PhD Course Works

I. Ph.D. Course Work in Physical Sciences

  II. Ph.D. Course Work in Chemical Sciences 

  III. Ph.D. Course Work in Mathematics 

   Courses in Physical Sciences

  Courses in Chemical Sciences

1. PS 427: Computational Physics (3 credits)
2. PS 603: Topics in Classical and Quantum Mechanics (3 credits)
3. PS 621: Advanced Statistical Physics (3 credits)
4. PS 633: Stochastic Phenomena (3 credits)
5. PS 651: Dynamical Systems and Chaos (3 credits)
6. PS 662: Experimental Methods in Physics (3 credits)
7. PS 664: Special Topics in Condensed Matter Physics (3 credits)
8. PS 682: Advanced Laser Physics (3 credits)
9. PS 721: Nonequilibrium Statistical Mechanics (3 credits)
10. PS 722: Phase Transitions and Critical Phenomena (3 credits)
11. PS 723: Introduction to String Theory (3 credits)
12. PS 761: Many-Body Theory (3 credits)
13. PS 762: Introduction to Computational Neuroscience (3 credits)
14. & 15. PS 699 & 799: Research Course – I and II (3 credits each)

Note: The Pre-Ph.D. students may take any of the courses listed under M.Sc. Semester IV, as recommended by the Ph.D. Student Advisor.

Core Courses:

1. PS 611C  Concepts in Chemistry (3 credits)

 Optional Courses:

1. PS 612C Analytical Methods in Chemistry (3 credits)

2. PS 613C    Computational Chemistry and its Applications (3 credits)

3. PS 614C    Advanced Spectroscopy and its Applications (3 credits)

4. PS 615C    Supramolecular Chemistry  (3 credits)

5. PS 616C    Molecular Materials (3 credits)
6. PS 617C    Research Course (3 credits)

Note: Students can take the necessary Physics courses from the courses in Physical Sciences according to their interests and needs.

Core Courses:

1. PS604 Research and Publication Ethics (2 credits)
2. PS641M Algebra (3 credits)
3. PS642M Analysis (3 credits)
4. PS643M Topology (3 credits)

  Optional Courses:

1. PS644M Topological Groups and Lie Groups (2 credits)
2. PS645M Functional Analysis and Operator Theory (2 credits)
3. PS646M Research Methodology Course I (3 credits)
4. PS647M Research Course II (3 credits)
5. PS648M Queueing Theory (3 credits)
6. PS649M Advanced Topics in Fluid Mechanics (3 credits)
7. PS712M Algebraic Number Theory (3 credits)

Selected courses are offered each semester from the following list:

PS 427: Computational Physics

• Overview of computer organization, hardware, software, scientific programming in FORTRAN and/or C, C++.

• Numerical Techniques
  Sorting, interpolation, extrapolation, regression, numerical integration, quadrature, random number generation, linear algebra and matrix manipulations, inversion, diagonalization, eigenvectors and eigenvalues, integration of initial-value problems, Euler, Runge-Kutta, and Verlet schemes, root searching, optimization, fast Fourier transforms.

• Simulation Techniques
  Monte Carlo methods, molecular dynamics, simulation methods for the Ising model and atomic fluids, simulation methods for quantum-mechanical problems, time-dependent Schrödinger equation, discussion of selected problems in percolation, cellular automata, nonlinear dynamics, traffic problems, diffusion-limited aggregation, celestial mechanics, etc.
• Parallel Computation 
  Introduction to parallel computation

Suggested Texts:

• V. Rajaraman, Computer Programming in Fortran 77.
• W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing. (Similar volumes in C, C++.)
• H.M. Antia, Numerical Methods for Scientists and Engineers.
• D.W. Heermann, Computer Simulation Methods in Theoretical Physics.
• H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods.
• J.M. Thijssen, Computational Physics.

PS 603: Topics in Classical and Quantum Mechanics

This course is intended as a refresher course for incoming Ph.D. students.  It reviews material that has been taught in M.Sc. courses, but with greater emphasis on problem-solving. The actual content and format of the course will depend upon the instructor and the composition of the class. The topics to be covered will be selected from the list presented below.

Course Outline:

Classical Mechanics

• Particle motion in 1, 2 and 3 dimensions; conservation laws; non-inertial frames.
• Generalized coordinates; Lagrangian method and examples; two-body problem; bound states and scattering.
• Small oscillations; Hamiltonian formalism; canonical transformations, Hamilton-Jacobi theory.
• Rigid-body motion; motion of tops; rotation matrices.
• Special relativity; relativistic kinematics.

Quantum Mechanics

• One-dimensional Schrödinger equation; particle in a square-well potential; bound states; transmission and reflection from step potentials; W.K.B. method for bound states; tunneling; harmonic oscillator; operator method of solution.
• Two-level and other finite-dimensional Hilbert space problems.
• Three-dimensional Schrödinger equation, angular problem algebra of angular momentum; square well in 3 dimensions; hydrogen atom.
• Atoms in electric magnetic fields; spin-orbit coupling; examples of time-dependent perturbation theory; scattering theory.

PS 621: Advanced Statistical Physics

This course is also intended as a refresher course for incoming Ph.D. students and reviews material that may have been taught in M.Sc. courses.  The topics to be covered will be selected from the list presented below:

PS 633:  Stochastic Phenomena

This course introduces students to the modeling and characterization of stochastic phenomena.  Theoretical concepts are illustrated via many physical examples.

Course Outline:

PS 651:  Dynamical Systems and Chaos

This course is intended for students planning to pursue research in the areas of nonlinearity, classical and quantum chaos and related subjects. Depending on the interests of the students/instructor, the topics to be covered will be chosen from the list below:

Course Outline:

Dynamical Systems

• One-dimensional dynamics; logistic map; Sharkovskii theorem; hyperbolic systems; symbolic dynamics; chaos and bifurcation theory; period-doubling and other routes to chaos; Hopf bifurcations; Ruelle-Takens scenario; strange attractors.

• Control of chaotic systems; synchronization and applications; experiments on chaotic systems; electronic circuits; chemical chaos; turbulence.

