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Courses for M.Sc. in Mathematics

New_M.Sc._Curriculum_as_per_NEP_2020

 

Course_Structure_for_M.Sc._in_Mathematics

 

Semester_I

(Total_Credits:_20)

Semester_II

(Total_Credits:_20)

Linear_Algebra_(PM414)_(4)Algebra_I_(4)Elementary_Number_Theory_(PM415)_(4)Measure_Theory_(4)Real_Analysis_(4)Complex_Analysis_(4)Basic_Topology_(4)Computational_Mathematics_(4)Ordinary_Differential_Equations_(4)Functional_Analysis_(4)Total_Credits_at_the_end_of_first_year_=_40
Semester_III

(Total_Credits:_20)

Semester_IV

(Total_Credits:_20)

Algebra_II_(4)Elective_I_(4)Partial_Differential_Equations_(4)Elective_II_(4)Probability_and_Statistics_(4)Elective_III_(4)Project_I_(8)Project_II_(8)Total_Credits_at_the_end_of_second_year:_40_+_40_=_80

 

List_of_Elective_CoursesCodeCourse_NameCodeCourse_NamePM_501Number_TheoryPM_506Advanced_AlgebraPM_502Differential_TopologyPM_507Algebraic_TopologyPM_503Harmonic_AnalysisPM_508Banach_and_Operator_AlgebrasPM_504Analytic_Number_TheoryPM_509Numerical_AnalysisPM_505ProofsPM_510Computational_Fluid_Dynamics

 

Details_of_the_courses Linear_Algebra_(PM414)   _4_Credits

Introduction:_This_course_is_addressed_to_the_first_year_students_pursuing_M.Sc._in_Mathematics_at_SPS._It_is_a_standard_course_in_which_various_kinds_of_decompositions_of_the_given_finite_dimensional_vector_space_and_different_matrix_representations_of_their_operators_are_studied._It_gets_more_interesting_when_this_theory_is_specialized_to_the_inner_product_spaces._Finally_a_very_basic_introduction_to_bilinear_forms_is_given_in_the_course._The_course

aims_partially_at_preparing_the_students_to_be_able_to_study_subjects_like_Representation_Theory,_Lie_algebras,_Functional_Analysis,_etc.

 

Prerequisites:_Material_covered_in_a_standard_undergraduate_Linear_Algebra_course.

 

    Polynomial_rings:_A_quick_revision_of_polynomial_ring_in_one_variable_over_a_field_(results_without_proof)._(1_hour)A_quick_review_of_basic_linear_algebra:_Important_definitions,_results_from_chapters_1_through_5_of_[1]._ (1_week)Fundamental_polynomials_associated_to_operators:_Annihilating_polynomial,_minimal_polynomial,_characteristic_polynomial_of_an_operator._(2_weeks)Diagonalization:_Direct_sum_decomposition,_invariant_subspaces,_primary_decomposition,_diagonalization,_triangulation_of_one_operator_and_that_of_a_family_of_operators._(2-3_weeks)D-N_Decomposition:_Decomposition_into_diagonalizable_and_nilpotent_operators_(1-2_weeks)Canonical_Forms:_Cyclic_decomposition_theorem,_rational_canonical_form,_Jordan_canonical_form._(3-4_weeks)Inner_product_spaces:_Inner_product_spaces,_orthogonality,_Gram-Schmidt_theorem,_orthogonal_projections,_adjoint_of_an_operator,_self-adjoint,_normal,_unitary_operators,_spectral_theorems._(3_weeks)Bilinear_forms:_Symmetric_and_skew-symmetric_forms,_groups_preserving_bilinear_forms._(1-2_weeks)

 

Suggested_Texts:

 

    K._Hoffmann_and_R._Kunze,_Linear_Algebra,_2nd_edition,Prentice_Hall_Inc.,_1971.M._Artin,_Algebra,Prentice_Hall_International,_1990.D._S._Dummit_and_R._M._Foote,_Abstract_Algebra,3rd_edition,_Wiley,_2003.S._H._Friedberg,_A._J._Insel_and_L._E._Spence,_Linear_Algebra,_5th_edition,_Pearson_Education,_2022.S._Lang,_Linear_Algebra,_Undergraduate_Texts_in_Mathematics,_Springer,_2004.P._R._Halmos,_Linear_Algebra_Problem_Book,_Mathematical_Association_of_America,_1995.N._Jacobson,_Basic_Algebra,Vol._1,_Hindustan_Publishing_Corporation,_1984.S._Axler,_Linear_Algebra_Done_Right,_Undergraduate_Texts_in_Mathematics,_Springer,_2014.

 

 

 

 Elementary_Number_Theory_(PM415)   _4_Credits

Course_Learning_Objectives:_This_course_introduces_the_students_to_number_theory_and_prepares_them_for_pursuing_more_specialized_and_advanced_topics_in_number_theory.

 

Prerequisites:_Mathematics_taught_in_the_secondary_school_(10+2)

 

    Divisibility:_Basic_properties_of_divisibility,_greatest_common_divisor,_the_Euclidean_algorithm,_primes,_unique_factorization,_infinitude_of_primes,_divergence_of_the_sum_of_reciprocals_of_primes.Congruences:_Basic_properties_of_congruences,_congruences_of_Fermat,_Euler_and_Wilson,_linear_congruences,_Chinese_remainder_theorem,_polynomial_congruences,_Hensel’s_lemma,_primitive_roots,_RSA_cryptosystems.Quadratic_residues:_Quadratic_residues_and_quadratic_nonresidues,_Legendre_symbol_and_Gauss’s_quadratic_reciprocity_law,_Jacobi_symbol_and_the_reciprocity_law.Arithmetic_functions:_Some_examples_of_arithmetic_functions,_Dirichlet_product,_multiplicative_and_completely_multiplicative_arithmetic_functions.Some_Diophantine_equations:_Linear_Diophantine_equations,_Pythagorean_triplets,_insolvability_of_some_Diophantine_equations_such_as_x4_+_y4_=_z2.Simple_continued_fractions:_Continued_fraction_expansion_of_real_numbers,_Dirichlet’s_theorem_on_Diophantine_approximation,_rational_approximations_of_real_numbers_by_convergents_in_the_continued_fraction_expansion,_Hurwitz’s_theorem,_periodic_continued_fractions,_Pell’s_equation.Distribution_of_prime_numbers:_Statements_of_Dirichlet’s_theorem_on_primes_in_arithmetic_progressions_and_the_Prime_Number_Theorem,_Chebyshev’s_functions,_some_elementary_results_on_the_distribution_of_prime_numbers,_Bertrand’s_postulate.

 

Suggested_Texts:

 

    I._Niven,_H._S._Zuckerman_and_H._L._Montgomery,_An_Introduction_to_the_Theory_of_Numbers,_5th_edition,_John_Wiley_&_Sons,_1991.G._E._Andrews,_Number_Theory,_Dover,_1994.A._Baker,_A_Concise_Introduction_to_the_Theory_of_Numbers,_Cambridge_University_Press,_1984.G._A._Jones_and_J._M._Jones,_Elementary_Number_Theory,_Springer,_1998.

 

 

Supplementary_References:

 

    T._M._Apostol,_Introduction_to_Analytic_Number_Theory,_Springer,_1976.H._Davenport,_The_Higher_Arithmetic,_8th_edition,_Cambridge_University_Press,_2008.G._H._Hardy_and_E._M._Wright,_An_Introduction_to_the_Theory_of_Numbers,_6th_edition_(revised_by_D._R._Heath-Brown_and_J._H._Silverman),_Oxford_University_Press,_2008.E._Landau,_Elementary_Number_Theory,_Chelsea,_1958.W._J._LeVeque,_Fundamentals_of_Number_Theory,_Dover,_1996.T._Nagell,_Introduction_to_Number_Theory,_Chelsea,_1981.H._Rademacher,_Lectures_in_Elementary_Number_Theory,_Blaisdell,_1964.I._M._Vinogradov,_Elements_of_Number_Theory,_Dover,_1954.

