 Physical Sciences
Courses for Aug  Nov 2019
The schedule of ICTS courses for Aug  Nov 2019 are given below

Introduction to General Relativity (Reading)
Instructor: Bala Iyer
Venue: Amal Raychaudhuri Meeting room, ICTS Campus, Bangalore
Class Timings: Monday 1:453:15 pm, Friday 1:453:15 pm
First Class: Monday, 12 August, 2019
Text Books:
 Introducing Einstein’s Relativity: Ray D’Inverno
 A first course in general relativity: B. Schutz
Structure of the course: The reading course has three components:
 Weekly Presentation and Participation
 Problem solving
 Final Oral Exam/Seminar
Presentations will be Twice a week (1.5 hrs each) where all students take turns in reading the assigned text and presenting them. I will start off the course with an Overview Lecture on GR and Information on Standard Texts they can consult. Problems on various modules will be evaluated by a TA. There will be an endsemester Oral Exam (which may be replaced by a Seminar)
Final Grades will be based on:
 Class presentation/participation: 30%
 Problems: 30%
 End term Oral Exam (or Seminar): 40%

String Theory I (Reading)
Instructor: R.Loganayagam
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Class Timings: Wednesday and Friday, 11:0012:30 AM(Tentative)
First Class: Wednesday (10:00 am), 7th August, 2019
Structure of the course: The reading course has three components : Presentation/Class participation, assignments and exams.
Presentations will be twice a week (1.52hrs each) where all students take turns in reading the assigned text and presenting them. I will start off the course with a set of 6 to 8 lectures (i.e, 3 − 4 weeks) giving a brief survey at the level of basic textbooks mentioned below.
Assignments will be a set of problems on various modules which need to be handed over by those who are crediting the course. Since I do not really have a TA for this course, I want the students who credit this course to grade each others’ assignments.
There will be a midsemester and an endsemester exam (the latter can be replaced by a termpaper, see below for details).
The grading policy will be based on the following weightage :
– Class presentation/participation : 20% – Assignments : 40% – Mid term Exam : 20% – End term Exam (or) Term paper : 20%
For more details, see the PDF

Classical Mechanics (Core)
Instructor: Manas Kulkarni
Class Timings: Wednesdays  3:30 to 5:00 pm and Fridays – 4 pm to 5:30 pm
Venue: Chern lecture hall, ICTS Campus, Bangalore
First Class: Wednesday (4:00pm), 7th August, 2019
Topics:
 Recap:
Recap of Newton's laws and their consequences
System of point masses, Rigid Bodies
Classical drivendissipative systems
 Lagrangian Formulation:
Principle of least action
Noether's Theorem, Symmetries
Small Oscillations, Applications
 Rigid body motion:
Euler Angles
Tops
 Hamiltonian formulation:
Liouville's Theorem
ActionAngle variables
HamiltonJacobi Equations
 Classical Integrable Models and Field Theory:
Lax Pairs
Toda Model
Calogero Family of Models
Integrable Field Theories
Integrable Partial Differential Equations and applications in physics.
Books:
 Landau Lifshitz course on theoretical physics: Vol 1: Classical Mechanics
 Classical Mechanics by Herbert Goldstein, Charles P. Poole, John L. Safko
 Analytical Mechanics by Louis N. Hand, Janet D. Finch
 classical integrable finitedimensional systems related to Lie algebras, M.A. Olshanetsky, A.M.Perelomov, Physics Reports, Volume 71, Issue 5, May 1981, Pages 313400

