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M.Sc. in Mathematics

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1. Complete 16 graduate courses of level 600 and above, totaling a minimum of 64 credits. Up to two of these courses may be replaced by level 500 courses, or by courses from other departments. All exceptions require the prior approval of the Graduate Advisor. 2. Maintain a CGPA of 5.0 (i.e. an average grade of C-). A student who does not have a CGPA of at least 5.0 at the start of the 3rd semester will be asked to leave the program
64

Total Credits

44

Core Credits

20

Major Electives

Core & Elective Courses

Core Courses

Course code
Title
Credit
MAT621
Analysis I
4

Core course for M.Sc. Mathematics

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 231/320 Real Analysis II (for undergraduates)

Overview: The aim of this course is to build a rigorous base for advanced topics such as Complex Analysis, Measure & Integration, Numerical Analysis, Functional Analysis, and Differential Equations. The course design also attempts to take into account the diverse backgrounds of our students.

Detailed Syllabus:

Topics under each section are divided in two parts. Part (a) contains topics that will be covered only briefly whereas topics in part (b) will be done in detail.

  1. Real number system
    • Archimedean property, density of rationals, extended real numbers, countable sets, uncountable sets.
    • Cauchy completeness of reals, Axiom of Choice, Zorn’s Lemma, equivalence of AC & ZL.
  2. Metric spaces
    • Definitions and examples, open sets, closed sets, limit points, closure,equivalent metrics, relative metric, product metric, convergence, continuity, connectedness, compactness.
    • Uniform continuity, completion of a metric space, Cantor’s intersection property, finite intersection property, totally bounded spaces, characterization of compact metric spaces.
  3. Sequences and Series of Functions
    • Pointwise and uniform convergence, uniform convergence and continuity, uniform convergence and integration, differentiation, Weierstrass M-test.
    • Power series, exponential and logarithmic functions, Fourier series, equicontinuous family of functions, Stone-Weierstrass approximation theorem, Arzela-Ascoli theorem.

References:

  1. Principles of Mathematical Analysis by Walter Rudin, Tata McGraw-Hill
  2. Mathematical Analysis by Tom M. Apostol, Narosa
  3. Topology of Metric Spaces by S. Kumaresan, Narosa
  4. Introduction to Topology & Modern Analysis by G. F. Simmons, Tata McGraw-Hill
  5. Real Analysis by N. L. Carothers, Cambridge University Press
MAT622
Topology
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 231/320 Real Analysis II for undergraduates, MAT 621 for graduate students.

Overview: This course concerns 'General Topology' which can be characterized as the  abstract framework in which the notion of continuity can be framed and studied. Thus topology provides the basic language and structure for a large part of pure and applied mathematics.

We will take up the following topics: Open and closed sets, continuous functions, subspaces, product and quotient topologies, connected and path connected spaces, compact and locally compact spaces, Baire category theorem, separability axioms.

Detailed Syllabus:

  1. Review: Operations with infinite collections of sets, axiom of choice, Zorn's lemma, real line, metric spaces.
  2. Topological Spaces: Definition and examples of topological spaces, Hausdorff property, fine and coarse topologies, subspace topology, closed sets, continuous functions, homeomorphisms, pasting lemma, product topology, quotient topology.
  3. Connectedness and Compactness: Connected spaces and subsets, path connectedness, compact spaces and subsets, tube lemma, Tychonoff theorem, local compactness, one-point compactification, Baire category theorem.
  4. Separation Axioms: First and second countability, separability, separation axioms (T1 etc.), normal spaces, Urysohn lemma, Tietze extension theorem.
  5. Topics for Student Presentations: Order topology, quotients of the square, locally (path) connected spaces, sequential and limit point compactness, topological groups, nets, applications of Baire category theorem.