 

 

 

Algebra_I_(PM401)   _4_Credits

 

Introduction:_It_is_wished_that_the_students_appreciate_the_beauty_and_challenges_in_classification_of_finite_abelian_groups_of_small_orders._Moreover,_their_contrast_with

infinite_non-abelian_groups_can_be_observed._This_course_equips_a_student_to_learn_representation_theory_of_groups_without_much_difficulty._Both_commutative_and_non-commutative_rings_are_studied_once_again_with_analogy_and_contrast_in_mind.

 

Prerequisites:_Material_covered_in_a_standard_undergraduate_Algebra_course.

 

Part_I:_Group_Theory

    A_quick_review_of_basic_group_theory,_revision_of_results_in_permutation_groups._(2_hours)Group_actions,_example_of_linear_groups_such_as_GL(n),_SL(n),_O(n),_SO(n),_U(n),_SU(n);_counting_formula,_class_equation,_Sylow’s_theorems_(3-4_weeks)Generators_and_relations,_free_groups,_semidirect_product_of_groups,_structure_of_finite_abelian_groups,_classification_of_finite_groups_of_small_orders_(2-3_weeks)

 

Part_II:_Ring_Theory

    A_brief_review_of_ring_theory:_Basic_definitions_(rings,_subrings,_ideals,_prime_ideals,_maximal_ideals)_and_results_in_the_theory_of_rings._(1_week)Examples_of_rings_(matrix_rings,_group_rings,_quaternions,_rings_of_holomorphic_functions,_rings_of_convergent_sequences,_a_few_number_rings),_properties_of_ideals,_integral_domains_and_their_rings_of_fractions,_radical_ideal,_nilradical,_Jacobson_radical,_ideal_quotient,_Chinese_remainder_theorem_(2-3_weeks)Concept_of_divisibility,_gcd_and_its_properties,_prime_elements,_irreducible_elements._(1_week)Euclidean_domains,_principal_ideal_domains_(PIDs),_unique_factorization_domains_(UFDs)_with_special_emphasis_on_polynomial_rings,_Gauss’s_lemma,_Eisenstein’s_criterion_of_irreducibility_of_polynomials._(4-5_weeks)

 

Suggested_Texts:

 

    I._N._Hersein,_Topics_in_Algebra,_Wiley_Student_edition,_2006.M._Artin,_Algebra,_Prentice_Hall_International,_1990.T._W._Hungerford,_Algebra,_Springer,_1980.D._S._Dummit_and_R._M._Foote,_Abstract_Algebra,_3rd_edition,_Wiley,_2003.N._Jacobson,_Basic_Algebra,_Vol._1,_Hindustan_Publishing_Corporation,_1984.S._Lang,_Algebra,_3rd_edition,_Graduate_Texts_in_Mathematics_211,_Springer,_2005.

 

 

 

 

 

 

 

 

 

 

 

Complex_Analysis_(PM402)   _4_Credits

 

Prerequisites:_Basic_knowledge_of_Real_Analysis

 

    Quick_Review_of_Complex_numbers:_Basic_operations,_conjugate,_modulus,_argument,_exponential_function,_rootsHolomorphic_functions:_Continuity,_derivative,_holomorphic_functions,_Cauchy-Riemann_differential_equations,_harmonic_functionsElementary_functions:_Polynomial_and_rational_functions,_exponential_function,_logarithm,_trigonometric_and_hyperbolic_functionsComplex_integration:_Paths_and_contours,_integration,_estimation_theorem,_Cauchy's_integral_formula,_Cauchy's_theorem,_Liouville's_theorem,_fundamental_theorem_of_algebra,_maximum_modulus_principle,_Schwarz's_lemmaSeries:_(absolute_and_uniform)_Convergence_of_series,_power_series,_Taylor_series,_Laurent_series,_the_identity_principleZeros,_singularities_and_residues:_Classification_of_singularities,_orders_of_poles_and_zeros,_winding_number,_meromorphic_functions,_Cauchy's_residue_theorem,_argument_principleMappings:_Linear_fractional_transformations,_conformal_mappingsApplication_of_complex_integration:_Computation_of_indefinite_integralsAdditional_Topics:_Branch_points,_doubly_periodic_functions,_construction_of_sine,_cosine_as_an_inverse_of_a_multi-valued_function,_Riemann_mapping_theorem,_Dirichlet_problem,_analytic_continuation,_multivariable_complex_analysis

 

Main_Text_Book:

 

    L._V._Ahlfors,_Complex_analysis,_An_introduction_to_the_theory_of_analytic_functions_of_one_complex_variable,_Third_edition,_International_Series_in_Pure_and_Applied_Mathematics,_McGraw-Hill_Book_Co.,_New_York,_1978.J._B._Conway,_Functions_of_one_complex_variable,_Graduate_Texts_in_Mathematics,_159,_Springer-Verlag,_New_York,_1995.

 

Supplementary_References:

 

    W._Rudin,_Real_and_complex_analysis,_Third_edition,_McGraw-Hill_Book_Co.,_New_York,_1987.E.M._Stein,_R._Sharkarchi,_Complex_analysis,_Princeton_Lectures_in_Analysis,_2,_Princeton_University_Press,_Princeton,_NJ,_2003.E._Goursat,_A_Course_in_Mathematical_Analysis,_Functions_of_a_complex_variable,_Part_I_of_Vol._II,_Ginn_and_Company,_1916.

 

 

 

 

 

 

 

 

 

 

Real_Analysis_(PM403)   _4_Credits

 

Prerequisites:_Basic_knowledge_of_Real_Analysis_and_Linear_Algebra

 

    Quick_review_of_basic_Real_Analysis:_Construction_of_real_numbers,_order_on_real_numbers_and_the_least_upper_bound_property,_convergence_of_sequence_and_series,_power_series,_multiplication_of_series,_absolute_and_conditional_convergence,_rearrangements_(with_proof_of_Riemann's_Theorem)._Continuity,_uniform_continuity,_compactness_and_connectedness_in_metric_spaces._Differentiation:_L'Hospital's_rule,_derivatives_of_higher_orders,_Taylor's_theorem,_differentiation_of_vector-valued_functionsThe_Riemann-Stieltjes_Integral:_Definition_and_existence_of_the_integral,_properties_of_the_integral,_integration_and_differentiation,_integration_of_vector-valued_functions,_rectifiable_curvesSequences_and_Series_of_Functions:_Pointwise_and_uniform_convergence,_uniform_convergence_and_continuity,_uniform_convergence_and_integration,_uniform_convergence_and_differentiation,_equicontinuity,_Arzela-Ascoli_theorem,_Stone-Weierstrass_theoremCalculus_of_Several_Variables:_Differentiation_of_functions_of_several_real_variables_(directional_derivatives,_partial_derivatives,_differentiability_and_the_total_derivative,_chain_rule,_Jacobian,_higher_derivatives,_interchange_of_the_order_of_differentiation,_Taylor's_theorem),_inverse_function_theorem,_implicit_function_theorem,_rank_theorem,_differentiation_of_integrals,_derivatives_of_higher_orderAdditional_Topics:_Integration_of_differential_forms:_Integration,_primitive_mappings,_partition_of_unity,_change_of_variables,_differential_forms,_Stokes'_theorem,_closed_and_exact_forms._Some_special_functions:_Power_series,_exponential_and_logarithmic_functions,_trigonometric_functions,_Gamma_function,_Fourier_series

 

Main_text_book:

 

    W._Rudin,_Principles_of_Mathematical_Analysis,_Third_edition,_McGraw_Hill_Book_Company,_New_York,_1976.

 

Supplementary_References:

 

    T._M._Apostol,_Mathematical_Analysis,_2nd_edition,_Addison-Wesley_Publishing_Company,_Reading,_Massachusetts,_1974.T._Tao,_Analysis_I_and_II,_Third_editions,_Texts_and_Readings_in_Mathematics,_Hindustan_Book_Agency,_New_Delhi,_2006.M._Spivak,_Calculus_on_Manifolds:_A_modern_approach_to_classical_theorems_of_advanced_Calculus,_West_View_Press,_27th_printing,_1998.K._Janich,_Vector_Analysis,_Undergraduate_Texts_in_Mathematics,_Springer,_2001.S._Lang,_Undergraduate_Analysis,_Second_edition,_Springer,_2005.H._L._Royden_and_P._M._Fitzpatrick,_Real_Analysis,_Fourth_Edition,_Pearson_Education,_Inc.,_2010.