Physics of Living Matter (Elective)
Instructor: VijayKumar Krishnamurthy
Prerequisites: A first course on statistical physics
Outline: Basic phenomenology of living systems. Bionumbers. Statistical physics in biology (active particles, chemical kinetics, feeding by diffusion, membrane potentials). Molecular machines (molecular motors, polymerases, synthases, enzymes, ionpumps, mitochondria). Macromolecular assemblies (polymers, membranes). Sensing and signalling (receptorligand interactions, MWC model, biochemical pathways, physical limits to sensing). Hydrodynamics (NavierStokes, low Reynolds number flows, swimming, generalized hydrodynamics, active matter, physics of the actomyosin cytoskeleton). Pattern formation (morphogen gradients, Turing patterns, mechanochemical patterns)
Time: Tuesdays and Thursdays 10:00 am  11:30 am
First Meeting: Thursday, 8th August 2019
Venue: Feynman Lecture Hall, ICTS, Bangalore
Webpage: https://biophysics.icts.res.in/teaching/physicsoflivingmatter/
Sign up: https://forms.gle/n5KDAzauZZBtXDbk8

Statistical Physics of Turbulent Flows (Elective)
Instructor: Samriddhi Sankar Ray
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Meeting Time: Wednesdays and Thursdays: 2.00 pm  3:30 pm
First Meeting: Wednesday, 7th August 2019
Course Outline:
 Basics of Fluid Dynamics
 Fourier Analysis
 Isotropic Turbulence: Phenomenology of ThreeDimensional Turbulence
 Analytical Theories (closures, etc) and Stochastic Models
 TwoDimensional Turbulence

Advanced Quantum Mechanics (Core)
Instructor: Suvrat Raju
Venue and Timings: 2:30 to 4:00 pm Feynman Lecture hall, Thursdays: 2:30 to 4:00 pm Chern Lecture hall
Course Outline
 Mathematical preliminaries of quantum mechanics: Linear Algebra; Hilbert spaces (states and operators)
 Heisenberg and Schrodinger pictures
 Symmetries: Role of symmetries and types (spacetime and internal, discrete and continuous); Symmetries and quantum numbers; Simple examples of symmetry (Translation, parity, time reversal); Rotations and representation theory of Angular momentum; Creation and annihilation operator formalism for a simple harmonic oscillator.
 Perturbation Theory
 Scattering
We will also study some additional topics, including some elements of quantum information theory.
Textbook: Modern Quantum Mechanics by Sakurai.

Lab Course (Core)
Instructors:
Abhishek Dhar,
Vishal Vasan
Timings for first meet: 2 pm Monday, 19th August 2019
Venue: J C Bose Lab
Course structure:
Students will rotate amongst 4 experiments, devoting two weeks to each experimental setup. Students are expected to devote 8 − 10 hours per week to each experiment. At the end of the allotted two week period for each experiment, students will give a short presentation to the instructors and rest of the class. Students will submit a report detailing the theory for their experiment, the experimental procedure, their data and analysis as well as their conclusions regarding the challenges, what remains to be investigated and their advice to the next team.
List of experiments:
 Exploring drag force on an object moving in a fluid
 Observing Brownian motion and estimating Avogadro’s number
 Surface gravity waves and dispersion relations
 Resonance of acoustic waves in cavities
Evaluation:
 (60%) Written report and presentation for each experiment
 (20%) Participation in discussions
 (10%) Ability to achieve openended goals of the experiment
 (10%) Final quiz: at the end of the final experiment each student will be individually quizzed on all experiments, for their understanding of the various concepts/ideas discussed throughout the term.
Courses for Jan  Apr 2019
The schedule of ICTS courses for Jan  Apr 2019 are given below

Classical field theory (Reading)
Instructor: VijayKumar Krishnamurthy
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Timings: Monday 11:0012:30Pm, Thursday 4:005:30 Pm
Topics:
Elasticity theory and fluid dynamics with rudiments of the dynamics of anisotropic fluids and pattern formation in biology. The course will also discuss developing finite element numerical codes in Python using FEniCS. The emphasis in the course will be on applications relevant to understanding the physics of living systems.
Prerequisites: Classical Mechanics and prior knowledge of the Python language. Exposure to numerical methods will be an advantage.
Evaluation: There will be two exams and around 4 assignments which will also include coding assignments. Both the assignments and exams will carry equal weight.
Interested people should send an email to <vijaykumar@icts.res.in> by 1700, 24th January 2019. Further details will be communicated by email.
References:
 Modern Classical Physics <https://press.princeton.edu/titles/10157.html>
 Elasticity and Geometry <https://global.oup.com/academic/product/elasticityandgeometry9780198506256 >
 Pattern Formation and Dynamics in Nonequilibrium Systems <https://doi.org/10.1017/CBO9780511627200>
 Soft Matter Physics <https://link.springer.com/book/10.1007/b97416>
FEniCS <https://fenicsproject.org/>

