Probability Theory - Core (MTH 124.4)

ICTS Course no.: MTH 124.4

Venue: Online

Class timings: Tue/Th 11:00-12:30

First meeting: Jan 11, 2022

Course description:

1. Review of discrete Probability
2. Review of Measure theoretic facts (distribution of a random variable, expectation, product measures, Fubini’s theorem)
3. Laws of Large numbers: independence, sums of an independent random variable, weak law of large numbers, Borel-Cantelli theorems, strong law of large numbers, random series
4. Central limit theorems: weak convergence of probability measures, characteristic functions,  central limit theorem, infinitely divisible distributions
5. Conditional expectation, Martingales (uniform integrability, Doob's up crossing lemma, martingale convergence and related topics)
6. Introduction to Brownian motion
7. Additional topics (if time permits): Random walks, Markov chains, ergodic theory.

Course Evaluation: The final grade will be based on the following three components:

1. Final exam: The final exam will count towards 40% of the final grade.
2. Homework: There will be four homework assignments throughout the semester and they will count towards 40% of the final grade.
3. Class performance: The students will be required to make short (~30-45 min) in-class presentations approximately once every two weeks on pre-specified topics. These will count towards 20% of the final grade.

Classical Mechanics - Core (PHY-204.5)

ICTS Course no.: PHY-204.5

Venue: Online

Class timings: Fridays and Saturdays 06:00-07:30

First meeting: 22nd January 2022

Course description:

1. Recap:

1. Recap of Newton's laws and their consequences - System of point masses, Rigid Bodies

2. Classical driven-dissipative systems

2. Lagrangian Formulation:

1. Principle of least action

2. Noether's Theorem, Symmetries - Small Oscillations, Applications

3. Rigid body motion:

1. Euler Angles - Tops

4. Hamiltonian formulation:

1. Liouville's Theorem

2. Action-Angle variables

3. Hamilton-Jacobi Equations

5. Dynamical Systems and Chaos

1. 1D Flows and Bifurcation

2. 2D Flows, Limit Cycles, Bifurcations

3. Chaos: Lorenz Systems, Fixed Points, Attractors, and Fractals - KAM theory (time permitting)

Books:

1. Landau Lifshitz course on theoretical physics: Vol 1: Classical Mechanics

2. Classical Mechanics by Herbert Goldstein, Charles P. Poole, John L. Safko

Other books/references especially for the last topic will be given in class periodically.

Evaluation40% Assignments + 30% Mid Semester exam  + 30% End SemesterExam

Lab Course - Core (PHY-108.5)

ICTS Course no.: PHY-108.5

Venue: Via Zoom for first and second experiments and subsequently in J C Bose Lab

Class timings: Mondays 02:00-03:30

First meeting: 17th January 2022

Course description:

List of experiments:

1. Surfactant spreading and fracture of interfacial particle rafts
2. Structure factor of disordered particle configurations
3. Measurement of Seebeck and Peltier coefficients.
4. Random Resistor Network

Evaluation:

1. (60%) Written report and presentation for each experiment
2. (20%) Participation in discussions
3. (10%) Ability to achieve open-ended goals of the experiment
4. (10%) Final quiz: at the end of the final experiment each

Students will be individually quizzed on all experiments, for their understanding of the various concepts/ideas discussed throughout the term.

ICTS Course no.: PHY-207.5

Venue: Online

Class timings: Tuesday/Thursday : 11:30 - 13:00 Hrs; Tutorials on Saturday : 11:30 - 13:00 Hrs

First meeting: Monday (16:00 - 19:00 Hrs), 24th January 2022

Course description: [PDF]

Course evaluation: The grading policy will be based on the following weightage :

– Assignments: 35% for PhD students, 40% for I.PhD students

– Midterm Exam: 30%

– End term Exam: 30%

– Term paper (a thorough review of a topic in electromagnetism not covered in the textbooks mentioned below. I will send in a list of term paper suggestions later.): 5% Extra credit (Compulsory for PhD Students)

Note that Assignments form a central part of this course since one of the main aims of this course is to train students to solve problems

Geometry and Topology in Physics -Topical (PHY-403.5)

ICTS Course no.: PHY-403.5

Venue: Online

Class timings: 11:00-12:30 Mondays and 04:00- 05:30 Tuesdays

First meeting: 11:00 am, 10th January 2022

Course description:

Prerequisites: Advanced Classical Mechanics, Quantum Mechanics, Statistical Mechanics, (all at the level of Landau and Lifshitz), complex analysis, some basic knowledge of group theory.