 

 

 

 

 

Basic_Topology_(PM404)   _4_Credits 

Prerequisites:_Basic_knowledge_of_Real_Analysis_and_Metric_Spaces

 

    Familiarity_with_Set_Theory:_Countable_and_uncountable_sets,_axiom_of_choice_and_its_variants.Topological_Spaces_and_continuous_functions:_Topology,_basis,_sub-basis,_Hausdorff_and_regular_spaces,_order_topology,_subspace_topology,_limit_points,_continuous_functions,_homeomorphisms,_product_topology_and_metric_topology.Quotient_Topology:_Quotient_map,_quotient_topology,_quotient_space.Nets:_Subnets,_convergence_of_netsConnectedness_and_Compactness:_Connectedness,_path-connectedness,_compactness,_comparison_with_compactness_in_metric_spaces_via_nets,_local_compactness_and_one-point_compactificationCountability_and_Separation_Axioms:_First_and_second_countability,_separability,_normality,_complete_regularity,_Urysohn's_lemma,_Tietze_extension_theorem,_Tychonoff_theorem_and_Stone-Cech_compactificationAdditional_Topics:_Urysohn_Metrization_theorem,_local_finiteness,_Nagata-Smirnov_metrization_theorem,_paracompactness_and_Smirnov_metrization_theorem

 

Main_Text_Books:

 

    J._R._Munkres,_Topology,_Second_Edition,_Pearson,_2000.G._E._Bredon,_Topology_and_Geometry,_Graduate_Texts_in_Mathematics,_139,_Springer,_1993.

 

Supplementary_References:

 

    C._O._Christenson_and_W._L._Voxman,_Aspects_of_Topology,_Second_edition,_B._C._S._Associates,_1998.K._Janich,_Topology,_Undergraduate_Text_in_Mathematics,_Springer,_1984.J._L._Kelley,_General_Topology,_Graduate_Text_in_Mathematics,_Springer,_1975.G._F._Simmons,_Topology_and_Modern_Analysis,_Tata_McGraw-Hill,_2004.J._Dugundji,_Topology,_McGraw-Hill_Inc.,_1988.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algebra_II_(PM405)   _4_Credits

 

Introduction:_The_primary_contents_are_field_theory_and_Galois_theory._A_very_brief_introduction_to_modules_is_given_towards_the_end._This_provides_sufficient_background_to_pursue_more_advanced_topics_such_as_Algebraic_Number_Theory,_Representation_Theory

and_Algebraic_Geometry.

 

Prerequisites:_Linear_Algebra_and_Algebra_I.

    Revision_of_polynomial_rings_(1_hour)Basic_definition_of_field_extension,_degree,_characteristic,_algebraic_extensions,_basic_results_(2-3_hours)Existence_and_uniqueness_(up_to_isomorphism)_of_the_splitting_field_of_a_polynomial,_existence_and_uniqueness_(up_to_isomorphism)_of_the_algebraic_closure_of_a_field_(1_week)Cyclotomic_fields_(1_week)Embeddings_of_an_algebraic_extension_of_the_base_field_into_a_fixed_algebraic_closure_of_and_over_the_base_field,_results_on_extending_the_embeddings_to_a_bigger_domain,_their_effect_on_the_polynomial_rings_(3_weeks)Separability_and_number_of_embeddings_(2_weeks)Normal_extensions_and_automorphisms_(1_week)Finite_fields,_main_theorem_of_finite_fields_(2_weeks)Galois_extensions,_fundamental_theorem_of_Galois_theory,_computing_various_Galois_groups_(2-3_weeks)Transcendental_extensions,_Luroth’s_theorem_(2-3_hours)Introduction_to_module_theory_(5-6_hours)

 

Suggested_Texts:

 

    I._N._Hersein,_Topics_in_Algebra,_Wiley_Student_edition,_2006.M._Artin,_Algebra,_Prentice_Hall_International,_1990.T._W._Hungerford,_Algebra,_Springer,_1980.D._S._Dummit_and_R._M._Foote,_Abstract_Algebra,_3rd_edition,_Wiley,_2003.N._Jacobson,_Basic_Algebra,_Vol._1,_Hindustan_Publishing_Corporation,_1984.S._Lang,_Algebra,_3rd_edition,_Graduate_Texts_in_Mathematics_211,_Springer,_2005.

 

Supplementary_References:

 

    S._Lang,_Algebra,_Graduate_Texts_in_Mathematics,_211,_revised_third_edition,_Springer-Verlag,_New_York,_2002.E._Artin,_Edited_and_supplemented_with_a_section_on_applications_by_Arthur_N._Milgram,_Second_edition,_with_additions_and_revisions,_Fifth_reprinting,_Notre_Dame_Mathematical_Lectures,_No._2,_University_of_Notre_Dame_Press,_South_Bend,_Ind._1959.N.S_Gopalkrishnan,_University_Algebra,_Second_edition,_New_Age_International,_New_Delhi,_1986.N.S_Gopalkrishnan,_Commutative_Algebra,_Oxonian_Press_Pvt._Ltd.,_New_Delhi,_1984.

 

 

 

Measure_Theory_(PM406)   _4_Credits

 

Prerequisites:_Real_Analysis,_Basic_Topology

 

    Quick_review_of_Riemann_IntegrationLebesgue_Measure_on_ℝ:_Outer_measure,_outer_regularity_of_outer_measure,_Lebesgue_measure,_regularity_of_Lebesgue_measure,_non-measurable_setsLebesgue_Integral:_Simple_functions,_almost_everywhere_property,_measurable_functions,_integrable_functions,_approximation_of_integrable_functions_by_step_and_continuous_functionsConvergence_of_sequence_of_functions:_Pointwise,_uniform,_Egorov's_theorem,_Lusin's_theoremAbstract_measure_spaces:_Sigma_algebras_and_measurable_spaces,_measures_and_measure_spaces,_completeness_of_a_measure,_measurable_functions_and_their_integration,_monotone_convergence_theorem,_Fatou's_lemma,_dominated_convergence_theorem,_modes_of_convergenceProduct_measure:_product_sigma_algebra,_sigma-finite_measure_spaces,_existence_of_product_measures,_Tonelli's_theorem,_Fubini's_theoremAdditional_Topics:_Lebesgue_differentiation_theorem,_almost_everywhere_differentiability,_absolute_continuity,_Caratheodory's_Extension_theorem_for_outer_measures,_Hahn-Kolmogorov_extension_theorem_for_pre-measures,_Lebesgue-Stieltjes_measure,_Radon_measure

 

Main_Text_book:

 

    T._Tao,_An_Introduction_to_Measure_Theory,_GTM_126,_American_Mathematical_Society,_2011.

