ICTS

Introduction to Dynamical Systems Theory (IISc course: MA 278; ICTS course: MTH 216.5)

Instructor: Amit Apte

Venue: google-meet, see link below for the google-classroom

Class timings: TBA

First meeting: 23rd February 2021 (10-11 am)

To join: https://bit D.O.T ly SLASH 3bg0P2l (hope you know what I mean!)

Course description:

Linear stability analysis, attractors, limit cycles, Poincare-Bendixson theorem, relaxation oscillations, elements of bifurcation theory: saddle-node, transcritical, pitchfork, Hopf bifurcations, integrability, Hamiltonian systems, Lotka-Volterra equations, Lyapunov function & direct method for stability, dissipative systems, Lorenz system, chaos & its measures, Lyapunov exponents, strange attractors, simple maps, period-doubling bifurcations, Feigenbaum constants, fractals. Both flows (continuous-time systems) & discrete-time systems (simple maps) will be discussed. Assignments will include numerical simulations.

Prerequisites, if any:

familiarity with linear algebra - matrices, and ordinary differential equations

Desirable:

Ability to write codes for solving simple problems.

Suggested books and references:

1. S. Strogatz, Nonlinear Dynamics and Chaos: with Applications to physics, Biology, Chemistry, and Engineering, Westview, 1994.

2. S. Wiggins, Introduction to applied nonlinear dynamics & chaos, Springer-Verlag, 2003.

3. K. Alligood, T. Sauer, & James A.Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, 1996.

4. M.Tabor, Chaos and Integrability in Non-linear Dynamics, 1989.

5. L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, AMS 1995.

6. Morris W. Hirsch, Robert L. Devaney, Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press 2012.

Academic Events for this course:

The course time will be adjusted in the first meeting based on any clashes that may arise, so those interested in registering should attend the first meeting without fail.

The mode of instruction will be completely online at least until 31st March. Further decisions will be announced in class.

Note:IISc semester schedule will be followed. Interested ICTS students, please contact the instructor for registration; Interested IISc students, please contact the IISc maths department for registration process (note: IISc students need to register for IISc course number MA 278).

google classroom link is https://classroom.google.com/u/0/c/Mjc1OTY1MTk3OTIw?cjc=jt67mhlp>

It is also mentioned on http://math.iisc.ac.in/course-schedule.html

Probability in High Dimensions (IISc course: MA 363; ICTS course: MTH 217.5)

Instructor: Anirban Basak

Venue: TBA

Class timings: 10-11:30 am on Tuesdays & Thursdays

First meeting: 2nd March 2021

Course description:

Prerequisites:

This is a graduate level topics course in probability theory. Graduate level measure theoretic probability will be useful, but not a requirement. Students are expected to be familiar with basic probability theory and linear algebra. The course will be accessible to advanced undergraduates who have had sufficient exposure to probability and linear algebra.

Course outline:

This course will be aimed at understanding the behavior of random geometric objects in high dimensional spaces such as random vectors, random graphs, random matrices, and random subspaces, as well. Topics will include the concentration of measure phenomenon, non-asymptotic random matrix theory, empirical processes, and some related topics from geometric functional analysis and convex geometry. Towards the latter half of the course, depending on students' interests, a few applications of the topics covered in the first half will be considered such as community detection, covariance estimation, randomized dimension reduction, and sparse recovery problems.

Suggested books and references:

1. Roman Vershynin, High-dimensional probability: An introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Series Number 47), 2018.

2. Roman Vershynin, Introduction to the non-asymptotic analysis of random matrices, Compressed sensing, 210-268, Cambridge University Press, 2012.

3. Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration Inequalities: A nonasymptotic theory of independence, Oxford University Press, 2013.

4. Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Springer Science & Business Media, 2013.

5. Avrim Blum, John Hopcroft, and Ravindran Kannan, Foundations of Data Science, Cambridge University Press, 2020.

6. Joel Tropp, An Introduction to Matrix Concentration Inequalities, Foundations and Trends in Machine Learning, Vol. 8, No. 1-2, pp 1-230, 2015.

Weekly schedule will be posted later.

Grading:

Students taking this course for credit are required to do a (reading) project, submit a report, and give a presentation on the same at the end of the semester. Depending on the number of registered students the grading scheme may change.

Email: anirban.basak@icts.res.in

Course webpage: Link

Office hours: By email appointment.

Note: IISc semester schedule will be followed (the start date and duration to be decided). Interested ICTS students, please contact the instructor for registration; Interested IISc students, please contact the IISc maths department for registration process (note: IISc students need to register for IISc course number MA 363).