Main Reference:

  1. Topology by James R. Munkres, 2nd Edition. Pearson Education, Indian Reprint, 2001.
  2. Basic Topology by M. A. Armstrong. Springer-Verlag, Indian Reprint, 2004.
  3. Topology by K. Jänich. Undergraduate Texts in Mathematics, Springer-Verlag, 1984.
  4. Introduction to Topology and Modern Analysis by G. F. Simmons. International Student Edition. McGraw-Hill, Singapore, 1963.
  5. Topology of Metric Spaces by S. Kumaresan. 2nd edition, Narosa, 2011.
MAT623
Analysis II
4

Core course for M.Sc. Mathematics

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 621 (Analysis I)

Overview: This course puts the concept of integration of a real function in its most appropriate setting. It is also a prerequisite for the study of general measures, which is the foundation for a large part of pure and applied mathematics – such as spectral theory, probability, stochastic differential equations, harmonic analysis, and partial differential equations.

Detailed Syllabus:

  1. Lebesgue Measure: Outer measure, measurable sets, Lebesgue measure, measurable functions, pointwise convergence, almost everywhere convergence.
  2. The Lebesgue Integral: Riemann integral, Lebesgue integral of a bounded measurable function over a set of finite measure, Lebesgue integral of a non-negative and a general measurable function.
  3. Differentiation and Integration: Differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolute continuity.
  4. The Classical Banach spaces: Lp spaces, Minkowski and Hölder inequalities, completeness of Lp spaces, bounded linear functions on Lp spaces.
  5. Introduction to General Topology: Open and closed sets, bases, separation properties, countability and separation, continuous maps, compactness, connectedness.

References:

  1. Real Analysis by H. L. Royden and P. Fitzpatrick. 4th edition, Prentice-Hall India, 2010.
  2. Measure Theory and Integration by G. de Barra, New Age International, reprint 2006.
  3. Real Analysis by N. L. Carothers, Cambridge University Press.
MAT624
Complex Analysis
4

Core course for M.Sc. Mathematics

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 621 (Analysis I). Undergraduates not allowed.

Overview: A graduate course of one variable complex analysis.

“The shortest path between two truths in the real domain passes through the complex domain” – Jacques Hadamard.

Detailed Syllabus:

  1. The complex number system: The field of Complex numbers, the complex plane, Polar representation and roots of complex numbers, Line and Half planes in the Complex plane, the extended plane and its Stereographic representation.
  2. Metric spaces and Topology of complex plane. Open sets in Complex plane, Few properties of metric topology, Continuity, Uniform convergence
  3. Elementary properties of Analytic functions. Analytic functions as mapping. Exponential and Logarithm
  4. Complex Integration: Basic review of Riemann-Stieltjes integral (without proof), Path integral, Power series representation of an analytic function, Liouville’s theorem and Identity theorem, Index of a closed curve, Cauchy theorem and Integral Formula, Open mapping theorem.
  5. Singularities: Removable singularity and Pole, Laurent series expansion, Essential singularity and Casorati-Weierstrass theorem Residues, Solving integral, Argument Principle, Rouche’s Theorem, Maximum modulus theorem.
  6. Harmonic Functions: Basic properties, Dirichlet problem, Green function.

References:

  1. Functions of One Complex Variable by John B Conway, 2nd edition, Narosa.
  2. Complex Analysis by Lars Ahlfors, 3rd edition, McGraw Hill Education India.
  3. Introduction to Complex Analysis by H A Priestley, Oxford University Press.
  4. Complex Function Theory by D Sarason, 2nd edition, TRIM Series, Hindustan Book Agency.
  5. Complex Analysis by T W Gamelin, Springer.
  6. Complex Variables by M J Ablowitz and A S Fokas, 2nd edition, Cambridge University Press.
MAT626
Functional Analysis
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 621 for graduate students, MAT 231/320 and MAT 360 for undergraduates.

Overview: This course introduces the tools of Banach and Hilbert Spaces, which generalize linear algebra and geometry to infinite dimensions. It is a prerequisite for advanced topics like Spectral Theory, Operator Algebras, Operator Theory, Sobolev Spaces, and Harmonic Analysis.  Functional Analysis is a vital component of applications of mathematics to areas like Quantum Physics and Information Theory.

