or suppressing crime in urban areas. In many of these cases, limited security resources prevent full security coverage at all times; instead, these limited resources must be scheduled, avoiding schedule predictability, while simultaneously taking into account different target priorities, the responses of the adversaries to the security posture and potential uncertainty over adversary types.
Computational game theory can help design such unpredictable security schedules. Indeed, casting the problem as a Bayesian Stackelberg game, we have developed new algorithms that are now deployed over multiple years in multiple applications for security scheduling: for the US coast guard in Boston and New York (and potentially other ports), for the Federal Air Marshals(FAMS), for the Los Angeles Airport Police, with the Los Angeles Sheriff's Department for patrolling metro trains, with further applications under evaluation for the TSA and other agencies. These applications are leading to real-world use-inspired research in the emerging research area of security games; specifically, the research challenges posed by these applications include scaling up security games to large-scale problems, handling significant adversarial uncertainty, dealing with bounded rationality of human adversaries, and other interdisciplinary challenges. This lecture will provide an overview of my research's group's work in this area, outlining key algorithmic principles, research results, as well as a discussion of our deployed systems and lessons learned.
The existence and stability of localized patterns of criminal activity is studied for the two-component reaction-diffusion model of urban crime that was introduced by Short et.~al.~[Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp.~1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems (NLEP's) are derived that determine the stability of these hot-spot patterns to O(1) time-scale instabilities. From an analysis of these NLEP's, and a further analysis of the spectrum associated with the slow translational instabilities, an explicit threshold for the minimum spacing between stable hot-spots is derived. The theory is confirmed via detailed numerical simulations of the full PDE system. Moreover, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur. Finally, in the 1-D context, we show how the existence and stability of hot-spot patterns is altered from the inclusion of a third component to the reaction-diffusion system that incorporates the effect of police. In the context of this extended model, the optimal strategy for the police is discussed.
Joint Work with Theodore Kolokolnikov (Dalhousie) and Juncheng Wei (Chinese U. of Hong Kong and UBC).
In this talk we will present some results for systems of equations modeling residential burglary.
For the parabolic system model proposed by Andrea Bertozzi et-al, we study the equilibrium case. By using bifurcation theory we show that this system does support pattern formation. We also give some results concerning stability of the bifurcating patterns. These results correspond to a joint work with Chris Cosner and Steve Cantrel from the University of Miami.
The model has been recently modified by Pitcher giving rise to a new parabolic system of equations. We show some results for this system that contain a condition for existence of global solutions. This work corresponds to a collaboration with Philippe Souplet and Quoc Hung Phan from Paris 13.