Bordism and topological field theory (Reading)
Instructor: Pranav Pandit
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Timings: Tuesday and Thursdays, 2:304:00pm
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:
 Cobordism as a generalized cohomology theory, basic homotopy theory, spectra
 The PontrjaginThom construction (reducing cobordism to homotopy theory)
 The AtiyahSegal axiomatization of topological quantum field theories
 The classification of 2d TQFTs in the AtiyahSegal framework.
 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:
 Extended 2d TFTs appearing in topological string theory; CalabiYau A∞categories.
 Constructing 3d TFTs from modular tensor categories; examples of interest in condensed matter physics.
 Factorization algebras (algebras of observables) and factorization homology.
For more details, see <PDF link>

Magnetism (Elective)
Instructor: Subhro Bhattacharjee
Venue: Emmy Noether Seminar Room, ICTS Campus, Bangalore
Timings: Wednesday and Friday, 6:00  7:30 Pm
First Class: Wednesday (4:00  5:30 pm), 16 January, 2019, Chern Lecture Hall, ICTS Campus, Bangalore
Topics:
 Introduction to magnetism
 Magnetic materials
 Mean eld theory for magnetic ordering and fluctuations
 Spin path integral
 Magnetism in one dimensional spin systems
 Quantum spin liquid and topological order
 Quantum Phase transitions in Magnetic systems
For more details, see <PDF link>
References:
 Reference material will be mentioned in class topicwise. General references include
 Quantum phase transition, Subir Sachdev
 Interacting electrons and quantum magnetism, Assa Auerbach
 Lectures on Manybody physics, P. Fazekas
Grading:
Assignments (50 %) : Typically one assignment every 2 weeks.
End semester Exam (50%)

Classical Electromagnetism (Core)
Instructor: R.Loganayagam
Tutors: Akhil Sivakumar and Srikanth Pai
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Timings : Wednesday  10:00  11:30 Am
Tutorials : Friday  2:30  3:30 Pm
First Class: Wednesday (2:30 pm), 2nd January, 2019,(Preliminary Test I);Emmy Noether Seminar Room, ICTS Campus; Bangalore For more details, see <PDF link>
The grading policy will be based on the following weightage :
 Quiz/Tests during Tutorials : 15% for Int.PhDs, 10% for PhD. students
 Assignments : 25%
 Mid term Exam : 30%
 End term Exam : 30%
 Term paper (a thorough review of a topic in electromagnetism not covered in textbooks below, see below for suggestions) : 5% Extra credit (Compulsory for PhD Students)

Mathematical Methods for physics (Core)
Instructor: Parameswaran Ajith
Teaching Assistant: Rahul Kashyap (ICTS)
Venue: Chern Lecture Hall, ICTS Campus, Bangalore
Timings : Tuesday 10:00  11:30 Hrs and Thursday 16:00  17:30 Hrs
First Class: Tuesday, 8th January, 2019
Topics:
Vector analysis in general coordinates, tensor analysis. Matrices, operators, diagonalization, eigenvalues and eigenvectors. Infinite series, convergence, Taylor expansion. Complex analysis, Cauchy’s integral theorem, Laurent expansion, singularities, calculus of residues, evaluating integrals. Partial differential equations, separation of variables, series solutions, Green’s function. SturmLiouville theory. Fourier and Laplace transforms.
References:
G. Arfken & H. Weber : Mathematical Methods for Physicists (Academic)
B. F. Schutz, A First Course in General Relativity (Cambridge)
Evaluation:
Assignments: 40%
Mid term test: 30%
Final test: 30%
Course web page