Detailed Syllabus:

  1. Banach Spaces
    • Spaces: Some inequalities, Banach Spaces, finite dimensional spaces, compactness and dimension, quotient spaces, bounded operators, sums of normed spaces.
    • Theorems: Baire Category Theorem, Open Mapping Theorem, Closed Graph Theorem, Principle of Uniform Boundedness.
    • Spaces: Hahn-Banach Theorem, Spaces in Duality, Adjoint operator.
    • Topologies: Weak topology induced by seminorms, weakly continuous functionals, Hahn-Banach separation theorem, weak*-topology, Alaouglu's Theorem, Goldstine's Theorem, reflexivity, extreme points, Krein-Milman Theorem.
  2. Hilbert Spaces
    • Inner products: Inner product spaces, Hilbert spaces, orthogonal sum, orthogonal complement, orthonormal basis, orthonormalization, Riesz Representation Theorem.
    • Operators on Hilbert spaces: Adjoint operators and involution in B(H), Invertible operators, Self adjoint operators, Unitary operators, Isometries.
    • Spectrum: Spectrum of an operator, Spectral mapping theorem for polynomials.

 

Main References:

  1. E. Kreyszig: Introductory Functional Analysis with Applications, Wiley India.
  2. G. F. Simmons: Topology and Modern Analysis, Tata McGraw-Hill, 2004.
  3. Gert K. Pedersen: Analysis Now, Springer, 1988.
  4. John B. Conway: A Course in Functional Analysis, Springer International Edition, 2010.
  5. V. S. Sunder: Functional Analysis - Spectral Theory, Hindustan Book Agency, 1997.
  6. S. Kesavan: Functional Analysis, Hindustan Book Agency, 2009.
  7. G. Bachman and L. Narici: Functional Analysis, 2nd edition, Dover, 2000.
  8. Sterling K. Berberian: Lectures in Functional Analysis and Operatory Theory, Springer, 1974.
MAT630
Differential Equations
4

Core course for M.Sc. Mathematics

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 230/430 ODE (for undergraduates)

Overview:

Detailed Syllabus

(a) Review of Solution Methods for first order and second order linear equations.

(b) Existence and Uniqueness of Initial Value Problems: Lipschitz and Gronwall's inequality, Picard’s Theorem, dependence on initial conditions, continuation of solutions and maximal interval of existence.

(c) Higher Order Linear Equations and Linear Systems: fundamental solutions, Wronskian, variation of constants, matrix exponential solution, behaviour of solutions.

(d)* Two Dimensional Autonomous Systems and Phase Space Analysis: critical points, proper and improper nodes, spiral points, saddle points, Limit cycles, and periodic solutions.

(e)* Asymptotic Behavior: Stability (linearized stability and Lyapunov methods).

(f) Sturm-Liouville Boundary Value Problems: Sturm-Liouville problem for 2nd order equations, Green's function, Sturm comparison theorems and oscillations, eigenvalue problems.

Sections (d) and (e) will also be explored by computer implementation using MATLAB or other software.

References:

  1. M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and Introduction to Chaos, Academic Press, 2004.
  2. L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Vol. 7, 2nd edition, Springer Verlag, New York, 1998.
  3. G. F. Simmons and S. G. Krantz, Differential Equations, Theory, Technique, and Practice, 4th edition, McGraw Hill Education, New Delhi, 2013.
  4. William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Wiley, New York, 1992.

Past Instructors: Ajit Kumar, Samit Bhattacharyya

MAT640
Graduate Algebra I
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: Linear Algebra, Group Theory

Overview:  An overview of graduate algebra with an emphasis on commutative algebra. The instructor may choose 3 or 4 topics from the following, depending on the background and interest of the students.

Detailed Syllabus:

  1. Group Actions: Orbit-stabilizer theorem, Centralizers, Normalizers, Class Equation, Sylow Theorems.
  2. Rings: Homomorphisms and ideals, quotient, adjunction, integral domains and fraction fields, maximal ideals, factorization, unique factorization domains, principal ideal domains, Euclidean domains, factoring polynomials.
  3. Modules: Submodules, quotient modules, homomorphisms, isomorphism theorems, direct sums, simple and semisimple modules, free modules, finitely generated modules, Schur’s Lemma, Jordan-Holder Theorem, Modules over a matrix algebra.
  4. Fields: Algebraic and transcendental elements, degree of field extension, adjunction of roots, finite fields, function fields, transcendental extensions, algebraic closure.
  5. Galois Theory: Galois group,  Galois extension, cubic equations, symmetric functions, primitive elements, quartic equations, Kummer extensions, quintic equations.
  6. Homological Algebra: Categories, monomorphisms and epimorphisms, projective and injective modules, left/right exact functors, additive functors, the Hom functor, diagram chasing, push-outs and pull-backs, tensor product, natural transformations, adjoint functors, flat modules.