 

Supplementary_References:

 

    G._B._Folland,_Real_Analysis:_Modern_Techniques_and_their_Applications,_expanded_and_revised_edition,_John_Wiley_and_Sons,_2013.W._Rudin,_Real_and_Complex_Analysis,_Third_Edition,_McGraw-Hill,_1987.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Functional_Analysis_(PM407)   _4_Credits

 

Prerequisites:_Real_Analysis,_Basic_Topology

 

    Hilbert_spaces:_Inner_product_spaces,_Hilbert_spaces,_orthogonality,_Riesz_representation_Theorem,_orthonormal_sets,_orthogonalization,_unconditional_sum,_orthonormal_bases,_isomorphisms_of_Hilbert_spaces,_separable_Hilbert_spaces,_direct_sums_of_Hilbert_spacesOperators_on_Hilbert_spaces:_Examples,_adjoint_of_an_operator,_invertible_operators,_self-adjoint_operators,_unitary_operators,_isometries,_projections,_compact_operatorsBanach_Spaces:_Normed_spaces,_equivalence_of_norms,_some_inequalities,_Banach_spaces,_finite_dimensional_spaces,_quotient_and_products_of_normed_spaces,_bounded_linear_operators_and_functionalsDual_Spaces:_Hahn-Banach_theorem,_dual_of_a_quotient_space_and_a_subspace,_reflexive_spacesCategory_Theorems:_Baire_category_theorem,_open_mapping_theorem,_closed_graph_theorem,_principle_of_uniform_boundednessOperators_on_Banach_spaces:_Adjoint_of_an_operator,_annihilators,_compact_operatorsAdditional_Topics:_Weak_topology,_weak-*_topology,_Banach-Alaoglu_theorem,_Goldstine's_theorem,_reflexivity_in_terms_of_weak_topology,_separable_Banach_spaces

 

Main_Text_Books:

 

    John_B._Conway,_A_Course_in_Functional_Analysis,_Graduate_Texts_in_Mathematics,_96,_Second_edition_1990,_corrected_fourth_printing,_Springer,_1994.S._Kesavan,_Functional_Analysis,_Texts_and_Readings_in_Mathematics_(TRIM_series),_52,_Corrected_reprint,_Hindustan_Publishing_Agency,_2017.

 

Supplementary_References:

 

    G._F._Simmons,_Introduction_to_Topology_and_Modern_Analysis,_Tata_McGraw-Hill,_2004.W._Rudin,_Real_and_Complex_Analysis,_Third_Edition,_McGraw-Hill,_1987.W._Rudin,_Functional_Analysis,_McGraw-Hill_Education,_Third_Edition,_1986.G._K._Paderson,_Analysis_Now,_Graduate_Texts_in_Mathematics,_118,_Springer,_2012.H._L._Royden_and_P._M._Fitzpatrick,_Real_Analysis,_Fourth_Edition,_Pearson_Education,_Inc.,_2010

 

 

 

 

 

 

 

 

 

 

 

 

Probability_and_Statistics_(PM409)   _4_Credits

 

Prerequisites:_Real_Analysis,_basic_knowledge_of_combinatorics

 

    Probability_and_random_variables:_Repeated_experiments_and_empirical_definition_of_probability._Sample_space,_events._Probability_as_a_set_function_on_a_$\sigma$-algebra._Random_variables,_distribution_functions_and_probability_density_functions._Expectation_value,_variance_and_higher_moments._Moment_generating_functions,_inequalities_of_Markov_and_Chebyshev.Conditional_probability_and_independence:_Conditional_probability,_marginal_distributions_and_conditional_distributions._Covariance_and_correlation,_stochastic_independence.Some_probability_distributions:_Binomial,_Poisson_and_normal_distributions._Properties_of_their_moments.Distributions_of_functions_of_random_variables:_Sampling._Transformations_of_random_variables,_Student's_t_and_F_distributions._Distributions_of_mean_and_variance_of_a_sample._Expectations_of_functions_of_random_variables.Limiting_distributions:_Stochastic_convergence_of_random_variables._Weak_and_strong_laws_of_large_numbers_(without_proofs)._Central_limit_theorem.Additional_topics:}_Hypothesis_testing_-_Examples_and_definitions._Uniformly_most_powerful_tests._Likelihood_ratio_tests._Statistical_significance.

 

Main_Text_Books:

 

    R._V._Hogg_and_A._T._Craig,_Introduction_to_Mathematical_Statistics,_Fourth_edition,_McMillan_Publishing_Company,_1978.S._Ross,_A_First_Course_in_Probability,_8th_Edition,_Prentice_Hall/Pearson,_2010.

 

Supplementary_References:

 

    W._Feller,_An_Introduction_to_Probability_Theory_and_Its_Applications,_Vol._I,_Third_edition,_Wiley,_2008.W._Feller,_An_Introduction_to_Probability_Theory_and_Its_Applications,_Vol._II,_Second_edition,_Wiley,_2008.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Computational_Mathematics_(PM410)   _4_Credits

 

Prerequisites:}_Algebra_I,_Real_Analysis,_Complex_Analysis,_Linear_Algebra,_additionally_some_knowledge_of_elementary_number_theory_and_ODEs_will_help,_but_is_not_essential.

 

Goal:_The_aim_is_to_introduce_the_students_to_algorithmic_way_of_learning,_to_teach_them_the_importance_of_computations_and_to_the_use_of_computers_for_implementation_of_a_few_algorithms._We_will_be_using_a_few_programming_tools_such_as_C++,_MATLAB/SciLab,_SAGE_to_implement_a_few_well-known_algorithms_from_Calculus,_Number_Theory,_Linear_Algebra,_Algebra,_Graph_Theory,_Discrete_Mathematics_and_possibly_Differential_Equations_and_Statistics_depending_on_the_preparation_and_inclination_of_the_students.

           _This_course_will_have_a_practical_component_and_labwork.

 

    Algorithms:_Introduction_to_algorithms_with_a_few_standard_examplesBrief_Introduction_to_Programming:}_Introduction_to_programming_languages_and_computational_systems_such_as_C++,_SAGE,_MATLAB/SciLabAlgorithms_in_Calculus:_Newton-Raphson_iteration_method_for_finding_real_root,_numerical_integrationAlgorithms_in_Linear_Algebra:_Solving_systems_of_linear_equations,_diagonalizationAlgorithms_in_Differential_Equations:_Solution_of_ordinary_differential_equations,_Runge-KuttaAlgorithms_in_Number_theory:_Sieve_of_Eratosthenes,_primality_tests,_Euclidean_algorithm,_greatest_common_divisor,_solution_to_Pell's_equation_using_continued_fractionsAlgorithms_in_Graph_Theory_and_Discrete_Mathematics:_Kruskal's_algorithm,_finding_Eulerian_cycles,_sorting_and_searching_algorithmsAdditional_topics:_Computation_of_Galois_groups,_Grobner_bases

 

Main_References:

 

    Thomas_H._Cormen,_Charles_E._Leiserson,_Ronald_L._Rivest,_Clifford_Stein,_Introduction_to_Algorithms,_Third_edition,_MIT_Press,_2009.S._Pemmaraju_and_S._Skiena,_Computational_Discrete_Mathematics,_First_Reissue_edition,_Cambridge_University_Press,_2009.Henry_Cohen,_A_Course_in_Computational_Algebraic_Number_Theory,_Springer,_1993.SAGE_http://www.sagemath.org/,_open-source_mathematics_software_system

 

Supplementary_References:

 

     D._F._Holt,_B._Eick,_E._A._O'Brien,_Handbook_of_Computational_Group_Theory,_Chapman_and_Hall/CRC_Press,_2005.F._Villegas,_Experimental_Number_Theory,_Oxford_Graduate_Texts_in_Mathematics,_Book_13,_2007.D._E._Knuth,_The_Art_of_Computer_Programming_Volumes_1_to_4,_Addison-Wesley_Professional,_First_edition,_2011.M._Petkovsek,_H._Wilf_and_D._Zeilberger,_A_=_B,_A._K._Peters/CRC_Press,_1996.