Advanced Statistical Physics (Core)
Instructor: Anupam Kundu
Venue: Emmy Noether Seminar Room, ICTS Campus, Bangalore
Timings: Tuesday 4:00  5:30pm and Friday 3:00  4:30 pm (Tentative)
First Class: Wednesday (4:00  5:30Pm), 2nd January, 2019
Topics:
 Brief overview of the statistical mechanics
 Interacting systems: Thermodynamic limits, fields, Collective phenomena
 Phenomenological description of phase transition and critical phenomena
 Statistical fields: Meanfield theory, Variational problem, LandauGinzburg theory, Saddle point approximations, Continuous and discrete symmetry breaking, domain walls.
 Correlations and fluctuations, Distribution functions
 Lattice systems, exact and approximate methods (Series expansions, BethePierls approximation, Duality in two dimension)
 Monte Carlo Simulations
 Scaling hypothesis (Homogeneity assumptions, divergence of correlation length, self similarity)
 Renormalisation Group theory (Conceptual, Gaussian model, Perturbative RG)
 Dissipative dynamics
Books:
 Statistical Physics of fields, Mehran Karder
 Lectures on phase transitions and Renormalisation group, N. Goldenfeld
 Statistical field theory, G. Mussardo
Condensed Matter Physics 1 (Elective)
Instructor: Chandan Dasgupta and Subhro Bhattacharjee
Venue: Chern Lecture Hall, ICTS Campus, Bangalore
Timings: Tuesday and Thursdays, 2:304:00 pm
First Class: Thursday (2:30 pm), 3rd January, 2019
Description: This course is aimed to introduce the basics of condensed matter physics. These ideas and techniques form the building blocks for studies in quantum manybody physics and a large class of quantum field theories that form the basis of our present understanding of materials around us. A detailed outline is attached and students interested in aspects of quantum manybody physics are strongly encouraged to credit/audit the course.
Helpful Prerequisites
Quantum Mechanics II, Statistical Mechanics I.
Tentative Topics
 Topic 0 : Introduction to quantum condensed matter (34 lectures)
 Topic 1: Electron Gas (7 lectures)
 Topic 2 : Lattice (8 lectures)
 Topic 3 : Electrons in crystalline solids (6 lectures)
 Topic 4 : Magnetism (2 lectures)
 Topic 5 : Superconductivity (4 lectures)
For more details, see <PDF link>
Grading
 Assignments (50%): Typically one assignment every 2 weeks.
 End semester Exam (50%)
Courses for Aug  Nov 2018
The schedule of ICTS courses for Aug  Nov 2018 are given below:

Course on Fluid Mechanics (Elective: Aug Nov 2018)
Instructor: Rama Govindarajan
Venue: Chern lecture hall, ICTS Campus, Bangalore
Time: Tuesdays and Thursdays, 11:00 AM
First Meeting: Tuesday, 21st August, 2018