Classical Chaos

• Qualitative analysis of ordinary differential equations; dissipative and conservative systems; Lagrangian and Hamiltonian formulation; integrability and Hamilton-Jocobi equations; perturbation methods; K.A.M. theory; chaotic dynamics.

• Attractors – simple and strange; Lyapunov exponents; Lorenz system.

Quantum Chaos

• Semiclassical mechanics; W.K.B. and E.B.K. quantization; periodic orbit theory; random matrices and applications to the study of eigenvalue spectra; periodic quantum systems; Floquet theory and recurrence; quantum maps.

Suggested Texts:

• R.L. Devaney, Introduction to Dynamical System
• E. Ott, Chaos in Dynamical Systems, Cambridge University Press (1992).
• A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, Springer-Verlag (1992).
• V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1979)

PS 662:  Experimental Methods in Physics

This course introduces the student to various important experimental techniques in physics.

Course Outline:

PS 664: Special Topics in Condensed Matter Physics

The topics to be covered will be chosen from the list below.

Course Outline:

• Quantization of free electromagnetic field; coherent states of electromagnetic field; blackbody radiation; density operator r; photon statistics; diagonal coherent-state representation of r.

• Interaction of two-level atom with classical field; Bloch equations; Rabi problem.

• Semi-classical theory of single-mode laser; Lamb equation; Langevin equation for field amplitude; Fokker-Planck equation; photon statistics.

• Relationship between quantum and semi-classical theories; coherent state representation of laser field; steady-state solution of master equation.

Suggested Texts:

PS 721: Non-equilibrium Statistical Mechanics

Course Outline:

• Liouville equation; response function and susceptibility; fluctuation-dissipation theorem; two-level systems; irreversibility and the master equation; Boltzmann equation; conductivity; Kubo formula.

• Brownian motion; Langevin, Fokker-Planck and diffusion equations; hydrodynamic fluctuations and structure factor.

PS 722: Phase Transitions and Critical Phenomena

Course Outline:

• Criteria for thermodynamic stability; first-order phase transitions; Van der Waals’ theory; Gibbs phase rule.

• Examples of phase transitions and phase diagrams; criticl points; second-order phase transitions; order parameter; critical exponents.

• Universality; Landau theory for phase transitions.

• Ising model; mean-field approximation; transfer matrix method; Onsager solution of 2-dimensional Ising model; Yang-Lee theorem.

PS 723: An Introduction to String Theory

Prerequisite: Lagrangian mechanics, Special theory of relativity, Electromagnetism, Quantum Mechanics.

Course outline:

• Motivation.

• Motion of a non-relativistic string, normal modes.

• Review of Special Theory of Relativity, light cone coordinates, Lorentz invariance in diverse dimensions, small and compact dimensions, square-well and quantum mechanics of extra dimensions, motion of a relativistic particle, world-line.

• Review of electromagnetism, electric field in diverse dimensions, point particle with electric charge.

• Newton’s law of gravitation in diverse dimensions, Planck units of mass, length and time, Newton’s constant, gravity and geometry.

• Relativistic string, world-sheet, invariants on the worldsheet, area of embedded surfaces, Nambu-Goto action, equations of motion, boundary conditions and branes, static gauge, tension and energy of strings.

• Classical motion of a string, dynamics on the world-sheet, conserved quantities.

• Point particle in light-cone gauge, quantization.

• String in light-cone gauge, string as oscillators, normal modes and particle spectrum (open and closed string), quantization.

• String thermodynamics: counting of states, partition function.

• Overview of results.

Suggested Texts:

1. B. Zwiebach, A first course in string theory (Cambridge University Press)
2. H. Goldstein, Classical mechanics (Addison-Wesley)
3. J. Hartle, Gravity: An introduction to Einstein’s general relativity (Pearson education)
4. J. Sakurai, Modern quantum Mechanics (Pearson education)
5. L. Schiff, Quantum mechanics (McGraw Hill)

PS 761: Many-Body Theory

Course Outline:

• Methods of second quantization; Green’s function; adiabatic switching and interaction picture; Wick’s theorem; diagrammatic analysis of perturbation theory; linear response theory.

• Fermi systems; Hartree-Fock approximation; hard sphere Fermi gas; uniform electron gas polarization and screening; correlation energy.

• Finite-temperature formalism; thermal Green’s functions; perturbation theory and diagrammatic analysis; electron gas at finite temperature; linear response at finite temperature.

• Linear theory of phonons; phonon-phonon and electron-phonon interaction; field theory for coupled electron-phonon system.

PS 762: Introduction to Computational Neuroscience

A major effort is currently under way to understand the operation of the central nervous system, and more specifically, of neuronal networks in the brain. This is of great importance at not only the theoretical level, but also for the possibility of understanding the causes and cures for diseases such as Alzheimer’s and Parkinson’s.

The approach presently being taken includes both experimental studies and theoretical and computational modeling to jointly address questions that arise in this area of research. There is an increasing need for scientists trained at the interface of these disciplines who possess a strong analytic background together with a solid understanding of biological phenomena.

The present course will teach students the basic set of mathematical and computational techniques required for them to pursue higher level research in the field of neuroscience. It would also prepare them, in part, to be able to move on to various industry jobs that require quantitative and analytic skills. For example, several pharmaceutical companies are actively seeking employees with the background to model and simulate processes on the computer prior to production and testing. The set of lectures will cover necessary techniques to be able to understand various biological questions, to address them mathematically and computationally and then to translate results into a language that is accessible to experimentalists. It is envisioned that students who pass this class will be able to immediately utilize their course work in either an experimental lab or on the way towards a PhD. Further the mathematical content of the course is sufficiently general that it will also allow students to work in modeling of biological problems outside of neuroscience, in fields such as genomics, protein signaling networks and even ecology.

This Pre-Ph.D. course will be accessible to final year students of the M.Sc.(Physics) program as well as those of the M.Sc.(Life Sciences) and that it could be an optional course in the M Tech. (Systems Biology) programme as well.

Pre-requisites: Calculus of many variables, a basic understanding of differential equations, ability to use computer software such as MATLAB and the ability to code in Fortran, C, C++ etc.