 

 

 

 

Ordinary_Differential_Equations_(PM411)   _4_Credits

 

 Prerequisites:_Calculus,_Linear_Algebra,_Real_Analysis

 

    Quick_Review_of_some_basic_methods_of_solving_first_order_ODEs:_Motivation,_Order_and_degree_of_ODEs._Method_of_separation_of_variables._Exact_Differential_Equations._Integrating_Factors.Existence,_Uniqueness_and_Continuity_Theorems_for_first_order_ODEs:_Picard’s_existence_and_uniqueness_Theorem,_Picard’s_successive_approximation_method,_Continuity_of_solutions_with_respect_to_initial_conditions,_Gronwal’s_inequality.Second_order_linear_ODEs:_Vector_space_of_space_of_solutions._Wronskian_and_linear_independence_of_solutions._Linear_ODE_with_constant_coeffecients._Variation_of_parameters._Method_of_undetermined_coefficients._Sturm_Separation_Theorem._Sturm_Comparison_Theorem.Solution_in_series_of_second_order_ODEs:_Ordinary_and_singular_points._Power_series_solution_at_an_ordinary_point._Legendre’s_equation._Solutions_at_a_regular_singular_point_using_the_Frobenius_method._Bessel’s_equation.Systems_of_first_order_ODEs:_System_of_first_order_ODEs_versus_n-th_order_ODE._Existence_and_uniqueness_Theorem_for_system_of_first_order_ODEs._Existence_and_uniqueness_Theorem_for_n-th_order_ODEs._Picard’s_succesive_approximation_method._Homogeneous_linear_systems_of_first_order_ODEs._Fundamental_matrix_and_solution_matrix._Nonhomogeneous_linear_systems._Linear_systems_with_constant_coefficients.Boundary-value_problems_and_self-adjoint_eigenvalue_problems:_Two_point_boundary_value_problems,_Green’s_functions._Sturm-Liouville_systems._Eigenvalues_and_eigenfunctions.Additional_topics:Stability_Analysis:_Linear_systems,_Stability_for_linear_systems_with_constant_coefficients,_Stability_of_linear_plane_systemsLaplace_transform_:_Properties_of_the_Laplace_transform,_Convolution_Theorem,_Step_function,_Impulse_function.Cauchy-Peano_Existence_Theorem:_Arzela-Ascoli_Theorem._Existence_of_solution_of_ODEs_not_satisfying_the_Lipshitz_condition.

 

Suggested_texts:

 

    Coddington,_E.,_An_Introduction_to_Ordinary_Differential_Equations,_Dover_Publications,_2012._(Original:_Prentice-Hall,_1961.)Coddington,_E.,_and_Levinson_N.,_Theory_of_Ordinary_Differential_Equations,_Tata-McGraw-Hill,_1990.Myint-U,_T.,_Ordinary_Differential_Equations,_North-Holland,_New_York,_1978.Rabenstein,_A._L.,_Introduction_to_Ordinary_Differential_Equations,_Elsevier_Science,_2014.Ross,_S._L.,_Introduction_of_Ordinary_Differential_Equations,_4_th_Ed.,_John_Wiley_and_Sons,_2007.Simmons,_G._F.,_Differential_Equations_with_Applications_and_Historical_Notes,_CRC_Press,_2017.

 

 

 

 

Partial_Differential_Equations_(PM412)   _4_Credits

 

Prerequisites:_Calculus,_Linear_Algebra,_Complex_Analysis,_Ordinary_Differential_Equations

 

    Motivation:_PDE_in_natural_science._First_order_PDE,_examples.Second_order_PDE:_Classification_and_reduction_to_canonical_forms._Well-posed_problem._Characteristics._Green's_function.Laplace_equation_(elliptic):_Boundary_value_problem,_Dirichlet_and_Neumann_boundary_conditions._Harmonic_functions._Mean_value_theorem._Solution_by_separation_of_variables.Heat_equation_(parabolic):_Initial_and_boundary_value_problem._Solution_by_separation_of_variables._Duhamel's_principle.Wave_equation_(hyperbolic):_D'Alembert's_solution._Cauchy_problem,_existence_and_uniqueness_of_solutions._Solution_by_separation_of_variables.Additional_topics:_Fourier_transform_method._Laplace_and_Mellin_transforms.

 

Main_Text_Book:

 

    I._Sneddon,_Elements_of_Partial_Differential_Equations,_Dover_reprint,_Dover,_2006.

 

Supplementary_References:

 

    L._C._Evans,_Partial_Differential_Equations,_Second_edition,_American_Mathematical_Society,_2010.J._Fritz,_Partial_Differential_Equations,_Fourth_edition,_Springer,_1991.E._L._Ince,_Ordinary_Differential_Equations,_reprint_edition,_Dover,_1956.V._I._Arnold,_Lectures_on_Partial_Differential_Equations,_Third_edition,_Springer,_2006.T._Amaranath,_An_Elementary_Course_in_Partial_Differential_Equations,_Second_edition,_Narosa,_2014.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Project_I_(PM4XX)   _8_Credits

 

Prerequisites:_Reasonably_good_understanding_of_M.Sc._first_year_courses;_especially_those

related_to_the_project_topic.

 

Goal:_A_student_will_choose_a_topic_related_to_but_beyond_the_first_year_courses._The_student_will_then_learn_the_topic_under_the_supervision_of_a_teacher.

           _It_is_expected_that_the_student_will_meet_the_supervisor_regularly_and_present_the_material_that_he/she_has_learnt_and_keep_his/her_supervisor_updated_with_his/her_progress._The_student_has_to_then_write_an_expository_report_of_about_15_pages_on_the_project_topic_and_also_present_it_to_a_panel_of_examiners_at_the_end_of_the_term.

 

Grading_scheme:_The_student’s_performance_will_be_evaluated_based_on_the_presentations_(made_during_the_meetings_with_the_supervisor),_project_report_and_the_final_presentation.

           _Apart_from_the_main_component_of_the_course,_the_students_are_expected_to_learn_the_basics_of_Latex_which_will_be_useful_in_writing_the_report_towards_the_end_of_the_semester.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Project_II_(PM4XX)   _8_Credits

 

Prerequisites:_Project_I

 

Project_II_can_be_a_continuation_of_Project_I._The_student_can_study_some_advanced_topics_and_do_some_literature_survey._The_student_may_also_read_some_research_papers._It_is_expected_that_the_student_will_meet_the_supervisor_regularly_and_present_the_material_that_he/she_has_learnt_and_keep_his/her_supervisor_updated_with_his/her_progress.

           _The_student_is_also_expected_to_write_an_M.Sc._thesis_on_the_topics_covered_in_the_project,_which_he/she_will_have_to_present_to_a_panel_of_examiners_at_the_end_of_the_term.

 

Grading_scheme:_The_student’s_performance_will_be_evaluated_based_on_the_presentations_(made_during_the_meetings_with_the_supervisor),_the_thesis_and_the_final_presentation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number_Theory_(PM501)   _4_Credits

 

Prerequisites:_Algebra_I,_Algebra_II,_Real_Analysis,_Complex_Analysis

 

Goal:_To_provide_an_introduction_to_Number_Theory_that_is_beyond_an_undergraduate_elementary_number_theory_course_and_having_algebraic,_algebraic_geometric_and_also_analytic_components._This_course_will_cover_a_few_topics_which_will_illustrate_that_number_theory_uses_tools_from_all_the_various_disciplines_of_mathematics.

 

    Unique_factorization_and_applications:_ℤ,_k[x],_unique_factorization_in_a_principal_ideal_domain,_study_of_ℤ[i],_ℤ[ω]_and_ℤ[1+-52]Congruences,_structure_of_(ℤ/nℤ)*Quadratic_reciprocity,_Gauss_and_Jacobi_SumsEquations_over_finite_fields,_Hasse-Davenport_relation,_zeta_function_as_a_generating_function_of_number_of_solutionsRiemann_zeta_function._Definition_of_Dirichlet_L-functions_attached_to_a_character_with_the_possibility_of_deeper_study_of_L-function_attached_to_a_character_of_order_2Diophantine_equations_of_genus_0_over_the_rationals:_Pythagorean_triplets,_Pell's_equationIrrationality_and_transcendence_of_e_and_πAdditional_Topics:_Brief_introduction_to_algebraic_number_theory,_a_very_brief_introduction_to_elliptic_curves,_the_group_law_on_an_elliptic_curve_possibly_without_proof_of_the_associativity_property,_a_few_important_concrete_examples_and_computations,_connection_to_congruence_number_problem_and_Fermat's_last_theorem

 

Main_Text_Book:

 

    K._Ireland_and_M._Rosen,_A_Classical_Introduction_to_Modern_Number_Theory,_Graduate_Texts_in_Mathematics,_84,_Springer-Verlag,_1990.