Textbooks: There are no fixed textbooks for the course. We will be drawing on many sources from the published literature and the internet. We will draw on standard texts like Nakahara and Nash and Sen for some parts of the course. We will also have student reviews of important papers towards the end of the course.

Structure of the course: The course will cover a number of applications of geometry and topology in the context of physical examples. The emphasis will be on the examples rather than on rigour. This course will be complementary to mathematics courses on geometry and topology. Exposure to such courses will be helpful, but not a prerequisite to follow the course.

What students will gain from the course: an appreciation of the commonality between different areas of physics; the unifying nature of geometric and topological ideas in physics. An opportunity to develop and hone their presentation skills in a friendly environment.

How the course will achieve its goals: We will take specific examples of systems from different areas of physics and analyse them from a geometric perspective. Make connections wherever possible between the different examples. The course will start with simple examples and graduate to more advanced ones. The choice of examples will depend on the feedback I get from the students.

Assessment: Assessments will be based on assignments (50%) and term paper presentations (50%) by the students. Students have to choose from a list of classic papers (one paper per student) and make a presentation to the class.

It is not presently clear if the pandemic will permit in-person classes. We hope that the situation will improve. If this does not happen, classes will be conducted online. In this case, students taking the course will be required to have a good internet connection.

Non equilibrium Statistical Physics -Topical (PHY-422.5)

ICTS Course no.: PHY-422.5

Venue: Online

Class timings: Wednesdays and Fridays 11:00-12:30

First meeting: 21st January 2022

Course description:

1. Random walks including method of generating functions and the first passage problem
2. Basics of stochastic procceses and Markov processes.
3. Brownian motion, classical and quantum Langevin equations.
4. Path integral approaches and Fokker Planck equations
5. Correlation Functions and Spectral Densities; the Wiener-Khintchine theorem; light spectra; noise in a gravitational-wave detector
6. Linear response theory and fluctuation dissipation relations
7. Interacting particle systems: Glauber dynamics and Monte-Carlo simulations.

The course will be aimed at understanding formalism through examples.

Requirements: Students should have a solid basic knowledge of statistical physics and quantum physics

References:

1. Stochastic processes in physics and chemistry: van Kampen
2. Nonequilibrium Statistical Physics: Noelle Pottier
3. Random Walks and random environment: Barry D. Hughes (some relevant chapters)
4. Stochastic processes in physics and astronomy, S. Chandrasekhar
5. Brownian Functionals in physics and computer science, S. N. Majumdar
6. Blandford and Thorne: Classical Physics: Chapter on Random Processes

Other relevant references will be provided during the course.

Assessment:  Assignment (50%) + Examination (50%)

Quantum Field Theory - Topical (PHY-481.5)

ICTS Course no.: PHY-481.5

Venue: Online

Class timings: Tuesdays and Thursdays 02:00-03:30

First meeting: 20th January 2022

Course description: This is an intermediate-level quantum field theory (QFT) course. It is designed to introduce students to QFT but also cater to students who have had some previous exposure to the object. So we will start with the basics of QFT but also seek to cover some advanced material.

Syllabus: The need for fields; Canonical quantization of fields; Particles; Perturbation theory; Feynman diagrams; Properties of the S matrix; Renormalization; Functional Methods; Gauge theories; New methods for scattering amplitudes.

Textbooks:
1) Introduction to quantum field theory, Peskin and Schroeder
2) Quantum Theory of Fields vol I and vol II, Weinberg

References to additional textbooks and notes will be provided during the lectures.