Course Outline [Approximate number of lectures per topic]

Introduction to neuroscience with description of some specific neuronal systems. [2]

Mathematical background – Introduction to dynamical Systems, review of basics of differential equations, introduction to phase plane analysis, dimensional reduction techniques including timescale separation ideas. [5]

Computational techniques – Introduction to relevant computer software such as XPP and Matlab. Classes during this time to be held in a computer lab in a tutorial manner with demonstrations of software usage. [5]

Models of single neurons – Derivation of the Hodgkin-Huxely equations and various reductions such as the FitzHugh-Nagumo and Morris-Lecar models. Analysis of these and other basic models such as the Integrate and Fire model. [6]

Models of synaptic interactions – Description of synapses and neurotransmitter release. Mathematical models for excitatory and inhibitory synapses. Models for short-term synaptic plasticity. [6]

Small network dynamics – Focus on understanding and characterizing the dynamics of small networks of excitatory, inhibitory or mixed-type neurons. Detailed analysis of conditions leading to complete synchronization, phase locking or chaotic behavior in such networks. [8]

Case studies –The dynamics of several specific biological examples will be explored including problems from the following areas: place cells in the hippocampus, sleep rhythms and oscillations of the thalamus, irregular activity in the basal ganglia, working memory models of the cortex and phase lag models of central pattern generators. [8]

Textbooks:

The following are list of suggested textbooks, although the course will initially be taught from a set of lecture notes.

1. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, by Peter Dayan and Larry F. Abbott. The MIT Press, 2001. ISBN 0-262-04199-5

2. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, by Eugene M. Izhikevich. The MIT Press, 2007. ISBN 0-262-09043-8

3. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, by Bard Ermentrout, SIAM 2002 ISBN 0-89871-506-7

1. PS 611C  Concepts in Chemistry (3 credits)

Course outline:

General Introduction

• Different types of forces- Covalent bonding, ionic bonding, H-bonding, Van der Waals forces; Theory of chemical bonding; Symmetry and Group theory.

Physical Chemistry

• Thermodynamics- Conservation of energy; Review of Enthalpy, Entropy, Free energy with examples involving chemical systems.
• Chemical kinetics and catalysis- Discussion of reaction rate theory; Collision and transition state theory; Potential energy surfaces; Catalysis; Enzyme catalysis.

Organic Chemistry

• Delocalized chemical bonding- Aromaticity, Hyperconjugation, Tautomerism.
• Acids and Bases.
• Stereochemistry- Optical activity; Chirality; Topicity and prostereoisomerism; Conformations of acyclic and cyclic molecules.
• Mechanism, reactivity and reactions of- Aliphatic and aromatic nucleophilic substitution; Aliphatic and aromatic electrophilic substitution; Addition to C-C and C-hetero multiple bonds; Elimination and Rearrangements; Oxidation and reductions.
• Named reactions; New reagents; Catalysts and their applications.

Inorganic Chemistry

• Structure of molecules- VSEPR theory; Bents rule; Berry Pseudorotation; Molecular orbital treatment for homonuclear, heteronuclear and delocalized molecules.
• Acids and Bases- Measures of acid-base strength; Hard and soft acids and bases.
• Redox reactions- Standard electrode potentials; Electromotive forces; Electrochemical series; Use of reduction potentials.
• Coordination chemistry- Bonding in coordination compounds; VBT, CFT and MOT; Electronic spectra of complexes; Magnetic properties of complexes.

Suggested Texts:

1. Peter Atkins and de Paula, Physical Chemistry, 7th edition, Oxford University Press Inc., New York.
2. J. E. Huheey, E. A. Keiter; R. L. Keiter, Inorganic Chemistry- Principles of Structure and Reactivity, 4th edition, Pearsons education.
3. J. March, Advanced Organic Chemistry, 5th edition, John Wiley and Sons.
4. F. Albert Cotton, Chemical Applications of Group Theory, 3rd edition, John Wiley and Sons.

2. PS 612C  Analytical Methods in Chemistry (3 credits)

Analytical chemistry is one of the important divisions of chemistry that aids researchers in experimental (classical and applied) chemistry to characterize chemical compounds. This course has been designed to introduce students to the methods and concepts of analytical chemistry and would give an overview of the instruments involved for characterization. 

In this course, students would be taught how to characterize molecules using various molecular spectroscopic and electro-analytical methods and bulk or aggregated structures using various surface and microscopic techniques.

Course outline:

Molecular Spectroscopy

• Infra-red spectroscopy- Theory; Instrumentation; Sample Handling; Interpretation of Spectra; Characteristic group absorptions of molecules.
• Mass spectroscopy- Instrumentation; Mass spectrum; Determination of molecular formula; Recognition of molecular ion peak; Ionization techniques; Fragmentation and rearrangements; Quantitative applications of mass spectrometry.   
• Nuclear Magnetic Resonance spectroscopy- Continuous-Wave (CW) NMR spectrometry; Relaxation; Pulsed Fourier Transform Spectrometry; Rotating Frame of Reference; Instrumentation and sample handling; Chemical shift; Spin coupling; Chemical shift equivalence; Magnetic equivalence; Geminal and vicinal coupling; Nuclear Overhauser Effect; 13C NMR spectroscopy; Correlation NMR spectrometry; 1H-1H COSY; Double-Quantum filtered 1H-1H COSY; 1H-13H COSY.

Electroanalytical Chemistry

• Potentiometry, Coulometry and Voltammetry.

Separation Methods

• An introduction to chromatographic separations; Gas chromatography; Gas chromatographic columns and stationary phases; Principles and applications of gas-liquid chromatography; High-Performance Liquid Chromatography (HPLC); Thin Layer and column Chromatography; Ion-Exchange Chromatography; Size-Exclusion chromatography.

Surface Characterization by Spectroscopy and Microscopy

• Reciprocal space map: 
  X-ray diffraction- Lattice; Lattice symmetry; Characterization of powder and thin films; Line shape analysis.
• Real space map: Near Field- AFM, STM
  Far field: SEM, TEM.

Thermal Methods

• Thermogravimetric methods; Differential thermal analysis; Differential Scanning Calorimetry.

Statistical Analysis- Evaluating Data

• Confidence limits, statistical aids to hypothesis testing; Detecting gross errors; Least square methods.

Suggested Texts:

1. Principles of Instrumental Analysis by Douglas A. Skoog, F. James Holler, Timothy A. Nieman; Saunders Golden Sunburst Series.
2. Modern Analytical Chemistry by David T Harvey; McGraw-Hill Science.
3. Spectrometric Identification of Organic Compounds by R. M. Silverstein, F. X. Webster; John Wiley and sons
4. Analytical Chemistry by Gary D. Christian; John Wiley and sons
5. 200 and More NMR Experiments: A Practical Course by Stefan Berger, Siegmar Braun; Wiley

3. PS 613C Computational Chemistry and its Applications (3 credits)

This course is intended for the incoming PhD students to provide the basic knowledge about computational chemistry methods and its use in connection to the experimental research. The aim of this course is to provide students with basic background on computational methods and molecular modeling, including some hands-on experiences to get started in modeling the physicochemical properties of molecules. The basic theoretical background will be provided in this course, and the emphasis will be given on hand-on application of the computational methods to model molecular properties. The topics to be covered will be selected from the list presented below depending on the availability of resources and time.

Course Outline:

Overview of Basic Quantum Chemistry:

• Introduction; Operators; Eigenvalues; Time-dependent Schrödinger equation; Particle in a box; Variation and perturbation methods; Born-Oppenheimer approximation.

Molecular Mechanics:

• Fundamentals; Potential energy functions; Force fields; Electrostatic interactions; van der Waals interactions; Geometry optimization; Energy minimization methods.

Simulation Methods:

• An introduction; Overview of Molecular Dynamics and Monte Carlo simulations; Introduction to application of simulation methods to explore bio-macromolecules; Brief on conformational analysis; Basic hands-on experiences on simulating bio-molecules (like protein, DNA, lipids) using academic and/or commercially available simulation packages (depending on availability of resources).

Overview of Molecular Orbital (MO) Theory:

• Trial wave functions; Hückel MO theory; Hartree-Fock (HF) theory; MO-LCAO formalism.

Semiempirical Implementation of MO Theory:

• Extended Hückel theory; CNDO, INDO and NDDO formalisms; Ongoing developments; Computation of electronic structures; Hands-on experience in running semiempirical jobs and extracting various molecular properties using academic and/or commercially available quantum chemical packages (depending on availability of resources).

Ab Initio Hartree-Fock Theory:

• Review of HF equation and variational principle; Basis sets; Practical issues; Electron correlation techniques; Configuration Integration; Geometry optimization; Electric dipole moment; Electric polarizability and Hyperpolarizability; Hands-on experience in running jobs to explore electronic structure of molecules using quantum chemical packages (depending on availability of resources).

Overview of Density Functional Theory (DFT):

• Introduction; Hohenberg-Kohn theorem; Kohn-Sham theory; Exchange-correlation functionals; Advantages and disadvantages of DFT compared to MO theory; Computing electronic structure; Hands-on experience in running DFT jobs using quantum chemical packages (depending on availability of resources); Comparison of performances of various methods.

Brief Overview of Condensed-phase Calculations:

• Condensed-phase effects in general; Continuum model and its implementation; Effect of continuum salvation on electronic spectra of molecules; Solvent models; Brief on Molecular Dynamics and Monte Carlo simulations; Introduction to hybrid QM/MM methods and its applications. 

Suggested Texts:

• Ira. N. Levine, Quantum Chemistry, 5th ed., Prentice Hall, NJ.

• Donald A. McQuarrie, Quantum Chemistry, University Science Books, Mill Valley, CA.

• Mark A. Ratner and George C. Schatz, Introduction to Quantum Mechanics in Chemistry, Prentice-Hall, NJ.

• Christopher J Cramer, Essentials of Computational Chemistry, 2nd edition, John Wiley & Sons Ltd., England.

• Frank Jensen, Introduction to Computational Chemistry, 2nd edition, John Wiley & Sons Ltd., England.

• Andrew R. Leach, Molecular Modelling: Principles and Applications, 2nd edition, Longman Group, United Kingdom.

• M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford.

• Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry, Introduction to Advanced Electronic Structure Theory, 1st ed., revised, Dover.

4. PS 614C  Advanced Spectroscopy and its Applications(3 credits)

            
This course is designed as a refresher course for incoming Ph.D. students with Physics or Chemistry background. It reviews some materials that may have been taught in the M.Sc. courses, and also includes material designed to familiarize students with some of the advanced spectroscopic techniques and their applications. This course will be beneficial also for students of Biophysics or Biochemistry who want to gain further knowledge about the application of some advanced spectroscopic techniques to study biological macromolecules. The topics to be covered will be selected from the list presented below.

Course Outline:

• Fundamentals of Photochemistry:

Overview
Laws of photochemistry; Interaction of radiation with matter; Transition between states; Fermi-Golden rule; Electronic transitions, dipole approximation and two-photon transitions.

Radiative Transitions – Absorption and Emission of Light  
Types of photophysical processes; Absorption of light; Frank-Condon principle; Emission spectra; Fluorescence and phosphorescence; Excited state dipole moments.

Non-radiative Transitions
Internal conversion; Intersystem crossing; Energy gap law; Isotope effect; Temperature effect.

Various Photophysical Processes
Fluorescence Resonance Energy Transfer (FRET); Förster theory; Overlap integral; Solvent effect on absorption and emission; Solvation energy; Lippert equation; Solvent relaxation dynamics; Fluorescence anisotropy; Fluorescence anisotropy decay; Excited state proton and electron transfer.

• Spectroscopic Techniques and Applications

Laser Fundamentals
Introduction to Lasers; Coherence; Population inversion; Laser cavity modes; Pulsed laser operations; Q-switching and mode-locking; Pulsed laser sources (Diode lasers, Ti:Sapphire lasers); Frequency multiplication of lasers and nonlinear optical effects.

Some Spectroscopic Techniques
Uv-vis spectrophotometer; Fluorescence spectrometer; Absorption, emission and excitation spectra; Time-resolved techniques; Time-correlated Single Photon Counting (TCSPC); Brief introduction to time-resolved fluorescence up-conversion technique; Microscopic techniques and imaging; Single molecule spectroscopy; Fluorescence correlation spectroscopy (FCS); Raman Spectroscopy.

Applications
Application of steady-state and time-resolved fluorescence techniques to study FRET, Solvation relaxation dynamics, Fluorescence anisotropy, Excited state charge transfer in molecular assemblies and biomacromolecules (e.g. DNA, protein, lipid etc.); Introduction to the application of fluorescence and Raman microscopic techniques to study microscopic and single molecular systems; Summery of applications of FCS; Brief introduction to the application of Coherent anti-Stokes Raman Spectroscopy (CARS) and microscopy; One- and multi-photon fluorescence imaging of biological systems.
            
Suggested Texts:

• Nicholas J. Turro, Modern Molecular Photochemistry, The Benjamin/Cummings Publishing Co., Inc.
• K. K. Rohatgi-Mukherjee, Fundamentals of Photochemistry, New Age International (P) Limited.
• J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd edition, Kluwer Academic, New York.
• Colin N. Banwell, Elaine M. McCash, Fundamentals of Molecular Spectroscopy, Tata McGraw-Hill.
• William T. Silvast, Laser Fundamentals, Cambridge University Press.
• D. V. O’Connor and D. Phillips, Time Correlated Single Photon Counting, Academic Press, New York.
• Robert M. Silverstein, Francis X. Websterand, David J. Kiemle, Spectrometric Identification of Organic Compounds.

5. PS 615C   Supramolecular Chemistry(3 credits)

This course is designed to introduce students to the interdisciplinary science of supramolecular chemistry. Thermodynamic and kinetic parameters involved in designing supramolecular systems would be taught in detailed in this course. The course would also give insight into the role of supramolecular chemistry of life and designing artificial mimics pertaining to nature.

Course Outline:

Principles of molecular recognition

• Quantification of non-covalent forces and medium effects; Host design; Preorganization; Enthalpy and entropic contributions; Cooperativity and allosteric effects; Induced fit; Complexation selectivity.

Supramolecular Chemistry of Life

• Alkali metal cations in biochemistry; Porphyrins and tetrapyrrole macrocycles; Plant photosynthesis; Uptake and transport of oxygen in Haemoglobin; Coenzyme B12; Neurotransmitters and Hormones, DNA; Biochemical self-assembly

Cation Binding Hosts

• Lariat ethers and podands, crown ethers, cryptands, calyx[n]arenes, cucurbit[n]urils, spherands; Selectivity of cation complexation; Macrocyclic, macrobicyclic and template effects

Anion Binding Hosts

• Concepts in anion host design; Guanidinium-based receptors; Organometallic receptors; Neutral receptors; Hydride sponge; Anticrowns; Biological Anion receptors

Binding of neutral molecules

• Binding by cavitands, cyclodextrins, cucurbit[n]urils, dendrimers, molecular clefts and tweezers, cyclophane  Hosts

Supramolecular reactivity and catalysis

• Catalysis by cation, anion and neutral receptors; Supramolecular metallocatalysis; Cocatalysis; Biomolecular and abiotic catalysis

Transport processes and carrier design

• Anion and cation carriers; Coupled transport processes; Electron coupled transport; Proton coupled transport; Light driven transport; Transport via transmembrane channels

Self Processes

• Programmed supramolecular systems; Self-assembly; Self-organization; Self-recognition; Self-replication systems

Suggested Texts:

• Supramolecular Chemistry by J. W. Steed and J. L. Atwood, John Wiley and Sons, Ltd.
• Supramolecular Chemistry-Concepts and Perspectives by Jean –Marie Lehn, VCH.
• Principles and Methods in Supramolecular Chemistry by Hans-Jorg Schnider and Anatoly K. Yatsimirsky, John Wiley and Sons, Ltd.
• Supramolecular Chemistry of anions by Antonio Bianchi, Kristin Bowman James and Enrique Garcia-Espana, Wiley-VCH.

6. PS 616C   Molecular Materials(3 credits)

This course is designed to introduce students to the basic concepts of chemical interactions and the principles and theories involved in the design of new-age applicative materials e.g., molecular sensors and switches, organic light emitting diodes, electrochromic materials etc. This course will also emphasize on how the change in molecular design and molecular interactions can tune or modulate the properties of these materials. Principles underlying the organic/inorganic synthesis and purification of materials through well-known named-reactions will also be covered in a lucid manner. 

This course will be inter-disciplinary in nature and would help understand the basic theoretical as well as practical aspects (design, synthesis and properties) of new materials.

[Introductory classes would be planned for physics students]

Course Outline:

Nature of Chemical Interactions:

• Covalent bond Vs ion-ion, ion-dipole, dipole-dipole, H-bonding (weak, moderate and strong), cation-p, anion-p, p-p interactions, van der Waals forces, hydrophobic effects. Examples and comparison of energy parameters.

How can we design giant structures using weak interactions? Self-assembling systems:

• Carboxylic acid dimers, alcohol-amine, amides etc. Designing molecular squares and boxes, giant self-assembling capsules, molecular tennis-balls, rosettes, self-assembly of metal arrays etc.

Soft Materials

Micelles, Vesicles:

• Amphiphiles, thermodynamic principles of self-association, DLVO (Derjaguin, Landau, Verwey, Overbeek) Theory. Factors affecting changes from one structure to another.

Liquid crystals

• Designing principles and classifications. Thermotropic and lyotropic liquid crystals. Nematic, smectic, cholesteric and discotic phases. Liquid crystalline polymers, Metallo-organic liquid crystals. Application towards liquid crystal displays.

Organogels, Hydrogels

• Designing principles and effects of H-bonding, p-p stacking, van der Waals interactions. Molecular visualization and characterization by SEM, TEM, AFM and XRD techniques. Application towards drug delivery and controlled release.

Glasses

• Design, properties and characterization.

Molecular devices

Molecular sensors and switches:

• Design and principles of photochemical sensors, PET (Photo induced electron transfer) systems, ON-OFF molecular switches, Molecular logic gates (AND, NOT, OR, NAND etc.); Electrochemical sensors.

Organic Light Emitting Diodes (OLED’s):

• Definition of electroluminescence and electroluminescence quantum effieciency, power efficiency. Design and characterization, examples of OLEDS: conjugated oligomers and polymers, low molecular weight materials. Application, commercialization and optimization of OLED’s.

Electrochromic materials:

• Definition and designing principles, conducting polymers, metallo polymers, metallophthalocyanines, visible and infrared electrochromism. Applications to practical materials.

Nonlnear Optical (NLO) materials:

• Definition of First-order and Second-order hyperpolarizabilities, Experimental techniques (Electric field induced second harmonic generation, Hyper Rayleigh Scattering) to determine b and c. Molecular designing principles and characterization of dipolar, multipolar and octupolar molecules. Imparting tunability in NLO molecules. Designing SHG active molecules (Influence of chirality, H-bonding, steric effects).

How to synthesize molecules?

• A brief idea on the principles involved in organic synthesis, purification and characterization. Importance of reactions like Sonogashira, Suzuki and Heck coupling and their applications to synthesize materials.

Suggested Texts:

1. The design of organic solids by G. R. Desiraju, Elsevier: Amsterdam.
2. The weak hydrogen bond in structural chemistry and biology by Gautam Desiraju and Thomas Steiner, Oxford University Press.
3. Supramolecular Chemistry by J. W. Steed and J. L. Atwood, John Wiley and Sons Ltd.
4. Molecular Fluorescence: Principles and Applications by B. Valeur, Wiley-VCH.
5. Organic electroluminescent materials and devices, by S. Miyata and H. S. Nalwa, Gordon and Breach Publishers, Amsterdam.
6. Non Linear Optical properties of organic molecules and crystals, by D. S. Chemla and J. Zyss, Academic Press, Inc.

Core Courses

PS604: Research and Publication Ethics (2 credits)

Introduction (1 hour)

• Meaning, objective and usefulness of research. Falsifiability, impact and consequences.

Scientific Ethics (10 hours)

• Ethics as a branch of philosophy, etymology of the word, ethics in society, individual and collective.

• Ethical scientific conduct, integrity and honesty, different roles (student, teacher, advisor, science administrator etc.).

• Ethical research practice, broader perspective and demands of different disciplines (maintenance and accessibility of raw data, computer codes etc.).

• Malpractices: fabrication, manipulation, cherry-picking, selective reporting etc.

• Conflict of interest. Confidentiality. Discrimination.

• Sustainability, sustainable developments goals (SDGs), sustainable consumption and production. Policies. Environmental impact (safety, waste disposal, hazard etc.)

Publication Ethics (10 hours)

• Scientific communication, reporting novel results. Types: term papers, project reports, short and long papers, review articles, science popularisation, research proposal etc.

• Author order, roles of contributors, variations among disciplines.

• Attributions and citations.

• Peer review, preprints, arXiv and open access publication.

• How to find an appropriate journal for publication?

• Predatory journals and publishers.

• Redundant and duplicate publications.

• Plagiarism: definition, types of plagiarism (including self-plagiarism), plagiarism detection software. Acceptable practice. Inadvertent mistakes, erratum, retraction.

• Consequences of scientific misconduct.

Database and Research Metrics: (3 hours)

• Indexing databases.

• Citation databases (Web of science, scopus, inspirehep etc.).

• Research metrics: impact factor, h-index, i10-index, g-index, altmetrics etc.

Suggested references

1. S. Holm and E. Stokes, Precautionary Principle, in Encyclopedia of Applied Ethics, (Ed.: R. Chadwick), 2nd edition, Elsevier, London, 2012

2. A. Briggle, Scientific responsibility and misconduct, in Encyclopedia of Applied Ethics, (Ed.: R. Chadwick), 2nd edition, Elsevier, London, 2012

3. C. Blackmore, Sustainability, in Encyclopedia of Applied Ethics, (Ed.: R. Chadwick), 2nd edition, Elsevier, London, 2012

4. H. Jonas, The Imperative of Responsibility: In Search of an Ethics for the Technological Age, University of Chicago Press, Chicago, USA 1984

5. K Muralidhar, Ethics in science, education, research and governance, INSA publication, 2019, https://insaindia.res.in/ebooks/

6. The United Nations Conference on Environment and Development held in Rio de Janeiro in 1992 (Rio 92). Agenda 21: A blueprint for a global partnership for sustainable development in the 21^st century

7. F. Moser and F. Dondi, On the road to Rio + 20: The role of international environmental governance and of the evolution of environmental ethics for a safer world, Toxicol. Environ. Chem. 2012, 94, 807–813

8. K. Sharma, Two decades of scientific misconduct in India: Retraction reasons and journal quality among inter-country and intra-country institutional collaboration, Scientometrics 129, 7735–7757 (2024), https://doi.org/10.1007/s11192-024-05192-z, https://arxiv.org/abs/2404.15306

9. B. Bhatt, A multi-perspective analysis of retractions in life sciences. Scientometrics 126, 4039–4054 (2021), https://doi.org/10.1007/s11192-021-03907-0

Additional Resources:

1. Institutional Ethics Review Board, JNU New Delhi, https://www.jnu.ac.in/ierb

2. Scientific values: ethical guidelines and procedures, Indian Academy of Sciences,

https://www.ias.ac.in/About_IASc/Scientific_Values:_Ethical_Guidelines_And_Procedures/

3. IISER Pune Guidelines on Scientific Ethics, https://www.iiserpune.ac.in/policies-and-guidelines/guidelines-on-academic-ethics

4. Plagiarism, Oxford University, https://www.ox.ac.uk/students/academic/guidance/skills/plagiarism

5. Types of plagiarism, iThenticate, https://www.ithenticate.com/resources/infographics/types-ofplagiarism-research

6. https://retractionwatch.com

7. Wikipedia page on Scientific Plagiarism in India,

https://en.wikipedia.org/wiki/Scientific_plagiarism_in_India

8. Academic ethics: What we must do, Sunil Mukhi at ICTS, Bengaluru (2017),

https://www.youtube.com/watch?v=zdfUWFNWQaw

9. Scientific writing and ethics in research, IPA lecture 2022, Sunil Mukhi.

https://www.youtube.com/watch?v=L-pwV6uQHAY

PS641M: Algebra (3 credits)

• Groups:  Nilpotent and solvable groups, Sylow’s theorems, free groups.
• Representation theory of finite groups, Peter-Weyl theorem.
• Rings and Modules: Commutative rings, Noetherian and Artinian rings and modules, principal ideal domains (PID), unique factorization domain, modules over PID, tensor products.
• Field Theory:  Algebraic and transcendental extensions, introduction to Galois theory.

Suggested Texts:

• M. Artin.  Algebra.
• I.N. Herstein.  Topics in Algebra.
• S. Lang. Algebra.

PS642M: Analysis (3 credits)

Prerequisites: Real Analysis, Basic Lebesgue Measure Theory, Basic Topology and Basic Functional Analysis.

Some topics from metric spaces (10 hours)

Compactness, Separability, sequence and series of functions, equicontinuous families, Ascoli-Arzela Theorem, Stone-Weierstrass Theorem. 

Functions of several variables (10 hours)

Operator norm on L(ℝⁿ, ℝ^m), differentiability of maps between ℝⁿ and ℝ^m, total derivative, partial derivatives, Jacobian, Inverse Function Theorem, Implicit Function Theorem, Rank Theorem. 

General Measure Theory (12 hours) 

Measurable sets, functions and spaces: sigma algebras, measurable functions, simple functions, measures, extended real line and its topology and arithmetic.

Lebesgue’s integration theory: Lebesgue’s monotone and dominated convergence theorems and some consequences, , completion of a measure space, product of measures, Fubini and Tonelli Theorems

L^p spaces (1 ≤p ≤∞): Holder’s and Minkowski’s inequalities, Completeness, Denseness of simple functions.

And, at least any one of the following three units:

Geometry of Hilbert spaces (12 hours)

Inner product spaces, examples of separable and non-separable Hilbert spaces, Completion, Best approximation in Hilbert spaces, Orthogonal projections, Riesz-Representation Theorem, unconditional sums (via nets), Bessel’s inequality, Orthonormal bases, (Hilbert) dimension, Direct sums of Hilbert spaces. 

Riesz-Markov Theorem (12 hours)

Complex Measures, Lebesgue-Radon-Nikodym Theorem, Riesz-Markov Theorem (positive version), Duals of L^p spaces, Riesz-Markov Theorem (complex version).

Operators on Hilbert spaces (12 hours) 

Operator norm, Adjoint of an operator, normal operators, compact operators, diagonalization of compact self-adjoint operators.

Suggested References:

1. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976

2. W. Rudin, Real and Complex Analysis, W. Rudin, McGraw-Hill, 1987.

3. S. Kantorovitz, Introduction to Modern Analysis, Oxford Graduate Texts in Mathematics, Oxford University Press, 2003.

4. J. B. Conway, A course in Functional Analysis, GTM 96, Springer, 2000.

5. G. B. Folland, Real Analysis: Modern Techniques and Their Applications, J. Wiley and Sons, 1999.

6. H. L. Royden and P. M. Fitzpatrick, Real Analysis, Prentice Hall, 2010.

PS643M: Topology (3 credits)

• General Topology:  Introduction, metric topology, separation axioms, compactness, Connectedness, product topology, introduction to manifolds, submanifolds.
• Homotopy Theory.  Covering spaces, homotopy maps, homotopy equivalence,Contractible spaces, deformation retraction.
• Fundamental Groups: Universal cover and lifting problem for covering maps, Fundamental groups of S1 and Sn.
• Introduction to Homology Theory.

Suggested texts:

• C.O. Christenson and W.L. Voxman.  Aspects of Topology.
• J.R. Munkres.  General Topology.
• I.M. Singer and J.A. Thorpe.  Lecture Notes in Elementary Topology and Geometry.

Optional Courses

PS644M: Topological Groups and Lie Groups (2 credits)

• Topological Groups:  Introduction, integration on locally compact spaces, Haar Measure, Character groups, group action.
• Lie groups and lie algebras:  Basic theory, linear groups.

Suggested texts:

• K. Chandrasekharan.  A Course on Topological Groups.
• W. Fulton and J. Harris.  Representation Theory.
• F.W. Warner.  Foundations of Differentiable Manifolds and Lie Groups.

PS645M: Functional Analysis and Operator Theory (2 credits)

• Functional Analysis
  Topological vector spaces: Separation properties, Linear Mappings, Finite dimensional spaces, Metrizability, Seminorms and local convexity.
  Completeness: Baire Category Theorem, Banach-Steinhauss Theorem, Open Mapping Theorem, Closed Graph Theorem.
  Convexity: Hahn-Banach Theorems, Weak topologies, Banach-Alaouglu Theorem, Exteme points, Krein-Milman Theorem.
  Some Applications: Stone-Weirstrass Theorem, Kakutani-Markov fixed point Theorem and Haar measure for compact groups
• Operator Theory
  Compact operators on Banach spaces.
  Bounded operators on a Hilbert space: Riesz Representation Theorem, Bounded operators, Adjoints, Normal Operators, Unitary Operators.
  Spectral Theorem: Spectrum of an operator, Resolution of Identity, Spectral Theorem, Functional Calculus of normal operators, Spectral theorem for compact normal operators.

Suggested texts:

• W. Rudin, Functional Analysis, ISPAM, McGraw-Hill, 2006.
• J. B. Conway, A Course in Functional Analysis, GTM, Springer, 1990.
• R. J. Zimmer, Essential Results of Functional Analysis, Chicago Lecture in Mathematics, 1990.
• G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2004.

PS646M, 647M: Research Methodology Course I & Research Course II (3 credits each)

The research courses are advanced courses designed to prepare students to work in a specific area.  The details of these courses are usually decided by the instructors.

PS648M Queueing Theory (3 credits)

Pre-requisites: Basics of probability theory

Course outline:

1. Quick review of probability, Stochastic processes: Definition and examples; Classification of stochastic processes with illustrations. 

2. Poisson Process:Inter-arrival and waiting time distributions, conditional distributions of the arrival times, nonhomogeneous Poisson process, compound Poisson random variables and Poisson processes, conditional Poisson processes. 

3. Markov chains: Introduction and examples, Chapman-Kolmogorov equations and classification of states, limit theorems, transitions among classes, mean time in transient states, branching processes, applications of Markov chain. 

4. Markov process, Classification of Markov process, Discrete-parameter Markov chains, Continuous-parameter Markov chains, Imbedded Markov chains, Long-run behavior of Markov process.

5. Characteristics of queueing processes, System performance measures, Little's law, Birth-death processes, M/M/1, M/M/c, M/M/1/K, M/M/c/K queueing systems, State-dependent service systems, Finite source queues, Queue with impatience. 

6. Bulk input queues (M^[X]/M/1), Bulk service queues (M/M^[Y]/1), Erlangian models, Preemptive and non-preemptive, priority, and retrial queues.

Additional Contents:

Pollaczek-Khinchin formula, M/G/1 queues: waiting times and busy period, G/M/1 queues, Vacation queues: Introduction, M/M/1 queues with vacations, N-policy and F-policy queueing models.

Suggested Texts:

1. Trivedi, K.S.: Probability and Statistics with Reliability, Queueing and Computer Science Applications, 2nd Edition, John Wiley Sons, 2002.

2. Gross, D., Shortle, J.F, Thompson, J.M and Harris, C.M.: Fundamentals of Queueing Theory", Wiley Student 4th Edition, 2014.

3. William J. Stewart: Probability, Markov Chains, Queues, and Simulation, Princeton University Press, 6 Oxford Street, New Jersey 08540, 2009.

4. Sheldon M. Ross: Stochastic Processes, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, 2^nd Edition, 1996.

5. Brzezniak, Z. and Zastawniak, T.: Basic Stochastic Processes: A Course through Exercises, Springer 1992. 

6. Medhi, J.: Stochastic Processes, New Age Science 2009.

7. Chakravarthy, S.R.: Introduction to Matrix-Analytic Methods in Queues 1, Analytical and Simulation Approach – Basics, John Wiley & Sons, Inc., USA, 2022.

PS649M Advanced Topics in Fluid Mechanics (3 credits)

Pre-requisites: Calculus, Basic Analysis, Differential Equations.

Course outline:

1. Fundamental and Governing Equations for Fluid Motion: Basic concept of fluids

and it’s physical properties, general theory of stream function, complex-potential,

stress, rate of strain, source, sink, and doublets, equation of conservation of mass

(continuity equation), equation of conservation of momentum, Euler's equation of

motion, Bernoulli’s equation, Navier Stokes equation, equation of moments of

momentum, equation of energy. (10 Hrs.)

2. Boundary Layer Theory: Concept of the boundary layer, velocity and thermal

boundary layer, derivation of boundary layer equations for two and three-dimensional

incompressible flows, non-dimensionalization of two and three-dimensional

incompressible flow models. Complex fluids: Non-Newtonian theory, Viscoelastic

fluids, Introduction to turbulent flow. (10 Hrs.)

3. Convection in Porous Media: Darcy flow model and Forchheimer modification,

Brinkman’s Equation, Oberbeck-Boussinesq Approximation, the first law of

thermodynamics, the second law of thermodynamics, natural and forced convection

boundary layers enclosed porous media heated from the side, enclosed porous media

heated from below. (10 Hrs.)

4. Hydrodynamics Stability Analysis:

Mechanism of instability, fundamental concepts of hydrodynamics stabilities,

Thermal instability analysis (or Benard Problems), Techniques of stability analysis:

Normal mode analysis method, Perturbation method, Energy method. Thermal instability of a layer of fluid heated from below. (12 Hrs.)

Additional Topics:

5. Computational methods to analyze flow models:

The relaxation technique and its use with low-speed inviscid flow, the pressure

correction technique, SIMPLE Algorithm, Staggered grid, Second order Upwind

schemes, Thomas algorithm for the tridiagonal system, Homotopy analysis method,

Adomian decomposition method, Shooting Method, Chebyshev spectral quasilinearization

method, bvp4c and bvp5c techniques. (16 Hrs.)

Main Textbooks:

1. F. Chorlton, Textbook of Fluid Dynamics, CBS Publishers & Distributors, 2004.

2. C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid

Dynamics, Springer Publishers, 2006.

3. Nield and Bejan, Convection in Porous Media, 4th Edition, Springer Science, 2013.

Supplementary References:

1. H. Schlichting and K. Gersten, Boundary Layer Theory, Springer publisher, 2000.

2. Adrian Bejan, Convection Heat Transfer, Wiley-India, 2012.

3. T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective Heat

Transfer, Springer-Verlag, 1988.

4. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, Inc. New

York, 1961.

5. P. G. Drazin and W. H. Reid, Hydrodynamics Stability, 2nd Edition, Cambridge University

Press, 2004.

6. C. A. J. Fletcher, Computational Techniques for Fluid Dynamics: Fundamental and General

Techniques: 001, Springer Verlag, 2nd ed. 1998.

7. J. D. Anderson, Computational Fluid Dynamics, McGraw-Hill, 2017.

8. T. J. Chung, Computational Fluid Dynamics, Cambridge University Press, 2002.

PS712M: Algebraic Number Theory (3 credits)

• Number fields, number rings and their structure as Dedekind domains.
• Factorization of prime ideals, quadratic and cyclotomic extensions.
• Decomposition group, inertia group.
• Group of units, ideal class group, theorems of Dedekind and Minkowski.
• Introduction to zeta function, Dirichlet character.

Suggested texts:

• D. Marcus.   Number Fields
• Borevich and Shafarevich.   Number Theory
• Esmonde and Murty.   Problems in Algebraic Number theory
• Frohlich and Taylor.   Algebraic Number Theory

Hasse.   Number Theory