 

Supplementary_References:

 

    G._H._Hardy_and_E._M._Wright,_An_Introduction_to_the_Theory_of_Numbers,_Sixth_Edition_(edited_by_R._Heath-Brown,_J._Silverman,_and_A._Wiles),_Oxford_University_Press,_2008.I._Niven,_H._S._Zuckerman,_H._L._Montgomery,_An_Introduction_to_the_Theory_of_Numbers,_Fifth_Edition,_John_Wiley_and_sons,_Inc.,_1991.N._Koblitz,_Introduction_to_elliptic_curves_and_modular_forms,_Graduate_Texts_in_Mathematics,_second_edition,_Springer,_1993.J._Silverman,_The_Arithmetic_of_Elliptic_Curves,_Graduate_Texts_in_Mathematics,_106,_second_edition,_Springer-Verlag_New_York,_2009.J._Silverman_and_J._Tate,_Rational_points_on_elliptic_curves,_Undergraduate_Texts_in_Mathematics,_Second_edition,_Springer_International_Publishing,_2015.J.-P._Serre,_A_course_in_Arithmetic,_Graduate_Texts_in_Mathematics_7,_Springer-Verlag_New_York,_1973.F._Villegas,_Experimental_Number_Theory,_Oxford_Graduate_Texts_in_Mathematics,_Book_13,_2007.

 

 

 

 

Differential_Topology_(PM502)   _4_Credits

 

Prerequisites:_Linear_Algebra,_Topology,_Real_Analysis

 

    Quick_review_of_multivariable_calculus:_Implicit_and_inverse_function_theoremsQuick_review_of_separation_axioms_and_paracompactness:_First_and_second_countability,_separability,_normality,_complete_regularity,_Urysohn_lemma_and_paracompactnessTopological_manifolds:_Definition_and_some_basic_propertiesSmooth_manifolds:_Smooth_atlas,_smooth_structure,_smooth_manifoldsSmooth_maps:_Smooth_maps,_diffeomorphisms,_bump_functions,_partitions_of_unityTangent_space_and_the_differential:_Derivations,_tangent_space,_tangent_vectors_to_curves,_differential_of_a_smooth_mapVector_fields:_Tangent_bundle,_vector_fields_on_manifolds,_orientation_on_a_manifoldImmersions,_submersions_and_embeddings:_Implicit_and_inverse_function_theorem_for_manifolds,_submersions,_immersionsSubmanifolds:_Embedded_submanifolds_and_their_tangent_spaces,_regular_and_critical_points_and_values,_level_sets,_immersed_submanifoldsAdditional_Topics:_Smooth_manifolds_with_boundary_and_associated_definitions,_,_submanifolds_of_manifolds_with_boundary,_embedded_submanifolds_and_their_tangent_spaces,_Lie_brackets,_Sard's_theorem,_differential_forms_and_integration_on_manifolds,_Stokes'_theorem

 

Main_Text_Books:

 

    J._M._Lee,_Introduction_to_Smooth_Manifolds,_GTM,_Springer,_2006.K._Janich,_Vector_Analysis,_Undergraduate_Texts_in_Mathematics,_Springer,_2001.

 

Supplementary_References:

 

    M._Spivak,_Calculus_on_Manifolds:_A_modern_approach_to_classical_theorems_of_advanced_Calculus,_West_View_Press,_27­th_printing,_1998.V._Guillemin_and_A._Pollack,_Differential_Topology,_AMS_Chelsea_Publishing,_1974.F._W._Warner,_Foundations_of_Differential_Manifolds_and_Lie_Groups,_Graduate_Texts_in_Mathematics,_94,_First_edition,_Springer,_1983.W._Rudin,_Principles_of_Mathematical_Analysis,_Third_edition,_McGraw_Hill_Book_Company,_New_York,_1976.G._E._Bredon,_Topology_and_Geometry,_Graduate_Texts_in_Mathematics,_139,_Springer,_1993.A._A._Kosinski,_Differential_Manifolds,_Dover_Publications_Inc.,_2007.J._R._Munkres,_Elementary_Differential_Topology,_Revised_Edition,_Annals_of_Mathematics_Studies_(AM-54),_Princeton_University_Press,_1967.

 

 

 

 

 

 

 

 

 

Harmonic_Analysis_(PM503)   _4_Credits

 

Prerequisites:_Real_Analysis,_Complex_Analysis,_Measure_Theory

 

    Fourier_series:_Fourier_development_of_periodic_functions,_examples,_Dirichlet_and_Fejer_kernels,_convergence_of_Fourier_series,_Gibb's_phenomenon,_Parseval's_equationReview_of_measure_theory_and_integration:_Lebesgue_measure,_Lebesgue_integral,_Lp_spaceFourier_transform:_Motivation_and_definition,_examples,_Fourier_inversion_formula,_uniform_continuity_and_Riemann-Lebesgue_lemma,_Plancherel's_theorem,_Poisson_summation_formula,_convolution_theorem,_differentiation_of_Fourier_transforms,_Hermite_functions,_Laplace_transformApplications:_Filtering,_differential_equations,_central_limit_theoremAdditional_Topics:_Spherical_harmonic_analysis,_harmonic_analysis_on_topological_groups,_Pontryagin_duality

 

Main_Text_Book:

 

    E.M._Stein_and_G._Weiss,_Introduction_to_Fourier_analysis_on_Euclidean_spaces,_Princeton_Mathematical_Series,_No._32,_Princeton_University_Press,_Princeton,_N.J.,_1971.

 

Supplementary_References:

 

    Y._Katznelson,_An_introduction_to_harmonic_analysis,_Third_edition,_Cambridge_Mathematical_Library,_Cambridge_University_Press,_Cambridge,_2004.G._B._Folland,_Fourier_analysis_and_its_applications._The_Wadsworth_and_Brooks/Cole_Mathematics_Series,_Wadsworth_and_Brooks/Cole_Advanced_Books_and_Software,_1992.L._Grafakos,_Classical_Fourier_analysis,_Third_edition,_Graduate_Texts_in_Mathematics,_249,_Springer,_New_York,_2014.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Analytic_Number_Theory_(PM504)   _4_Credits

 

Prerequisites:_Complex_Analysis

 

    Arithmetic_functions_and_Dirichlet_series:_The_ring_of_arithmetic_functions,_Dirichlet_series,_important_arithmetic_functions,_average_estimatesCharacters:_Group_characters,_Dirichlet_characters,_detection_of_residue_classes,_Gauss_sumsPrime_number_distribution:_Infinitude_of_primes,_Chebyshev's_bounds,_Riemann_zeta_function,_Perron's_formula,_prime_number_theorem,_Dirichlet_L-functions,_primes_in_arithmetic_progressionsCircle_method:_General_set_up,_ternary_Goldbach_problem,_partitionsSieve_methods:_Selberg's_sieve,_large_sieve,_estimates_for_twin_primes,_estimates_for_twins_of_almost-primesAdditional_Topics:_Modular_forms,_exponential_sums

 

Main_Text_Book:

 

    T.M._Apostol,_Introduction_to_Analytic_Number_Theory,_Undergraduate_Texts_in_Mathematics._Springer-Verlag,_New_York-Heidelberg,_1976.

 

Supplementary_References:

 

    H._L._Montgomery,_R.C._Vaughan,_Multiplicative_number_theory_I_-_Classical_theory,_Cambridge_Studies_in_Advanced_Mathematics,_97,_Cambridge_University_Press,_Cambridge,_2007.H._Davenport,_Multiplicative_number_theory,_Third_edition,_Revised_and_with_a_preface_by_H._L._Montgomery,_Graduate_Texts_in_Mathematics,_74,_Springer-Verlag,_New_York,_2000.R._C._Vaughan,_The_Hardy-Littlewood_method,_Second_edition,_Cambridge_University_Press,_1997.A._C._Cojocaru,_M._R._Murty,_An_introduction_to_sieve_methods_and_their_applications,_London_Mathematical_Society_Student_Texts,_66,_Cambridge_University_Press,_Cambridge,_2006.S._W._Graham,_G._Kolesnik,_van_der_Corput's_method_of_exponential_sums,_ London_Mathematical_Society_Lecture_Note_Series,_126,_Cambridge_University_Press,_Cambridge,_1991.N._Koblitz,_Introduction_to_elliptic_curves_and_modular_forms,_Second_edition,_Graduate_Texts_in_Mathematics,_97,_Springer-Verlag,_New_York,_1993.