References:

  1. Algebra by M Artin, 2nd edition, Prentice-Hall India, 2011.
  2. Abstract Algebra by D S Dummit and R M Foote, 2nd edition, Wiley, 1999.
  3. Algebra by S. Lang, 3rd edition, Springer, 2005.
  4. Algebra by T. W. Hungerford, Springer India, 2005.
MAT660
Linear Algebra
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: None. Not open to undergraduates.

Overview: The theory of vector spaces is an indispensable tool for Mathematics, Physics, Economics and many other subjects. This course aims at providing a basic understanding and some immediate applications  of the language of vector spaces and morphisms among such spaces.

Detailed Syllabus:

  1. Familiarity with sets: Finite and infinite sets; cardinality; Schroeder-Bernstein Theorem;  statements of various versions of Axiom of Choice.
  2. Vector spaces: Fields; vector spaces; subspaces; linear independence; bases and dimension; existence of basis; direct sums; quotients.
  3. Linear Transformations: Linear transformations; null spaces; matrix representations of linear transformations; composition; invertibility and isomorphisms; change of co-ordinates; dual spaces.
  4. Systems of linear equations: Elementary matrix operations and systems of linear equations.
  5. Determinants: Definition, existence, properties, characterization.
  6. Diagonalization: Eigenvalues and eigenvectors; diagonalizability;  invariant subspaces; Cayley-Hamilton Theorem.
  7. Canonical Forms: The Jordan canonical form; minimal polynomial; rational canonical form.

References:

  • Friedberg, Insel and Spence: Linear Algebra, 4th edition, Prentice Hall India
  • Hoffman and Kunze: Linear Algebra, 2nd edition, Prentice Hall India
  • Paul Halmos: Finite Dimensional Vector Spaces, Springer India
  • Sheldon Axler: Linear Algebra Done Right, 2nd edition, Springer International Edition
  • S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice Hall India
MAT680
Numerical Analysis & Computer programming
4

Credits: 4 (3 lectures and 2 lab hours weekly)

Prerequisites: None. Not open for undergraduates.

Overview: This course takes up the problems of practical computation that arise in various areas of mathematics such as solving algebraic or differential equations. The focus is on algorithms for obtaining approximate solutions, and almost half of the course will be devoted to their implementation by computer programs in MATLAB.

Detailed Syllabus:

  1. Solving equations: Iterative methods, Bisection method, Secant method, and Newton-Raphson method.
  2. Solving Linear systems: Gaussian Elimination and pivoting
  3. Computing eigenvalue and eigenvector: Jacobi method
  4. Curve fitting
  5. Solution of ODEs and systems: Runge-Kutta method, Boundary value problems, Finite Difference Method
  6. Solutions of PDEs

References:

  1. Numerical Methods using Matlab, by John H. Mathews and Kurtis D. Fink, 4th edition, PHI, 2009.
  2. An Introduction to Numerical Analysis, by E. Suli and D. Mayers, Cambridge University Press.
  3. Numerical Analysis, by Rainer Kress, Springer, 2010.
  4. Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch, 3rd edition, Springer, 2009.

Elective Courses

This is a sample of the electives that have already been offered at SNU. The actual courses offered in a semester depends on student interest and faculty availability. New courses are constantly being added.

Course code
Title
Credit
MAT632
Geometry
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 360 Linear Algebra II for undergraduates, MAT 660 Linear Algebra for graduate students

Overview: This course provides a bridge to modern geometry. It provides a unified axiomatic approach leading to a coherent overview of the classical geometries (affine, projective, hyperbolic, spherical), culminating in a treatment of surfaces that sets the stage for future study of differential geometry.

