The schedule of ICTS courses for Jan - Apr 2019 are given below:

  1. Bordism and topological field theory (Reading)

    Instructor Pranav Pandit

    Venue: Feynman Lecture Hall, ICTS Campus, Bangalore

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

    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.

    For more details, see <PDF link>

  2.  

  3. Introduction to Mechanics

    InstructorVishal Vasan

    Venue: CAM Lecture Hall 111, Bangalore

    TimingsTuesday & Thursday 9:00 - 10:30am 

    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

     

    Evaluation and homeworks

    • 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 homeworks.
    • 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.)

    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  

  4.  

  5. Dynamics Systems

    Instructor: Amit Apte

    Venue: Feynman Lecture Hall, ICTS Campus, Bangalore

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

    First Class: Wednesday (11:00am), 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


  6. The following is the list of courses offered at IISc. For the current list see:

    Core Elective Courses
    Course No. Course Title
    MA 212 Algebra I
    MA 219 Linear Algebra
    MA 221 Analysis I: Real Anaysis
    MA 231 Topology
    MA 261 Probability Models
    MA 223 Functional Analysis
    MA 232 Introduction to Algebraic Topology
    MA 242 Partial Differential Equations
    MA 213 Algebra II
    MA 222 Analysis II : Measure and Integration
    MA 224 Complex Analysis
    MA 229 Calculus on Manifolds
    MA 241 Ordinary Differential Equations

    Advanced Elective Courses
    Course No. Course Title
    MA 215 Introduction to Modular Forms
    MA 277 Advanced PDE and Finite Element Method
    MA 361 Probability Theory
    MA 368 Topics in Probability and Stochastic Processes
    MA 278 Introduction to Dynamical Systems Theory
    MA 313 Algebraic Number Theory
    MA 314 Introduction to Algebraic Geometry
    MA 315 Lie Algebras and their Representations
    MA 317 Introduction to Analytic Number Theory
    MA 319 Algebraic Combinatorics
    MA 320 Representation Theory of Compact Lie Groups
    MA 332 Algebraic Topology
    MA 364 Linear and Nonlinear Time Series Analysis
    MA 369 Quantum Mechanics

The schedule of ICTS courses for Aug - Nov 2018 are given below:


  1. Techniques in discrete probability (Elective)

Instructor: Riddhipratim Basu

Venue: Math department LH-5, IISc, Bangalore

Meeting Time: Tuesdays and Thursdays, 2:00-3:30 pm 

First Class: Thursday, 2nd August, 2018


MA 394: Techniques in discrete probability

Credits: 3:0

Pre-requisites:

  1. This course is aimed at Ph.D. students from different fields who expect to use discrete probability in their research. Graduate level measure theoretic probability will be useful, but not a requirement. I expect the course will be accessible to advanced undergraduates who have had sufficient exposure to probability.

We shall illustrate some important techniques in studying discrete random structures through a number of examples. The techniques we shall focus on will include (if time permits)

  1. the probabilistic method;
  2. first and second moment methods, martingale techniques for concentration inequalities;
  3. coupling techniques, monotone coupling and censoring techniques;
  4. correlation inequalities, FKG and BK inequalities;
  5. isoperimetric inequalities, spectral gap, Poincare inequality;
  6. Fourier analysis on hypercube, Hypercontractivity, noise sensitivity and sharp threshold phenomenon;
  7. Stein’s method;
  8. entropy and information theoretic techniques.

We shall discuss applications of these techniques in various fields such as Markov chains, percolation, interacting particle systems and random graphs.


Suggested books:

  1. Noga Alon and Joel Spencer, The Probabilistic Method ,Wiley, 2008.
  2. Geoffrey Grimmett, Probability on Graphs ,Cambridge University Press, 2010.
  3. Ryan O'Donnell, Analysis of Boolean Functions ,Cambridge University Press, 2014.

 

The following is the list of courses offered at IISc. For the current list see:

Core Elective Courses
Course No. Course Title
MA 212 Algebra I
MA 219 Linear Algebra
MA 221 Analysis I: Real Anaysis
MA 231 Topology
MA 261 Probability Models
MA 223 Functional Analysis
MA 232 Introduction to Algebraic Topology
MA 242 Partial Differential Equations
MA 213 Algebra II
MA 222 Analysis II : Measure and Integration
MA 224 Complex Analysis
MA 229 Calculus on Manifolds
MA 241 Ordinary Differential Equations

Advanced Elective Courses
Course No. Course Title
MA 215 Introduction to Modular Forms
MA 277 Advanced PDE and Finite Element Method
MA 361 Probability Theory
MA 368 Topics in Probability and Stochastic Processes
MA 278 Introduction to Dynamical Systems Theory
MA 313 Algebraic Number Theory
MA 314 Introduction to Algebraic Geometry
MA 315 Lie Algebras and their Representations
MA 317 Introduction to Analytic Number Theory
MA 319 Algebraic Combinatorics
MA 320 Representation Theory of Compact Lie Groups
MA 332 Algebraic Topology
MA 364 Linear and Nonlinear Time Series Analysis
MA 369 Quantum Mechanics

The schedule of ICTS courses for Jan - Apr 2018 are given below

  1. Introduction to PDEs (Reading)

    Instructor: Rukmini Dey

    Venue: S N Bose Meeting Room, ICTS Campus, Bangalore

    Meeting Time: Monday and Friday: 11:30 am - 1:00 pm

    First Class:  Monday, 15th January, 2018

    Course contents: First 5 chapters of Ian Sneddon's book: Elements of PDEs.

    Syllabus: Ordinary Differential Equations in more than 2 variables; Partial Differential Equations of the first order; Partial Differential Equations of the Second Order; Laplace Equation; Wave Equation. If time permits we will go through some chapters of "Fourier Series" by Georgi P. Tolstov.​
     

  2. Introduction to Dynamical Systems (Reading)

    Instructor: Vishal Vasan

    Venue: ICTS Campus, Bangalore

    Meeting Time: Friday: 4:30 pm - 6:00 pm

    First Class:  Friday, 19th January, 2018

    Course contents: Nonlinear Dynamics and Chaos by S Strogatz. Selected reading from Differential Equations and Dynamical Systems by L Perko and other suitable texts.

    Syllabus: The course will cover the entire content of Strogatz' book supplemented with more detailed mathematical treatments of selected theorems from other sources.

Courses for Aug - Nov 2017

Courses for Jan - Apr 2017