 

 

 

 

 

 

 

 

 

 

 

Proofs_(PM505)   _4_Credits

 

Prerequisites:_Algebra_I,_Real_Analysis,_Complex_Analysis,_Basic_Topology

 

Goal:_To_discuss_the_concept_of_a_proof_or_what_would_constitute_as_a_piece_of_an_evidence._To_introduce,_explain_and_discuss_proofs_in_mathematics,_computer_science_and_other_disciplines,_brief_discussion_about_kinds_of_proofs_using_examples,_The_concept/definition_of_zero_knowledge_proof_in_computer_science,_computer_assisted_proofs,_formal_proof._Another_important_goal_of_this_course_will_be_to_teach_the_students_how_to_write_mathematics_and_proofs_(using_examples_and_possibly_small_projects_as_assignments).

 

    What_is_a_proof_or_a_piece_of_evidence:_Proofs_in_different_walks_of_life_-_in_politics,_in_social_sciences,_in_history,_as_per_the_judiciary,_criminology,_in_biology,_medical_sciences,_weather_prediction,_in_physics_and_in_mathematics_-_illustrated_using_examples_in_brief.Proofs_in_mathematics:Language_of_proofs._Introduction_to_logicMethods_of_proving:_Numerical,_computational,_by_induction_(various_kinds),_by_contradiction_(reductio_ad_absurdum),_contrapositive,_types_of_proofs_-_direct,_indirect,_constructive,_non-constructiveHow_to_disprove:_how_to_construct_counterexamples?Proofs_in_computer_science:_The_concept_of_zero-knowledge_proof_in_brief_using_couple_of_the_following_or_similar_examples_(a)_Using_the_game_called_``where_is_Waldo''?_(b)_Graph_colouring_problem_and_it's_solution_by_Wigderson_et_al_(c)_``How_to_explain_zero_knowledge_protocols_to_your_children''._The_concept_of_a_proof_certificate_-_e.g._Pratt's_certificate_that_certifies_that_a_specific_large_number_is_a_prime_number_(based_on_Lucas's_theorem)._Size_and_complexity_of_a_proof.Computer_Assisted_Proofs:_One_or_two_examples_from_the_below_will_be_discussed_in_brief:_four_colour_theorem,_classification_of_finite_groups,_sphere_packing,_existence_of_Lorentz_attractorAdditional_Topics:_Role_of_proofs_in_mathematics:_Discussions_around_the_main_points_of_the_bulletin_of_AMS_article_of_W._Thurston,_errors_in_mathematical_research_papers,_role_of_speculation,_conjectures,_and_questions_in_the_progress_of_mathematics._Concept_of_a_formal_proof

 

Main_References:

 

    S._Krantz,_The_proof_is_in_the_pudding_-_The_changing_nature_of_mathematical_proof,_Springer-Verlag,_2011.Franklin_and_Daoud,_Proofs_in_Mathematics_-_An_Introduction,_Quakers_Hill_Press,_1996/Kew_Books,_2011.M._Aigner_and_G._Ziegler,_Proofs_from_THE_BOOK,_4th_edition,_Springer-Verlag,_2009.M._Petkovsek,_H._Wilf_and_D._Zeilberger,_A_=_B,_A_K_Peters/CRC_Press,_1996.T._C._Hales,_Formal_proof,_Notices_of_the_AMS,_Vol_55,_Number_11,_1370-1380_(and_references_therein).L._Lamport,_How_to_Write_a_21st_century_proof,_J._Fixed_Point_Theory_App._Vol._11,_Issue_1_(2012),_pp_43-63.B._Mazur,_The_faces_of_evidence_(in_Mathematics)},_Notes_for_the_presentation_and_discussion_at_Museion,_February_5,_2014.

 

Supplementary_References:

 

    A._Jaffe_and_F._Quinn,_``Theoretical_Mathematics'':_Toward_a_cultural_synthesis_of_mathematics_and_theoretical_physics,_Bulletin_of_the_AMS,_Vol._29,_No._1,_July_1993,_Pages_1-13.W._Thurston,_On_proof_and_progress_in_Mathematics,_Bull._of_the_AMS,_30_(1994),_161-177.T._C._Hales,_Jordan's_proof_of_the_Jordan_Curve_Theorem,_STUDIES_IN_LOGIC,_GRAMMAR_AND_RHETORIC_10_(23),_2007.B._Mazur,_Announcement_of_a_joint_undergraduate_course_(Law,_Harvard)_taught_by_Noah_Feldman_and_co-taught_by_B._Mazur:_``Nature_of_Evidence''.B._Mazur,_Shadows_of_Evidence_-_An_essay_on_science,_2013.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Advanced_Algebra_(PM506)   _4_Credits

 

Prerequisites:_Algebra_I,_Algebra_II

 

    Commutative_Algebra:_Projective_and_injective_modules,_projective_and_injective_resolutions,_functors,_chain_complex,_exact_sequences,_higher_derived_functors,_functorialityGroup_cohomology:_Ext_and_Tor,_group_cohomology,_group_extensions,_H1,_H2Central_simple_algebras:_Simple_modules,_Schur's_lemma,_semisimple_modules,_central_simple_algebras,_Wedderburn's_decomposition_theorem,_tensor_operationBrauer_group:}_Brauer_group_of_a_field,_relation_of_Brauer_group_to_Galois_cohomomology_of_the_field

 

Main_Text_Books:

 

    N._Jacobson,_Basic_Algebra_Vol._I_and_II,_Second_edition,_W._H._Freeman_and_Company,_1989.T._Y._Lam,_A_First_Course_in_Non-Commutative_Rings,_Second_edition,_Graduate_Texts_in_Mathematics,_131,_Springer-Verlag,_New_York,_2001.

 

Supplementary_References:

 

    N._S._Gopalkrishnan,_Commutative_Algebra,_Oxonian_Press_Pvt._Ltd.,_New_Delhi,_1984.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algebraic_Topology_(PM507)   _4_Credits

 

Prerequisites:_Basic_knowledge_of_Topology_and_Group_Theory

 

    The_Fundamental_group:_Homotopy_of_paths._Definition_of_the_fundamental_group,_covering_spaces,_the_fundamental_group_of_a_circle,_retractions,_Brouwer_fixed_point_theorem_for_a_discDeformation_Retracts_and_Homotopy_TypeFundamental_groups_of_n-spheres_and_some_surfacesThe_Seifert-van_Kampen_Theorem:_Free_products_of_groups,_free_groups,_Seifert-van_Kampen_theorem,_fundamental_groups_of_wedge_of_circles_and_of_toriClassification_of_Covering_spaces:_Covering_spaces,_equivalence_of_covering_spaces,_universal_covering_space,_covering_transformations,_existence_of_covering_spacesSingular_Homology:_Singular_complex,_singular_homology_groups,_homotopy_axiom,_Hurewicz_theorem_relating_the_fundamental_and_homology_groupsAdditional_Topics:_Brief_idea_about_simplicial_homology,_reduced_homology_groups,_homology_of_spheres

 

Main_Text_Books:

 

    J._R._Munkres,_Topology,_Second_Edition,_Pearson,_2000.J._J._Rotman,_An_Introduction_to_Algebraic_Topology,_Springer,_1988.