Detailed Syllabus:

  1. Affine geometry – finite planes, planes over fields, affine transformations, collineations, affine coordinates, triangles and parallelograms, classical theorems of Menelaus and others.
  2. Projective geometry – finite planes, projective completion of affine planes, homogeneous coordinates, projective transformations, collineations, projective line, poles and polars.
  3. Conics – affine and projective classifications, group actions.
  4. Euclidean geometry – isometries, triangles, parallelograms, length minimizing curves, geometry of plane curves.
  5. Hyperbolic geometry – Poincare upper half plane, Poincare metric, distance function, triangles and area, two-point homogeneity.
  6. Spherical geometry – Sphere, tangent space, great circles, triangles and area, two-point homogeneity
  7. Surfaces – Level surfaces, parametrized surfaces, curvature, Gauss theorem, introduction to manifolds.

 

References:

  1. An Expedition to Geometry by S Kumaresan and G Santhanam. Hindustan Book Agency, 2005.
  2. Geometry by M. Audin. Springer International Edition, Indian reprint, 2004.
  3. Geometry by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray. Cambridge University Press, 2nd edition, 2012.
  4. Geometry by Roger A Fenn, Springer International Edition, 2005.
MAT634
Differential Geometry
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 621 (Analysis I) or MAT 332/432 (Geometry of Curves and Surfaces)

Overview: Differential Geometry generalizes the calculus of several variables on Euclidean spaces to `differential manifolds’. This enables the use of analysis and linear algebra to study geometry. The two highlights of this course are the study of Lie groups and of differential forms, with the latter leading to the general Stokes’ theorem in integration.

Detailed Syllabus:

Part 1: Calculus in Rn

Rn as a normed linear space, derivative, chain rule, mean value theorem, directional derivatives, inverse mapping theorem, implicit function theorem, immersions and submersions, integration, higher derivatives, maxima and minima, existence of solutions of ODE.

Part 2: Differential Manifolds and Lie Groups

Differential manifolds, smooth maps and diffeomorphisms, Lie groups, tangent spaces, derivatives, immersions and submersions, submanifolds, vector fields, Lie algebras, flows, exponential map, Frobenius theorem, Lie subgroups and subalgebras.

Part 3: Differential Forms and Integration

Multilinear algebra, exterior algebra, tensor fields, exterior derivative, Poincare lemma, Lie derivative, orientable manifolds, integration on manifolds, Stokes' theorem.

References:

  1. An Introduction to Differentiable Manifolds and Riemannian Geometry, by William M Boothby, 2nd edition, Academic Press.
  2. A Course in Differential Geometry and Lie Groups, by S Kumaresan, TRIM Series, Hindustan Book Agency.
  3. Analysis on Manifolds, by James Munkres, Addison-Wesley.
  4. Calculus on Manifolds, by Michael Spivak, Addison-Wesley.
MAT642
Graph Theory
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: For undergraduates: MAT 140 (Discrete Structures), MAT 360 (Linear Algebra II). For MSc students: MAT 660 (Linear Algebra)

Overview: Combinatorial graphs serve as models for many problems in science, business, and industry. In this course we will begin with the fundamental concepts of graphs and build up to these applications by focusing on famous problems such as the Traveling Salesman Problem, the Marriage Problem, the Assignment Problem, the Network Flow Problem, the Minimum Connector Problem, the Four Color Theorem, the Committee Scheduling Problem , the Matrix Tree Theorem, and the Graph Isomorphism Problem. We will also highlight the applications of matrix theory to graph theory.

Detailed Syllabus:

  1. Fundamentals:  Graphs and Digraphs, Finite and Infinite graphs, Degree of a vertex, Degree Sequence, Walk, Path, Cycles, Clique, Operations on Graphs, Complement, Subgraph, Connectedness, Components, Isomorphism, Special classes of graphs: Regular, Complete, Bipartite, Cyclic and Euler Graphs, Hamiltonian Paths and Circuits. Trees and binary trees.
  2. Connectivity: Cut Sets, Spanning Trees, Fundamental Circuits and Fundamental Cut Sets, Vertex Connectivity, Edge Connectivity, Separability.
  3. Planar graphs, Coloring, Ramsey theory, Covering, Matching, Factorization, Independent sets, Network flows.
  4. Graphs and Matrices: Incidence matrix, Adjacency matrix, Laplacian matrix, Spectral properties of graphs, Matrix tree theorem, Automorphism group of a graph, vertex, edge and distance transitive graphs, Cayley graphs.
  5. Algorithms and Applications: Algorithms for connectedness and components, spanning trees, minimal spanning trees of weighted graphs, shortest paths in graphs by DFS, BFS, Kruskal's, Prim's, Dijkstra's algorithms.

