Complex Systems Related courses -- Winter 2009

This course provides an overview of the major classes of ecological models, with an emphasis on ecological dynamics. We will focus on principles guiding the formulation of models and on the mathematical techniques that can be used to analyze them. We will examine deterministic and stochastic models, structured and unstructured models, single- and multiple-species models. Because ecological systems are typically nonlinear, we cannot often "solve" model equations: we employ techniques of nonlinear analysis, stochastic analysis, and numerical analysis to obtain results. This course will introduce many of these techniques in the context of ecological theory.

An additional goal of the course is to develop students' skills in the use of mathematical software. We will make extensive use of Matlab and R for numerical computations and Mathematica for symbolic computation.

Prerequisites: To get the most out of the class, students should have a firm grasp of elementary linear algebra (e.g., MATH 214, 217, 417, or 419) and ordinary differential equations (e.g., MATH 214, 216, 256, 286, or 316), and have had some exposure to probability. Students without these prerequisites should consult the instructor before registering.

Background and Goals: We often try to understand phenomena by formulating and analyzing models consisting of differential equations with initial and/or boundary conditions. But often, and especially if the equations are nonlinear, explicit solutions are not available. And even if we are clever enough or lucky enough to find an explicit formula or integral representation, it may still be difficult to extract useful information. In practice we frequently study such problems via approximate solutions obtained by asymptotic analysis. Asymptotic analysis is a collection of mathematical methods developed to systematically produce accurate approximations together with reliable error estimates. This course is an introduction to asymptotic analysis with a focus on differential equations. The prerequisite of a complex variables course such as Math 555 is absolutely essential. (Note: Math 556 is not a prerequisite for 557.) The material in Math 557 is itself prerequisite to many advanced topics in applied mathematics, science and engineering.

Course content: Asymptotic sequences and (divergent) series; asymptotic expansions of integrals and Laplace's method; the methods of steepest descents and stationary phase; asymptotic evaluation of inverse Fourier and Laplace transforms; asymptotic solutions for linear (non-constant coefficient) differential equations; WBK expansions; singular perturbation theory; boundary, initial and internal layers; method of multiple scales and nonlinear oscillations; selected applications.

Mathematical biology is a fast growing and exciting modem application which has gained worldwide recognition. This course will focus on the devrivation, analysis, and simulation of partial differential equations (PDEs) which model specific phenomena in molecular, cellular, and population biology. A goal of this course is to understand how the underlying spatial variability in natural systems influences motion and behavior. If time permits, we will cover the use of Delay Differential Equations to study models of Infectious Diseases.

Mathematical topics covered include derivation of relevant PDEs from first principle; reduction of PDEs to ODEs under steady state, quasi-state and traveling wave assumptions; solution techniques for PDEs and concepts of spatial stability and instability. These concepts will be introduced within the setting of classical and current problems in biology and the biomedical sciences such as cell motion, transport of biological substances, biological pattern formation, and cancer Above all, this course aims to enhance the interdisciplinary training of advanced undergraduate and graduate students from mathematics and other disciplines by introducing fundamental properties of partial differential equations in the context of interesting biological phenomena.

This course will discuss aspects of the modern theory of nonlinear dynamics and ordinary differential equations as applied to problems in geometric mechanics, Hamiltonian and nonholonomic systems (systems with nonintegrable constraints), nonlinear stability theory and nonlinear control theory. The role of symmetry and reduction will be discussed as well as topics such as the least action principle, integrability, symplectic and Poisson geometry, and controllability and accessibility on manifolds. Text: A. Bloch, Nonholonomic Mechanics and Control, Springer Verlag. Other books will be referenced as well as the primary mathematical literature. Prerequisite: a course in differential equations. Grading: The course grade will be based mainly on completion various problem sets and general class participation.

We consider Language as a Complex Adaptive System. Recent research across a variety of disciplines in the cognitive sciences has demonstrated that patterns of use determine how language is acquired, is structured, is processed, and changes over time. However, there is mounting evidence that processes of language acquisition, use and change are not independent from one another but are facets of the same complex adaptive system. A conference on this theme was held in November 08 at UMich (see http://elicorpora.info/LLC for details)

Each week as a class we (i) view the podcast recording of their presentation, (ii) read their associated journal article, and (iii) discuss this theme.

Updated November 2008