Course evaluation: Grades will be calculated using 75% assignments + 25% final exam.

Open Quantum Systems - Topical (PHY-405.5)

ICTS Course no.: PHY-405.5

Venue: Online

Class timings: Wednesdays and Fridays 4:00 PM to 05:30 PM

First meeting: 19th January 2022

Course description:

Prerequisites: Quantum Mechanics, Statistical Physics

Syllabus:

1. General formalism and various approaches for Open Quantum Systems
2. Damped Quantum Harmonic Oscillator and multi-level systems
3. Exact results for Spin Boson Model (Dephasing) and some generalizations
4. Integrability of Jaynes-Cummings and quantum Rabi models
5. Driven-Dissipative Quantum Systems and applications
6. Hermitian and Non-Hermitian Dicke Model: Chaos and Connections to Random Matrix Theory
7. Matrix Product States for Open Quantum Many-Body Systems

References:

Below are some suggested references. I will also be making additional notes.

1. Howard Carmichael, Statistical Methods in Quantum Optics 1. Master Equations and Fokker-Planck Equations (Springer)
2. Girish S. Agarwal, Quantum Optics (Cambridge University Press)
3. Heinz-Peter Breuer and Francesco Petruccione, The theory of open quantum systems (Oxford University Press)
4. Marlan O. Scully and M. Suhail Zubairy, Quantum optics (Cambridge University Press)
5. Fritz Haake, Sven Gnutzmann, Marek Kuś, Quantum signatures of chaos (Springer)
6. Simulation methods for open quantum many-body systems, Hendrik Weimer, Augustine Kshetrimayum, and   Román Orús, Rev. Mod. Phys. 93, 015008  (2021)

Term paper (report + presentation) topics

Below are suggested topics for term paper (report + presentation). The suggested references for each of them will be updated. Students will need to finalize a topic (latest by February 23rd, 2022), make a report and then give a presentation (at the end of the semester).

1) Circuit-QED with non-trivial lattice geometry and connectivity
2) Parity-Time Symmetric Systems and exceptional points
3) Opto-mechanical Systems
4) Symmetries and spectral properties of Liouvillians

Homework – 40 %
Term paper (report and presentation) – 30 %
Final Exam – 30 %

Physics of Living Matter - Topical (PHY-411.5)

ICTS Course no.: PHY-411.5

Venue: Online [Zoom link will be sent separately]

Class timings: 10:00-11:30 am, Tuesday and Thursday

First meeting: 10:00 am, 25th January 2022

Webpage: https://biophysics.icts.res.in/teaching/physics-of-living-matter/

Course description:

This course will give an introduction to stochastic processes, self-organized pattern formation, active matter, spin models, and dynamical systems by giving biological examples at various different scales. Prior knowledge of biology is not necessary but familiarity with differential equations and a first course in statistical physics are requisites.

Fill out this Google Form before 17th Jan 2022.

Introduction to Numerical Relativity - Topical (PHY-433.5)

ICTS Course no.: PHY-433.5

Venue: Online

Class timings: Wednesday 2-3:30pm / Thursday 3:45-5pm

First meeting: January 19, 2022

Course description: Reading course based primarily on Baumgarte & Shapiro's book: "Numerical Relativity: solving Einstein's equations on the Computer". Following chapters in order, followed by additional readings:

1.  General Relativity preliminaries

2.  The 3+1 decomposition of Einstein's equations

4.  Choosing coordinates: lapse and shift

3.  Constructing initial data

12. Binary black hole initial data

11. Recasting the evolution equations

13. Binary black hole evolution

7.  Locating black hole horizons

6.2 Numerical Methods: Finite Difference

6.3 Numerical Methods: Spectral

A1. Numerical Methods: Discontinuous Galerkin

A2. Assignment: Literature Review of advanced topic

A3. Assignment: Evolve scalar waves in curved spacetime using open-source NR codes

Format: Two sessions a week each of 90 minutes with students presenting. Interim problems for tutorials, review assignments for advanced topics.

Evaluation: The final grade will be based on the following components: