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Harish-Chandra Research Institute


Research Project topics

A student can take up a research project under any one of the following broad areas. This may involve lectures in the second year, or a reading course with discussion, as decided by the thesis advisor 

Introduction to number theory: 


Divisibility, congruence, the Fundamental Theorem of Arithmetic, the Chinese Remainder Theorem. Elementary proofs of the infinitude of primes in certain arithmetic progressions, the quadratic reciprocity law.
The Bertrand postulate, the Euler and Abel summation formulas and applications, preparation for the Prime Number Theorem Combinatorial sieves including the Brun sieve and the Turan sieve, and their applications.Rational approximations, the Dirichlet Theorem, the Liouviile Theorem, Siegel’s Lemma, and their applications, Gaussian integers, sums of squares.

Textbooks:
1. D. M. Burton, Elementary number theory, Second edition, W. C. Brown Publishers, Dubuque, IA, 1989.
2. K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York Inc., New York, 1968.
3. A. C. Cojocaru, and M. R. Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts, 66, Cambridge University Press, Cambridge, 2006.
4.. G. H. Hardy, and E. M. Wright, An introduction to the theory of numbers, Sixth revised edition, Oxford University Press, Oxford, 2008.

 

Algebraic number theory :

p-adic numbers, p-adic valuations, absolute values, completions, local fields, henselian fields, extensions of valuations, ramification, higher ramification groups.

Galois extensions, projective and inductive limits, abstract class field theory, the Herbrand quotient.

The local reciprocity law, the norm residue symbol, formal groups, cyclotomic extensions, compatibility with the ramification filtration.

Ideles and idele classes, ideles in extensions, global class field theory, power reciprocity laws.

Textbooks:


1. A. Fröhlich, and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, 27, Cambridge University Press, Cambridge, 1993.

2. H. Hasse, Number theory, Translated from the third (1969) German edition, Reprint of the 1980 English edition, Classics in Mathematics, Springer-Verlag, Berlin, 2002.

3. J. Neukirch, Algebraic number theory, Translated from the 1992 German original, Grundlehren der Mathematischen Wissenschaften, 322. Springer-Verlag, Berlin, 1999

4. A. Weil, Basic number theory, Reprint of the second (1973) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. 

 

Analytic number theory: 


Arithmetical functions, Euler, Abel Summation formula and applications to summatory functions. The Riemann Zeta function, and Dirichlet L-functions

The Prime Number Theorem, and Dirichlet’s Prime Number Theorem

Sieve methods (Brun, Selberg, and Large), and their applications

Fourier techniques and applications in number theory.

Textbooks:
1. K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren dermathematischen Wissenschaften, Band 148, Springer-Verlag New York Inc., New York, 1968
2. H. Iwaniec, and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.
3. H. Iwaniec, and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.

 

Commutative algebra:

Modules and tensor products, prime ideals, the Zariski topology, rings and modules of fractions, flatness, valuation theory, integral extensions, discrete valuation rings, Dedekind domains, Artinian and Noetherian rings and modules, the Hilbert basis theorem, primary decomposition, Noether normalization, Hilbert’s Nullstellensatz, completions, the Krull dimension

Textbooks:
1. S. Bosch, Algebraic geometry and commutative algebra, Universitext, Springer, London, 2013.
2. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.

 

Algebraic varieties:

Spaces with sheaves of functions, affine algebraic varieties over an algebraically closed field, the category of algebraic varieties, subvarieties, products, projective varieties, separation, normality, dimension, rational maps, tangent spaces, smoothness, completeness, finite morphisms, constructible sets, divisors, curves, and the Riemann-Roch theorem.

1. G. R. Kempf, Algebraic varieties, London Mathematical Society Lecture Note Series, 172, Cambridge University Press, Cambridge, 19
2. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
3. J. S. Milne, Algebraic geometry, http://www.jmilne.org/math/ .

 

Introduction to Elliptic curves:


Elliptic curve and rational points, group law, elliptic curves over a finite field, elliptic curves over complex numbers, elliptic curves over local and global fields, Mordel-Weil theorem, Selmer group, and Tate-Shafarevich group.

Textbook:

The Arithmetic of Elliptic Curves, Silverman, Joseph H.

 

p-adic Numbers, p-adic Analysis, and Zeta-Functions:

p-adic numbers, p-adic interpolation of Riemann's zeta function, p-adic power series, rationality.

Textbook:

p-adic Numbers, p-adic Analysis, and Zeta-Functions, Koblitz, Neal.

 

Introduction to Iwasawa theory:

Cyclotomic fields, local units, Iwasawa algebras, and p-adic measures, cyclotomic unites, Euler system, Main conjecture.

Textbook:

1. Cyclotomic Fields and Zeta Values, Coates, John, Sujatha, R.

2. Introduction to Cyclotomic Fields, Washington, Lawrence C.

 

Function Field Arithmetic:

Number fields and Function fields, Drinfeld modules, Explicit class field theory of Drinfeld modules, Gamma functions, Zeta functions.

Textbook:
1. Function Field Arithmetic, Dinesh S Thakur.

 

Introduction to p-adic Galois representations:


Absolute Galois groups of non-archimedean local fields, classification of p-adic Galois representations in terms of certain objects from semilinear algebra, the so-called étale \varphi - and (\varphi, \Gamma)-modules.

Textbooks:
1. Theory of p-adic Galois representations, preprint, J.-M. Fontaine, Y. Ouyang.
2. An introduction to the theory of p-adic representations, Geometric aspects of Dwork theory. Vol. I, 255-292, Walter de Gruyter, 2004, L. Berger.

 

Local Fields :


Discrete valuation rings and Dedekind domains, completions, discriminant and different, ramification groups, the norm, Artin representation, group cohomology, Galois cohomology, class formations, Brauer groups, local class field theory.

Textbooks:

1. J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, 3, Cambridge University Press, Cambridge, 1986.

2. I. B. Fesenko, and S. V. Vostokov, Local fields and their extensions, Second edition, Translations of Mathematical Monographs, 121, American Mathematical Society, Providence, RI, 2002.

3. K. Iwasawa, Local class field theory, Oxford Science Publications, Oxford Mathe- matical Monographs, The Clarendon Press, Oxford University Press, New York, 1986.

4. J. P. Serre, Local fields. Translated from the French original, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.

 


Representations of finite groups:


Representations, irreducible and indecomposable representations, class functions, orthogonality relations for characters, character tables, Schur’s lemma, unitary representations, duals and tensor products of representations, regular representations, canonical decompositions, examples, induced representations, Mackey’s criterion, Frobenius reciprocity, group algebras, Maschke’s theorem, applications of the representation theory of finite groups. Artin’s theorem, Brauer’s theorem and applications.

Textbooks:
1. W. Fulton, and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.
2. J-P. Serre, Linear representations of nite groups, Translated from the second French edition, Graduate Texts in Mathematics, 42, Springer-Verlag, New York- Heidelberg, 1977.

 

Introduction to Riemannian manifolds:

Riemannian metrics: Definition, Methods for constructing Riemannian metrics, Lengths and Distances.

Model Riemannian manifolds: Euclidean spaces, Spheres, Hyperbolic spaces, Invariant metrics on Lie groups, Other homogeneous Riemannian manifolds, Symmetric spaces

Connections, The Levi-Civita Connections: The problem of differentiating vector fields, Connections, The Levi-Civita connections on abstract manifolds, Geodesics, Parallel transport, Pullback connections, The Exponential map, Geodesics of model spaces.

Geodesics: Geodesics and the minimizing curves, Uniformly normal neighbourhoods, Completeness, Distance function, Convex neighbourhoods.

Curvature: The curvature tensor, Flat manifolds, Symmetries of the curvature tensor, Jacobi fields: The Jacobi equation, Conjugate points, Cut points, The second variation formula.

Comparison theory: Jacobi fields, Hessians and Riccatti equations, comparisons based on sectional and ricci curvatures.

Curvature and Topology: Manifolds of constant curvature, Manifolds of nonpositive curvature, Manifolds of nonpositive curvature, Maniofolds of positive curvature.

Textbooks:
1. John M. Lee, Introduction to Riemannian manifolds, Second edition
2. Manfredo P. do Carmo, Riemannian Geometry
Peter Petersen, Reimannian Geometry, Third Edition

 

Lie algebras:


Linear Lie algebras, Lie algebras of derivations, homomorphisms of Lie algebras, representations of Lie algebras, solvable and nilpotent Lie algebras, Engel’s Theorem, Lie’s Theorem, the Jordan-Chevalley decomposition.
Cartan’s criterion, the Killing form, criterion for semi-simplicity, inner derivations, the abstract
Jordan decomposition, modules, the Casimir element, Weyl’s Theorem, preservation of Jordan decomposition, Representations of sl(2, F ), maximal toral subalgebras, and roots Orthogonality, integrality and rationality properties of roots, root systems, bases, Weyl chambers, the Weyl group, irreducible root systems, the Cartan matrix. Coxeter graphs, Dynkin diagrams, the Classification Theorem, construction of root systems & automorphisms, the theory of weights, dominant weights, the Isomorphism Theorem, Cartan subalgebras, the universal enveloping algebra, the Poincare-Birkhoff- Witt Theorem.

Textbooks:

1. R. W. Carter, Lie algebras of nite and affine type, Cambridge Studies in Advanced Mathematics, 96, Cambridge University Press, Cambridge, 2005.
2. J. E. Humphreys, Introduction to Lie algebras and representation theory, Second printing,
revised, Graduate Texts in Mathematics, 9, Springer-Verlag, New York- Berlin, 1978.
3.A. W. Knapp, Lie groups beyond an introduction, Second edition, Progress in Mathematics,
140, Birkhauser Boston, Inc., Boston, MA, 2002.

 

Fourier analysis:


Lp spaces, basic inequalities including Holder’s, Chebyshev, and Minkowski, inequalities for integrals, weak Lp spaces, the Riesz-Thorin and Marcinkiewicz Interpolation Theorems

Convolution, Young’s inequality, the generalized inequality of Young, approximations to the identity, the Fourier transform, Hausdorf-Young inequality, the Riemann Lebesgue Lemma the Fourier Inversion Theorem, the Plancherel Theorem, summability theorems, including those of Cesaro and Fejer, Fourier inversion on the torus, the Riemann localisation principle

Distributions, Sobolev spaces, Sobolev Embedding Theorem, Rellich’s theorem, applications to basic linear PDE

Textbooks:


1. G. B. Folland, Real analysis. Modern techniques and their applications, Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.


2. W. Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

 

Harmonic analysis:


Maximal Function, the Riesz-Thorin Interpolation Theorem. Singular integrals, the Calderon-Zygmund Decomposition, Singular integral operators which commute with dilations, vector-valued analogues, Riesz transforms, Poisson integrals, approximate identities, spherical harmonics
The Little wood-Paley g-function, multipliers, dyadic decomposition, the Hormander- Mihilin Multiplier Theorem, the Marcinkiewicz Multiplier Theorem.

Textbooks:


1. J. Duoandikoetxea, Fourier analysis, Translated and revised from the 1995 Spanish original,
Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI,
2001.
2. E. M. Stein, Singular integrals and differentiability properties of functions, Prince- ton
Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.

 


Partial Differential Equation:


Brief Introduction: A partial differential equation (PDE) is a type of differential equation that involves an unknown function of multiple variables and its partial derivatives. PDEs are widely used to model and analyze various physical phenomena and processes in fields such as physics, engineering, finance, and biology. They provide a mathematical framework for describing dynamic systems involving multiple variables and their ineractions.
The classification and properties of PDEs depend on various factors, such as their order, linearity, and the nature of the function being solved. Some common types of PDEs include the heat equation, wave equation, Laplace's equation, and the Schroedinger equation. Each of these equations captures specific behaviors and characteristics of the systems they represent.

Prerequisites: Real and Complex analysis, Measure theory, Functional Analysis, Ordinary differential equation
Topics to be covered:
Characteristics method for first order PDE, Classification of 2nd order PDEs, fundamental solutions and Green functions for Laplace, heat and wave equations, explicit solution formulas, harmonic functions, mean value property, maximum principles, uniqueness of solutions, method of separation of variables, similarity methods, transform methods, power series method, Cauchy-Kowalewsky theorem, Holmgren's uniqueness theorem.
Distribution Theory, Sobolev Spaces, Calculus of variation, Weak solutions, Lax-Milgram Lemma, Existence and regularity of solutions, Stability estimate.

References:
1. Evans, Lawrence C., Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. xviii+662 pp. ISBN: 0-8218-0772-2
2. John, Fritz, Partial differential equations, Fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, new York, 1982. x+249 pp. ISBN: 0-387-90609-6
3. Kesavan, S., Topics in Functional analysis and applications. John Wiley & Sons, Inc., New York, 1989. xii+267 pp. ISBN: 0-470-21050-8

 


Introduction to the Mathematical Fluid Mechanics


Brief Introduction : Mathematical fluid mechanics deals with the study of fluid flow using mathematical models and techniques. It provides a theoretical framework to understand and analyze the behavior of fluids, such as liquids and gases, in various physical systems. This involves PDEs (for example Navier-Stokes equation) which are usually derived from the principles of conservation of mass, momentum, and energy.
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to areas of active research in fluid dynamics.

Prerequisites: Real and Complex Analysis, Measure theory, Functional Analysis, Partial Differential Equation.
Topics to be covered:
Derivation of the governing equations: Navier-Stokes, Euler and some of its properties, Vorticity formulation, Low reynolds number flows, Stokes flow, Lubrication theory, Stokes Paradox, Boundary layers and asymptotic models, (non)-existence of solutions, Water waves, Instability analysis etc.
Depending on the specific interest, some of the advanced topics one may choose are the following: Theory of semigroup and its application in PDE, PDEs as gradient flows, Fluid-structure Interaction problem, Thin film flows, Models arising in geophysical fluid dynamics.

References:
1. Galdi, G. P. An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011. xiv+1018 pp. ISBN: 978-0-387-09619-3
2. Ockendon, H.; Ockendon, J. R., Viscous flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995. viii+113 pp. ISBN: 0-521-45244-9; 0-521-45881-1

 


Homogenization


Brief Introduction : Homogenization is a mathematical technique used to study the behavior of complex heterogeneous materials or structures in order to understand the effective or macroscopic behavior of a material or structure that consists of multiple microscopic components with different properties. It aims to derive mathematical models that capture the averaged or macroscopic behavior of such systems. This procedure typically involves two steps: the microscale analysis and teh macroscale analysis.

Prerequisites:Real and Complex Analysis, Measure theory, Functional Analysis, Partial Differential equation.
Topics to be covered In this course, we will learn about the microscale and the macroscale analysis from various point of view.
In the microscale analysis, the behavior of the individual microscopic components is analyzed using techniques such as asymptotic analysis, variational methods, or multiscale expansions. This step provides information about how the individual components interact with each other and with their surroundings. In the macroscale analysis, the informtaion obtained from the microscale analysis is used to derive effective equations that describe the behavior of the entire system on a larger scale.

Reference:
1. Tartar, Luc, The general theory of homogenization. A personalized introduction. lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. xxii+470 pp. ISBN: 978-3-642-05194-4
2. Allaire, Gregoire, Shape optimization by the homogenization method. Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002. xvi+456 pp. ISBN: 0-387-95298-5

 


Inverse Problem


Brief Introduction:Mathematical inverse problems refer to a class of problems where the goal is to determine the unknown causes or parameters of a system based on observed data or measurements. In other words, given the output or response of a system, inverse problems aim to reconstruct the input or the cause that generated the observed data.
The forward model describes how the system behaves and generated the output based on the input. the inverse model uses the information provided by the forward model and the observed data to estimate or infer the unknown quantities. Applications of inverse problems are widespreas. They include image reconstruction in medical imaging (e.g., computed tomography, magnetic resonance imaging), seismic imaging in geophysics, remote sensing, and many more. Inverse problems play a vital role in extracting valuable information from observed data and improving our understanding of complex systems.

Prerequisites: Real and Complex Analysis, measure theory, Functional Analysis, Partial Differential equation.
Topics to be covered:
Carleman estimate, Unique continuation, Introduction to Calderon problem for elliptic and parabolic equation, Scattering problem, Gelfand problem for the wave equation.

Reference:
1. Isakov, Victor, Inverse problems for partial differential equations. Applied Mathematical Sciences, 127. Springer-Verlag, New York, 1998. xii+284 pp. ISBN: 0-387-98256-6
2. Eskin, Gregory, Lectures on linear partial differential equations. Graduate Studies in mathematics, 123. American Mathematical Society, Providence, RI, 2011. xviii+410 pp. ISBN: 978-0-8218-5284-2
3. Katchalov, Alexander; Kurylev, Yaroslav; Lassas, Matti; Inverse boundary spectral problems. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123. Chapman & Hall/CRC, Boca Raton, FL, 2001. xx+290 pp. ISBN: 1-58488-005-8


Other elective courses: Apart from the list above, there may be elective courses in special Topics in Algebra/Analysis/Topology/ Algebraic Geometry/Differential Geometry based on the interest of students. If a faculty member opts to give such a course, the syllabus and text books will be announced well in advance.

 

 

 

 

 


Modified: 2024-03-20