Courses for Jan - Apr 2019

1. Bordism and Topological Field Theory (Reading)

Instructor: Pranav Pandit

Venue: Feynman Lecture Hall, ICTS Campus, Bangalore

Timings: Tuesday and Thursdays, 2:30-4:00 pm

First Class: Wednesday (6:00 - 7:30 pm), 15 January 2019, Feynman Lecture Hall, ICTS Campus, Bangalore 

Topics: The core topics for this course will be: 

  1. Cobordism as a generalized cohomology theory, basic homotopy theory, spectra 
  2. The Pontrjagin-Thom construction (reducing cobordism to homotopy theory)
  3. The Atiyah-Segal axiomatization of topological quantum field theories 
  4. The classification of 2d TQFTs in the Atiyah-Segal framework.
  5. The notion of an extended topological field theory, and the statement of the classification theorem for such theories (the cobordism hypothesis). 1 Possible advanced topics, depending on the time available and the interests of the participants, include: 
  6. Extended 2d TFTs appearing in topological string theory; Calabi-Yau A∞-categories.
  7. Constructing 3d TFTs from modular tensor categories; examples of interest in condensed matter physics.
  8. Factorization algebras (algebras of observables) and factorization homology.
  9. For more details, see <PDF link>

2. Introduction to Mechanics

Instructor: Vishal Vasan

Venue: CAM Lecture Hall 111, Bangalore

Timings: Tuesday & Thursday 9:00 - 10:30 am 

First Class: Tuesday, 8th January 2019

Required background: This course is meant to introduce a typical student of mathematics to certain PDE/ODE models as they arise in physics. As such, this course is targeted towards students with no prior physics background.Familiarity with ideas from ODE/PDE theory and functional analysis will be very useful. 

Tentative Topics 

       I. Classical Mechanics

              (a) Elements of Newtonian mechanics and formulations: Lagrangian, Hamiltonian

              (b) Principle of stationary action

              (c) Legendre transform

              (d) Noether’s theorem

              (e) Hamilton-Jacobi theory

       II. Continuum Mechanics

              (a) Conservation equations, strain and constraint tensors

              (b) Constitutive laws (solid and fluid), frame indifference, isotropy

              (c) Stokes, Navier-Stokes and Euler systems

              (d) Maxwell system

       III. Water-waves

              (a) Potential flow in a freely moving boundary

              (b) Hamiltonian formulation of water waves

              (c) Multiple scales and asymptotic models

              (d) Shallow-water waves

              (e) Quasi-geostrophic equations

       IV. Quantum mechanics

              (a) Quantum states

              (b) Observers and Observables

              (c) Amplitude evolution

              (d) Simple examples

              (e) Evolution of expectations and conservation laws

Reference books 

   The main reference will be An Brief Introduction to Classical, Statistical and Quantum Mechanics by O. B¨uhler. 

   In addition, the students may find the following list of texts useful throughout the course to supplement their understanding.

      (1) V.I Arnold, Mathematical Methods in Classical Mechanics 

      (2) G. Duvaut, Mechanics of continuous media 

      (3) H. Goldstein, C.P. Poole & J. Safko, Classical Mechanics 

      (4) R. Dautray & J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology 

      (5) A. Chorin & Marsden, A Mathematical Introduction to Fluid Mechanics. 

      (6) P. Kundu, Fluid Mechanics 

      (7) E. Zeidler, Nonlinear Functional Analysis and its Applications IV. 

      (8) T. Frankel, Geometry of Physiscs 

      (9) M. Peyrard & T. Dauxois, Physics of Solitons 

      (10) J. Pedlosky, Waves in the ocean and atmosphere 

      (11) C. Cohen-Tannoudji, Quantum Mechanics Vol. I  

Evaluation and homework

  • Homeworks will be assigned typically every other week and due in two weeks time. Home-works count for 50% of the final grade and there will likely be 4 − 5 homework.
  • Students will be expected to submit a report. Topics will be chosen after discussion with the instructor, but typically will be a specific PDE model. The report will discuss the relevance, derivation and open problems associated with the PDE model and any other related issues.
  • Each student will submit one draft (as a midterm) and a final draft (as a final exam). Writing is an essential part of the course and all reports must be prepared using LATEXor similar software.
  • Students may also expect to be assigned required reading materials (articles, book chapters, etc.)

3. Dynamics Systems

Instructor: Amit Apte

Venue: Feynman Lecture Hall, ICTS Campus, Bangalore

Timings: Monday and Wednesdays, 4:15 - 5:45 pm

First Class: Wednesday (11:00 am), 2nd January 2019

Topics:

1) Linear dynamical systems: 
     -  autonomous systems, 
     -  Floquet theory for periodic systems, 
     -  Lyapunov exponents and their stability, 
     -  numerical techniques for computing Lyapunov exponents

2) Nonlinear systems:

      - flows, stable and unstable manifolds
      - limit sets and attractors 

3)  Bifurcations and chaos  

       - normal forms, Lyapunov exponents (again!)

4) Ergodic theory and hyperbolic dynamical systems.
    Reference Texts

    1. Introduction to Linear Systems of Differential Equations by L. Ya. Adrianova; https://bookstore.ams.org/mmono-146
    2. Differential Equations and Dynamical Systems by Lawrence Perko
    3. Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch, Stephen Smale, and Robert L. Devaney
    4. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes
    5. Introduction to Smooth Ergodic Theory by Luis Barreira and Yakov Pesin
    N+1. Review articles and other papers as and when required

For the current list at IISc: http://math.iisc.ac.in/newcourse.html