Advanced Quantum Mechanics (Core)
Instructor: Suvrat Raju
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Timings: Tuesdays and Thursdays, 2:30  4:00 pm
First Class: Tuesday (2:30 pm), 7th August, 2018
Description: This is a core course covering some fundamental concepts in quantum mechanics. We will discuss some simple linear algebra, Hilbert spaces, the Heisenberg and Schrodinger pictures, discrete symmetries, continuous symmetries with a focus on the theory of angular momentum, perturbation theory, identical particles and some elements of scattering theory. If time permits, we will also discuss some selected topics from quantum information theory.
Textbook: The course will closely follow the textbook "Modern Quantum Mechanics" by Sakurai. Additional references, if required, will be provided in class.
Classical Mechanics (Core)
Instructor: Manas Kulkarni
Venue: Chern lecture hall, ICTS Campus, Bangalore
Timings: Wednesdays and Fridays, 4:00  5:30 pm
First Class: Wednesday (4:00pm), 1st August, 2018
Topics:
1) Recap:
 Recap of Newton's laws and their consequences
 System of point masses, Rigid Bodies
 Classical drivendissipative systems
2) Lagrangian Formulation:
 Principle of least action
 Noether's Theorem, Symmetries
 Small Oscillations, Applications
3) Rigid body motion:
 Euler Angles
 Tops
4) Hamiltonian formulation:
 Liouville's Theorem
 ActionAngle variables
 HamiltonJacobi Equations
5) Classical Integrable Models and Field Theory:
 Lax Pairs
 Toda Model
 Calogero Family of Models
 Integrable Field Theories
 Integrable Partial Differential Equations and applications in physics.

Statistical PhysicsI (Core)
Instructor: Abhishek Dhar
Venue: Chern Lecture Hall, ICTS Campus, Bangalore
Timings: Monday 4:00  5:30pm, Wednesday: 2:30  4:00pm
First Class:Friday, 3rd August, 2018. at 2:15pm
Details: The course on statistical physicsI will be based on the book Statistical Physics of Particles: Mehran Kardar
Topics to be covered:
 Thermodynamics,
 Probability,
 Kinetic theory of gases
 Classical statistical mechanics
 Interacting particles
 Quantum statistical mechanics
 Ideal quantum gases
 Biological Physics (Elective)
Course Title: Biological Physics (Elective)
Instructor: Vijay Kumar Krishnamurthy, Sriram Ramaswamy, Shashi Thutupalli
Venue: Physics department, IISc, Bangalore
Time: Tuesdays and Fridays, 2:00 pm 3:30 pm
First Meeting:Tuesday, 7th August, 2018
Outline
 the living state as a physicist sees it
 what a cell contains
 noise and biological information
 random walks, Brownian motion, diffusion
 fluid flow in cell and microbe biology
 entropic forces, electrostatics, chemical reactions, selfassembly
 macromolecules: statistics, forces, folding, melting
 molecular machines
 electrical transport across membranes: neurons, nerve impulses
 cell membrane mechanics: elasticity, order, shape, dynamics
 the cytoskeleton and cell mechanics
 collective motility
Prerequisites
Mechanics and Statistical physics at 1styear graduate student level
Evaluation
Homework assignments, midsemester & endsemester exams
Reference Texts
References in biophysics
Courses for Jan  Apr 2018
 Quantum and Statistical field theory (Core)
Instructor: Subhro Bhattacharjee
Tutor : Pushkal Shrivastava
Venue: Emmy Noether Seminar Room, ICTS Campus, Bangalore
Meeting Time: Tuesdays & Thursdays: 3:15 pm  4:45 pm
First Class: Thursday (3:00 pm), January 4, 2018
Course Outline :
 Statistical Mechanics  From the discrete to the continuum
 Quantum Mechanics of particles to continuum Quantum Fields
 Functional integral formulation of QM and QFT
 Parallels and Differences between continuum description of Stat. Mech. and Quantum systems.
 Spontaneous Symmetry breaking
 Wilsonian RG
 Additional reading topics selected by the instructor
 Electromagnetic Theory (Core)
Instructor: R Loganayagam
Tutor : Chandan Kumar Jana
Venue: Emmy Noether Seminar Room, ICTS Campus, Bangalore
Meeting Time: Tuesdays & Thursdays: 10:30 am  12:00 pm
First Class: Wednesday (4:30 pm), January 3, 2018
Course Outline : Please click here for more details
 Condensed Matter Physics  Interacting Systems (Elective)
Instructor: Chandan Dasgupta
Venue: Emmy Noether Seminar Room, ICTS Campus, Bangalore
Meeting Time: Mondays: 11:30 am  1:00 pm & Wednesdays: 1:45 pm  3:15 pm
First Class: Wednesday (11:30 am), January 10, 2018
Prerequisites: Courses on elementary solid state physics and statistical physics.
Course Outline :
 Classical systems of particles: Cluster expansion, van der Waals equation, liquidstate theory,classical density functional theory.
 Interacting electrons: HartreeFock approximation, exchange and correlation effects, quasiparticles, Fermi liquid theory, density functional theory. Dielectric function of electron gas  random phase approximation, plasmons, screening. The Hubbard model  metalinsulator transition, spin and charge density wave states.
 Interacting bosons: Weakly interacting bosonic systems, BoseEinstein condensation, superfluidity. Anharmonic effects in phonons.
 Electronphonon interaction: Phonons in metals, electron mass renormalization, effective interaction between electrons, polarons.
 Superconductivity: Cooper instability, BCS theory, GinzburgLandau theory, vortex lattice, Josephson effect.
 Magnetism: Microscopic mechanisms, models, magnetic phase transitions, spin waves.
 Numerical methods for Physics and Astrophysics (Elective)
Instructor: P. Ajith
Tutor : Ajit Kumar Mehta
Venue: Emmy Noether Seminar Room, ICTS Campus, Bangalore
Meeting Time: Wednesdays & Fridays: 3:30 pm  5:30 pm
First Class: Wednesday (3:30 pm), January 17, 2018
More info: click here
 Classical fields  elasticity theory and fluid dynamics (Reading)
Instructor: Abhishek Dhar
Summary: The course will be based almost entirely on the book "Applications of classical physics" by Roger D. Blandford and Kip S. Thorne. The focus will be on explaining the basics of elasticity theory and fluid dynamics, and their applications to understanding various physical phenomena, both from everyday life and from the laboratory.
Course contents: Chapters 1115 of "Applications of classical physics" available here Chapter 11: Elastostatics , Chapter 12: Elastodynamics, Chapter 13: Foundations of Fluid Dynamics, Chapter 14: Vorticity, Chapter 15: Turbulence
Courses for Aug  Nov 2017
Courses for Jan  Apr 2017
Courses for Aug  Nov 2016
Courses for Aug  Nov 2015

