GA.2830.001: ADVANCED TOPICS IN APPLIED MATHEMATICS
(Quantifying Uncertainty in Complex Turbulent Systems)

3 points, Fall Term 2011
Thursday, 3:20PM - 5:00PM

Prerequisite: Some knowledge of elementary stochastic diff. equations and nonlinear dynamics although the first two lectures will provide a review.

In many situations in science and engineering, the analysis and prediction of phenomena often occur through complex dynamical equations which have significant model errors compared with the true signal in nature. Clearly, it is important both to improve the imperfect model's capabilities to recover crucial features of the natural system and also to model the sensitivities of the natural system to changes in external or internal parameters. These efforts are hampered by the fact that the actual dynamics of the natural system are unkown. An important example with major societal impact is the Earth's climate. This course is a discussion of cutting edge mathematical developments in quantifying uncertainty. Mathematical ideas involving fluctuation dissipation theorems, empirical information theory, and uncertainty propagation will be developed in a suite of instructive elementary and more complex examples. This is a seminar style course where Prof. Majda will give about half the lectures; his post docs, Branicki, Giannakis, and Sapsis will give the other lectures and the graduate students attending will be invited (this is not require) to participate in these lectures and small class projects.

Grading: Seminar Style Class

Background text: Information Theory and Stochastics for Multi Scale Nonlinear Systems, by Majda, Abramov, Grote, CRM Series, Vol. 25, American Mathematical Society, 2005 

G63.2480.001: ADVANCED TOPICS IN APPLIED MATHEMATICS
(Vorticity and Imcompressible Flows)

3 points, Spring Term 2011
Thursday, 3:20PM - 5:00PM

Prerequisite: no background besides some familiarity with elementary PDE is needed.

The course will cover material in chapters 1-5 and 7 of the text by Majda and Bertozzi.  If time permits, there will also be some lectures on statistical theories for vortices and an introduction to hurricane dynamics.  The main goal is to introduce graduate students, post docs, and visitors  to the fascinating interplay among exact solutions, nonlinear analysis, numerical computing, and statistical ideas in developing intuition about fluid flow.

Text: Vorticity and Incompressible Flow, Majda & Bertozzi, Cambridge University Press, 2002

Recommended text: Introduction to PDE and Waves for the Atmosphere and Ocean, Majda, Courant Lecture Notes, #9, AMS
G63.2480.001: ADVANCED TOPICS IN APPLIED MATHEMATICS
(Real Time Filtering of Turbulent Signals in Complex Systems)

3 points, Spring Term 2010
Thursday, 3:20PM - 5:00PM

Grading: This course will be set up as a reading course.

G63.2830.001  ADVANCED TOPICS IN APPLIED MATHEMATICS  (Vorticity and Incompressible Flow) - CLOSED FOR THE SEMESTER

3 points. Fall Term 2009
Thursday, 3:15PM - 5:00PM

Prerequisite: no background besides some familiarity with elementary PDE is needed.

The course will cover material in chapters 1-5 and 7 of the text by Majda and Bertozzi.  If time permits, there will also be some lectures on statistical theories for vortices and an introduction to hurricane dynamics.  The main goal is to introduce graduate students, post docs, and visitors  to the fascinating interplay among exact solutions, nonlinear analysis, numerical computing, and statistical ideas in developing intuition about fluid flow.

Grading: this course will be graded as a seminar course.

G63.2840.001: ADVANCED TOPICS IN APPLIED MATHEMATICS (Fluctuation Dissipation Theorems and Climate Change)

3 points, Spring Term 2009
Thursday, 3:15PM - 5:00PM

Can one do climate change response by computing suitable statistics of the present climate: This is an applied challenge of obvious practical importance. This class focuses on these issues from the viewpoint of modern applied mathematics, where ideas from dynamical systems, statistical physics, information theory, and stochastic-statistical dynamics will be blended with suitable qualitative and quantitative models and novel numerical algorithms to attach these questions.

The course has no formal requirements, but familiarity with elementary ODE and SDE is useful background. Chapters 2 and 3 of the book, "Information Theory and Stochastic for Multiscale Nonlinear Systems" by Majda, Abramov and Grote (American Mathematical Society) will provide the introductory material for these topics. Additional material, such as coping with model error and ensemble predictions, will also be discussed.

Grading: This course will be graded as a seminar course.