Mathematics in economics: Achievements,difficulties, perspectives

I am deeply moved by this high honor which has fallen my lot to receive and I am happy to have the opportunity to appear here as a participant in this honorable series of lectures.In our time mathematics has penetrated into economics so solidly and widely and my theme is connected with such a variety of facts and problems that it brings me to cite the words of Kozma Prutkov, which are very popular in my country: One cannot embrace the unembraceable. The appropriateness of this wise sentence is not diminished by the fact that that great thinker is only a pen name.So, I want to restrict my theme to topics which are nearer to me, mainly to optimization models and their use in the control of the economy in order to best use resources for obtaining best results. I shall touch mainly on the problems and experiences of a planned economy, especially of the Soviet economy. Even within these limits I will succeed to consider only a few problems.Copyright © the Nobel Foundation 1975.     

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.     

Identity,Goals, and Learning: Mathematics in Cultural Practice

In this article, I explore and elaborate the relation between goals, identities, and learning and argue for their utility as a model by which to understand the nature of learning in general and to better understand the way in which race, culture, and learning become intertwined for minority students in American schools. Drawing on sociocultural perspectives on learning and development, including Wenger (1998) and Saxe (1999), I describe findings from 2 studies of learning among African American students outside of school, in the cultural practices of dominoes and basketball. This research shows that indeed, as players come to learn these practices, they both shift in regard to the goals they seek to accomplish within the practice and change as they come to define themselves vis-à-vis the practice. The implications for understanding the relation between race, culture, and learning are discussed.     

Democracy in action: Quantization, saturation, and compressive sensing

Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analog-to-digital converters and digital imagers in certain applications. To date, most of the CS literature has been devoted to studying the recovery of sparse signals from a small number of linear measurements. In this paper, we study more practical CS systems where the measurements are quantized to a finite number of bits; in such systems some of the measurements typically saturate, causing significant nonlinearity and potentially unbounded errors. We develop two general approaches to sparse signal recovery in the face of saturation error. The first approach merely rejects saturated measurements; the second approach factors them into a conventional CS recovery algorithm via convex consistency constraints. To prove that both approaches are capable of stable signal recovery, we exploit the heretofore relatively unexplored property that many CS measurement systems are democratic, in that each measurement carries roughly the same amount of information about the signal being acquired. A series of computational experiments indicate that the signal acquisition error is minimized when a significant fraction of the CS measurements is allowed to saturate (10–30% in our experiments). This challenges the conventional wisdom of both conventional sampling and CS.     

Metric structures in : dimension, snowflakes, and average distortion

We study the metric properties of finite subsets of L₁. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L₁.We present some new observations concerning the relation of L₁ to dimension, topology, and Euclidean distortion. We show that every n-point subset of L₁ embeds into L₂ with average distortion , yielding the first evidence that the conjectured worst-case bound of is valid. We also address the issue of dimension reduction in L_p for p(1,2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ₁ norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251–260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in ℓ₁, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514–523], cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.     

Fronts in reactive convection: Bounds,stability, and instability

This paper examines a simplified active combustion model in which the reaction influences the flow. We consider front propagation in a reactive Boussinesq system in an infinite vertical strip. Nonlinear stability of planar fronts is established for narrow domains when the Rayleigh number is not too large. Planar fronts are shown to be linearly unstable with respect to long‐wavelength perturbations if the Rayleigh number is sufficiently large. We also prove uniform bounds on the bulk burning rate and the Nusselt number in the KPP reaction case. © 2003 Wiley Periodicals, Inc.