 Mathematics
Courses for Aug  Nov 2019
The schedule of ICTS courses for Aug  Nov 2019 are given below:

Algebra: a categorical perspective
Instructor: Pranav pandit
Venue: Chern Lecture Hall, ICTS Campus, Bangalore
Course description: This will be an advanced course in algebra, emphasizing the categorical viewpoint and the methods of homological algebra. Topics that we will aim to discuss include categories and functors, (co)limits and Kan extensions, adjunctions and monads, derived categories, derived functors, algebras and their representation theory, and Galois theory.
Prerequisites: The equivalent of a oneyear graduate level course in algebra
First meeting: 11:00 am, Tuesday, 6th August 2019
For more details, see the PDF

MA 396: Theory of large deviations and related topics
Instructor: Anirban Basak
Email: anirban.basak@icts.res.in
Course webpage: https://home.icts.res.in/~anirban/MA3962019.html
Office hours: to be announced later.
Office location: to be announced later.
Class time and location: Tu Th 2.003.30 PM, LH3, IISc Mathematics department.
Prerequisite: This is a graduate level topics course in probability theory. Graduate level measure theoretic probability will be useful, but not a requirement. The course will be accessible to advanced undergraduates who have had sufficient exposure to probability.
Course outline: Large deviations provide quantitative estimates of the probabilities of rare events in (highdimensional) stochastic systems. The course will begin with general foundations of the theory of large deviations and will cover classical large deviations techniques. In the latter part of the course some recent developments, such as large deviations in the context of random graphs and matrices, and its application in statistical physics will be discussed.
Suggested books:
 Amir Dembo and Ofer Zeitouni, Large Deviations Techniques and Applications.
 Firas RassoulAgha and Timo Seppäläinen, A Course on Large Deviations with an Introduction to Gibbs Measures.
 Marc Mézard and Andrea Montanari, Information, Physics, and Computation.
 Sourav Chatterjee, Large Deviations for Random Graphs.
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.