References:

  1. D. West, Introduction to Graph Theory, Prentice Hall.
  2. Narsingh Deo, Graph Theory: With Application to Engineering and Computer Science, PHI, 2003.
  3. Chris D. Godsil and Gordon Royle, Algebraic Graph Theory, Springer-Verlag, 2001.
  4. Norman Biggs, Algebraic Graph Theory, 2nd edition, Cambridge Mathematical Library.
  5. Frank Harary, Graph Theory, Narosa Publishing House.
  6. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Addison Wesley.
  7. R. J. Wilson, Introduction to Graph Theory, 4th Edition, Pearson Education, 2003.
  8. Josef Lauri, Raffaele Scapellato, Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts.
MAT683
Computational Stat. using R
4

Computational Statistics using R

MAT684
Statistics I
4

Core course for M.Sc. Mathematics.

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 184 (Probability) for undergraduates.

Overview: This course builds on a standard undergraduate probability and statistics course in two ways. First, it makes probability more rigourous by using the concept of measure. Second, it discusses more advanced  topics such as multivariate regression, ANOVA and Markov Chains.

Detailed Syllabus:

  1. Probability: Axiomatic approach, conditional probability and independent events
  2. Random Variables – Discrete and continuous. Expectation, moments, moment generating function
  3. Joint distributions, transformations, multivariate normal distribution
  4. Convergence theorems: convergence in probability, Weak law of numbers, Borel- Cantelli lemmas, Strong law of large numbers, Central Limit Theorem
  5. Random Sampling & Estimators: Point Estimation, maximum likelihood, sampling distributions
  6. Hypothesis Testing
  7. Linear Regression, Multivariate Regression
  8. ANOVA
  9. Introduction to Markov Chains

References:

  • Statistical Inference by Casella and Berger. Brooks/Cole, 2007. (India Edition)
  • An Intermediate Course in Probability by Allan Gut. Springer, 1995.
  • Probability: A Graduate Course by Allan Gut. Springer India.
  • Measure, Integral and Probability by Capinski and Kopp. 2nd edition, Springer, 2007.
MAT721
General Measure Theory
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 623 Analysis II

Overview:

Detailed Syllabus:

  1. Measure and Integration
    • Measure Spaces
    • Measurable functions
    • Integration – Fatou’s Lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem.
    • General Convergence Theorems
    • Signed Measures – Hahn Decomposition Theorem, Jordan decomposition of a measure, Radon-Nikodym Theorem, Lebesgue Decomposition Theorem.
    • The Lp spaces – Riesz Representation Theorem.
  2. Measure and Outer Measure
    • Outer measure and measurability
    • The extension theorem – Caratheodory Theorem
    • The Lebesgue-Stieltjes Integral
    • Product measures – Fubini’s Theorem, Tonelli Theorem, Lebesgue Integral on Rn, change of variable.
    • Inner Measure
  3. Measure and Topology
    • Baire sets and Borel sets
    • Regularity of Baire and Borel measures
    • Construction of Borel Measure
    • Positive linear functionals and Borel Measures – Riesz Markov Theorem (Dual of Cc(X)).
    • Bounded linear functionals on C(X) – Riesz Representation Theorem

 

References:

  1. Real Analysis by H. L. Royden and P. Fitzpatrick. 4th edition, Prentice-Hall India, 2010.
  2. Measure Theory and Integration by G. de Barra, New Age International, reprint 2006.
  3. Real Analysis: Modern Techniques and their Applications by G. B. Folland, Wiley, 2nd edition, 1999.
  4. Measure Theory by Paul Halmos, Springer, 1974.
MAT722
Algebraic Topology
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 640, MAT 622. Undergraduates can substitute MAT 640 with MAT 240.