 

Supplementary_References:

 

    A._Hatcher,_Algebraic_Topology,_Cambridge_University_Press,_2002.G._E._Bredon,_Topology_and_Geometry,_Graduate_Texts_in_Mathematics,_139,_Springer,_1993.F._H._Croom,_Basic_concepts_of_Algebraic_Topology,_Undergraduate_Texts_in_Mathematics,_Springer,_1978.Anant_R._Shastri,_Basic_Algebraic_Topology,_First_edition,_Chapman_and_Hall/CRC,_2013.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Banach_and_Operator_Algebras_(PM508)   _4_Credits

 

Prerequisites:_Real_analysis,_Basic_Topology_and_Functional_Analysis

 

    Banach_algebras:_Banach_algebras,_ideals_and_quotients,_invertible_elements,_spectrum_and_spectral_radius,_Spectral_mapping_theorem,_Gelfand-Mazur_theorem,_commutative_Banach_algebras_and_their_Gelfand_representations,_holomorphic_functional_calculus,_quotients,_Stone-Weierstrass_theoremC*-algebras:_Banach_*-algebras_and_C*-algebras,_multiplier_algebra,_unitization,_Gelfand-Naimark_representation_of_commutative_C*-algebras,_functional_calculus,_spectral_mapping_theorem,_positive_elements_of_C*-algebrasOperators_on_Hilbert_spaces:_Spectrum_and_other_properties_of_normal,_self_adjoint,_projection_and_unitary_operators,_partial_isometry,_polar_decomposition,_finite-rank_and_compact_operators,_diagonalization,_Hilbert-Schmidt_operators,_trace-class_operatorsThe_Spectral_Theorem:_Spectral_measures,_spectral_theorem_for_normal_operatorsGelfand-Naimark_Representation:_Ideals_in_C*-algebras,_approximate_units,_quotients,_positive_linear_functionals,_Gelfand-Naimark_representation_of_C*-algebrasAdditional_Topics:_von_Neumann_algebras_-_Strong_and_weak_operator_topologies,_commutants,_von_Neumann_algebras,_double_commutant_theorem,_polar_decomposition,_projections,_Calkin_algebra,_pre-dual_of_a_von_Neumann_algebra,_Kaplansky_density_Theorem,_Abelian_von_Neumann_algebras

 

Main_Text_Book:

 

G._J._Murphy,_$C^*$-algebras_and_Operator_Theory,_Academic_Press_Inc.,_1990.

 

Supplementary_References:

    W._Arveson,_A_course_on_Spectral_Theory,_GTM,_Springer,_2002.R._G._Douglas,_Banach_algebra_techniques_in_Operator_Theory,_Second_Edition,_GTM,_Springer,_1998.J._Dixmier,_C*-algebras,_North-Holland_Publishing_Company,_1977.E._Kaniuth,_A_course_on_commutative_Banach_algebras,_Graduate_Texts_in_Mathematics,_Springer,_2009.M._Takesaki,_Theory_of_Operator_Algebras_I,_Springer,_2002.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Numerical_Analysis_(PM509)   _4_Credits

 

Prerequisites:_Basic_analysis_and_algebra.

 

    Quick_review_of_basic_numerical_computations_and_errors:_Floating_point_systems,_Exact_and_approximate_numbers,_Rounding_of_numbers,_Significant_digits,_accuracy,_and_various_types_of_errors_encountered_in_computations,_order,_and_rate_of_convergence._Roots_of_algebraic_and_transcendental_equations:_Direct_and_Iterative_methods_with_convergence_criteria_and_error_analysis.Solution_of_a_system_of_equations:_Direct_methods:_Pivoting,_Gauss_elimination_with_pivoting,_Tridiagonal_systems,_Thomas_algorithm,_Cholesky_factorizations,_SOR_methods._Eigen-value_problems:_Power_method,_Givens_method,_Jacobi_eigenvalue_method,_and_Householder_methods.Numerical_solutions_of_IVPs_and_BVPs:_Single-step_and_multistep_methods:_Euler’s_method,_Runge-Kutta_methods,_Adam-Bashforth_method,_Shooting,_and_finite_difference_methods.Solution_of_BVPs_by_Variational_Methods:_Introduction,_Variational_principle,_Ritz_Method,_Functional_for_a_differential_equation,_Galerkin’s_Method,_Collocation_Method.Solution_of_BVPs_by_Finite_Element_Method:_Introduction,_Ritz_finite_element_method,_Galerkin_finite_element_method,_Rayleigh-Ritz_Method,_and_application_on_BVPs.

 

Suggested_References:

 

Main_Textbooks:

    Richard_L._Burden_and_J._Douglas_Faires:_Numerical_Analysis,_Ninth_Edition,_Brooks/Cole,_Cengage_Learning,_2011.Kendall_E._Atkinson:_An_introduction_to_numerical_analysis,_Second_edition,_John_Wiley_and_Sons,_2008.G._D._Smith:_Numerical_solution_of_partial_differential_equations:_Finite_Difference_Methods,_Third_edition,_Oxford_University_Press,_1985.D._M._Causon,_C._G._Mingham,_and_L._Own:_Introductory_Finite_Volume_Methods_for_Partial_Differential_Equations,_Springer,_2009.

 

Supplementary_References:

    M._K._Jain,_S._R._K._Iyengar,_and_R._K._Jain:_Numerical_Methods_for_Scientific_and_Engineering_Computation._New_Age_International,_2012.R._S._Gupta,_Elements_of_Numerical_Analysis,_Macmillan_India_Ltd.,_2009.

 

 

 

 

 

 

 

 

 

 

 

 

Computational_Fluid_Dynamics_(PM510)   _4_Credits

 

Prerequisites:_Basic_Calculus_and_Analysis.

 

Unit_I:_Concept_of_fluids,_physical_properties_of_fluids,_continuum_hypothesis,_density,_specific_weight,_specific_volume,_kinematics_of_fluids.

Unit_II:_Eulerian_and_Lagrangian_method_of_description_of_flows,_equivalence_of_Eulerian_and_Lagrangian_methods,_streamline,_path_line,_streak_lines,_stream_function¸_vortex_lines,_rotational_and_irrotational_motion.

Unit_III:_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.

Unit_IV:_Mathematical_description_of_physical_phenomenon,_finite_control_volume,_Reynolds-transport_theorem,_Navier_Stokes_equation,_equation_of_moments_of_momentum,_equation_of_energy.

Unit_V:_Computational_techniques_to_solve_the_flow_models:_Non-dimensionalization,_Grid_formulation_and_appropriate_transformations,_Lax-Wendroff_Technique,_MacCormack's_Technique,_Quasi-linearization,_Spectral_methods,_Finite_difference_Method,_Finite_volume_Method_and_bvp4c,_bvp5c.

 

Practicals:_Practicals_will_be_performed_on_the_MATLAB_software.

           _Spectral_Method_to_solve_the_boundary_value_problem.

Finite_difference_Method_to_solve_the_boundary_value_problem.

Bvp4c_and_bvp5c_methods_to_solve_the_boundary_value_problem.

 

Suggested_References:

 

Main_Text_Books:

    R._H._Pletcher,_J._C._Tannehill_and_D._A._Anderson,_Computational_Fluid_Mechanics_and_Heat_Transfer,_CRC_Press,_Taylor_and_Francis,_2013.J._D._Anderson,_Computational_Fluid_Dynamics,_McGraw-Hill,_1995.S._V._Patankar,_Numerical_Heat_Transfer_and_Fluid_Flow,_CRC_Press,_Taylor_and_Francis,_Indian_Edition,_2017.

 

Supplementary_References:

    J._C._Strikwerda,_Finite_Difference_Schemes_and_Partial_Differential_Equations,_Second_Edition,_SIAM,_2004.J._W._Thomas,_Numerical_Partial_Differential_Equations:_Finite_Difference_Methods,_Springer,_2013.H._K._Versteeg,_and_W._Malalasekera,_An_Introduction_to_Computational_Fluid_Dynamics:_The_Finite_Volume_Method,_Second_Edition,_Pearson,_2008.S_K_Som,_Gautam_Biswas,_S_Chakraborty,_Introduction_to_Fluid_Mechanics_and_Fluid_Machines,_McGraw_Hill_Education,_3rd_Edition,_2017.C._A._J._Fletcher,_Computational_Techniques_for_Fluid_Dynamics:_Fundamental_and_General_Techniques:_001,_Springer_Verlag,_2nd_ed._1998.