Introduction to Topology and Geometry
Instructor: Rukmini Dey
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Meeting Time: Monday: 2:00 pm  3:00 pm and Friday: 2:00 pm  4:00 pm
First Meeting: 7th August 2019
Syllabus:
Topology
 Pointset topology: open sets, closed sets, notions of continuity, connected sets, compact sets etc, homeomorphism, homotopy etc.
 Covering spaces, Fundamental Group and Simplicial Homology basic defintions and examples and methods of computing them.
Topology
 Differential geometry of curves and surfaces: curvature of curves, SerreFrenet formula, tangent planes, Gauss map, prinicipal curvatures, Gaussian and mean curvature.
 Manifolds, vector fields on manifolds, Lie algbera, Lie group, their action on manifolds.
 Differential forms on manifolds; de Rham cohomology
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 
Courses for Jan  Apr 2019
The schedule of ICTS courses for Jan  Apr 2019 are given below:

Bordism and topological field theory (Reading)
Instructor: Pranav Pandit
Venue: Feynman Lecture Hall, ICTS Campus, Bangalore
Timings: Tuesday and Thursdays, 2:304:00pm
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:
 Cobordism as a generalized cohomology theory, basic homotopy theory, spectra
 The PontrjaginThom construction (reducing cobordism to homotopy theory)
 The AtiyahSegal axiomatization of topological quantum field theories
 The classification of 2d TQFTs in the AtiyahSegal framework.
 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:
 Extended 2d TFTs appearing in topological string theory; CalabiYau A∞categories.
 Constructing 3d TFTs from modular tensor categories; examples of interest in condensed matter physics.
 Factorization algebras (algebras of observables) and factorization homology.
For more details, see <PDF link>

Introduction to Mechanics
Instructor: Vishal Vasan
Venue: CAM Lecture Hall 111, Bangalore
Timings: Tuesday & 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) HamiltonJacobi theory
II. Continuum Mechanics
(a) Conservation equations, strain and constraint tensors
(b) Constitutive laws (solid and fluid), frame indifference, isotropy
(c) Stokes, NavierStokes and Euler systems
(d) Maxwell system
III. Waterwaves
(a) Potential flow in a freely moving boundary
(b) Hamiltonian formulation of water waves
(c) Multiple scales and asymptotic models
(d) Shallowwater waves
(e) Quasigeostrophic 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. Homeworks 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. CohenTannoudji, Quantum Mechanics Vol. I
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/mmono146
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
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 
Courses for Aug  Nov 2018
The schedule of ICTS courses for Aug  Nov 2018 are given below:
 Techniques in discrete probability (Elective)
Instructor: Riddhipratim Basu
Venue: Math department LH5, IISc, Bangalore
Meeting Time: Tuesdays and Thursdays, 2:003:30 pm
First Class: Thursday, 2nd August, 2018
MA 394: Techniques in discrete probability
Credits: 3:0
Prerequisites:
 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)
 the probabilistic method;
 first and second moment methods, martingale techniques for concentration inequalities;
 coupling techniques, monotone coupling and censoring techniques;
 correlation inequalities, FKG and BK inequalities;
 isoperimetric inequalities, spectral gap, Poincare inequality;
 Fourier analysis on hypercube, Hypercontractivity, noise sensitivity and sharp threshold phenomenon;
 Stein’s method;
 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:
 Noga Alon and Joel Spencer, The Probabilistic Method ,Wiley, 2008.
 Geoffrey Grimmett, Probability on Graphs ,Cambridge University Press, 2010.
 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 
Courses for Jan  Apr 201
The schedule of ICTS courses for Jan  Apr 2018 are given below
 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.
 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