Overview: Algebraic topology is a tool which captures key information about geometrical properties of a topological space. We shall focus on two basic algebraic invariants of topological spaces: the fundamental group and the simplicial homology groups. If time permits, we shall also get a glimpse of singular homology.

Detailed Syllabus:

  1. The Fundamental Group
    • Homotopy, Fundamental Group, Introduction to Covering Spaces, The Fundamental Group of the circle S 1 , Retractions and fixed points, Application to the Fundamental Theorem of Algebra, The Borsuk-Ulam Theorem, Homotopy Equivalence and Deformation Retractions, Fundamental group of a product of spaces, torus, n-sphere, and the real projective n-space.
    • van Kampen’s Theorem: Free Products of Groups, The van Kampen Theorem, Fundamental Group of a Wedge of Circles, Definition and construction of Cell Complexes, Application of Van Kampen Theorem to Cell Complexes, Statement of the Classification Theorem for Surfaces.
  2. Covering Spaces: Universal Cover and its existence, Unique Lifting Property, Homomorphisms and automorphisms of Covering Spaces, Action of the fundamental group on the fibers, Deck Transformations, Group Actions, Covering Space Actions, Normal or Regular Covering Spaces.
  3. Simplicial Homology (If time permits): Finite Simplicial complexes, Polyhedra and Triangulations, Simplicial approximation, Barycentric subdivision.  Orientation of simplicial complexes, Simplical chain complex and homology. Invariance of homology groups. Computations and applications.

References:

  1. A.Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.
  2. S. Deo, Algebraic Topology, Hindustan Book Agency, 2006.
  3. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin, 1991.
  4. W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.
  5. J.R. Munkres, Elements of Algebraic Topology, Addison Wesley, 1984.
  6. J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004.
  7. J.W. Vick, Homology Theory, Springer-Verlag, 1994.
  8. E. H. Spanier, Algebraic Topology, Springer, 1994.
MAT724
Hardy-Hilbert Spaces & Apps.
4

Hardy-Hilbert Spaces and Applications 

MAT741
Analytic Number Theory
4

Analytic Number Theory

MAT742
Complex Networks
4

Complex Networks

MAT744
Representation Theory
4

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 634 (Differential Geometry) and MAT 640 Graduate Algebra I

Overview: Representations of groups realize the group elements as linear transformations on vector spaces, or even more concretely, as matrices. This enables the use of linear algebra to study algebra, and connects group theory with other areas such as geometry, harmonic analysis and number theory. In this course, we will first study the representation theory of finite groups, and then that of compact groups.

Detailed Syllabus:

Part 1: Representations of Finite Groups

Review of group actions, representations,  unitarizability and unitary equivalence of finite dimensional representations of finite groups, complete reducibility, group algebra as a *-algebra, regular representations, matrix coefficients, Schur’s lemmas, tensor products of representations, orthogonality of matrix coefficients, orthogonality of characters, direct sum decompositions, projection formulas, dimension theorem, character tables, Frobenius-Schur theorem on real and quaternionic representations, Fourier analysis on finite groups, subgroups of index 2, induced representations, Frobenius character formula, Frobenius reciprocity, Mackey irreducibility criterion.

Part 2: Representations of Compact Groups

Review of manifolds and Lie groups, the classical compact groups, topological properties of G and G/H, invariant forms and integration, Haar measure, examples of Haar measure for matrix groups, matrix coefficients, characters, Schur orthogonality, review of spectral theory, Schur’s lemma, regular representations, Frobenius reciprocity, Peter-Weyl theorem, representations and harmonic analysis of SU(2), Fourier theory.

References:

  1. Representation Theory of Finite Groups by Benjamin Steinberg, Springer.
  2. Representations of Finite and Compact Groups by Barry Simon, Graduate Studies in Mathematics, American Mathematical Society.
  3. Representation Theory – A First Course by William Fulton and Joe Harris, Springer.
  4. A First Course on Representation Theory and Linear Lie Groups by S C Bagchi, S Madan, A Sitaram, and U B Tewari, Universities Press.
  5. Compact Lie Groups by Mark R Sepanski, Springer.
  6. Short Courses in Mathematics by S Kumaresan, Universities Press.