Undergraduate Courses
Graduate Courses

2004
Calculus 1, Fall 2004
Analysis 1, Fall 2004

Instructor:
M. R. Pournaki
 
Textbook:
Gröbner Bases in Commutative Algebra by Viviana Ene and Jürgen Herzog

Click on the cover to view the whole book:

Syllabus:
Polynomial rings, Definition of the polynomial ring, Some basic properties of polynomial rings, Ideals, Operations on ideals, Residue class rings, Monomial ideals and Dickson's lemma, Operations on monomial ideals, Monomial orders, Examples and basic properties of monomial orders, Construction of monomial orders, Initial ideals and Gröbner bases, The basic definitions, Macaulay's theorem, Hilbert's basis theorem, The division algorithm, Buchberger's criterion, Buchberger's algorithm, Reduced Gröbner bases, Elimination of variables, Elimination orders, The elimination theorem, Applications to operations on ideals, Intersection of ideals, Ideal quotient, Saturation and radical membership, K-algebra homomorphisms, Homogenization, Zero dimensional ideals, Ideals of initial forms.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Syllabus:
Complex numbers, Limits and continuity, Differentiation, Transcendental functions, Some applications of derivatives, Integration, Techniques of integration, Applications of integration, Sequences, series, and power series.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking and its solutions by clicking .

Instructor:
M. R. Pournaki
 
Textbook (Under Preparation):
Homological Methods in Commutative Algebra by S. Yassemi and M. R. Pournaki



Syllabus:
Rings and modules of fractions, Support of a module, Primary decomposition, Associated primes, Krull dimension theory, System of parameters, Regular local rings, Regular sequences, Grade, Depth, Cohen-Macaulay rings, Complexes and homology, Free resolutions, Projective resolutions, Injective resolutions, Derived functors, Ext and Tor, Homological dimensions theory, Depth revisited, Auslander-Buchsbaum formula, Bass formula, Classification of regular rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking.
You may view the Midterm Exam 2 by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki
 
Textbook:
Introduction to Module Theory by S. Yassemi and M. R. Pournaki

Syllabus:
Modules and submodules, Homomorphisms and exact sequences, Products and sums of modules, Hom, Tensor, Category and functor, Free modules, Projective modules, Injective modules, Flat modules, Jacobson radical of rings, Chain conditions on modules, Noetherian and Artinian rings, Simple and semisimple modules, Semisimple and primitive rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Algebra 1 (22-217) (Fall 2015, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Hamid Aminian and Ramin Mousavi

Textbooks:
1) Algebra by Thomas W. Hungerford
2) Abstract Algebra by David S. Dummit and Richard M. Foote
3) Contemporary Abstract Algebra by Joseph A. Gallian

Click on the covers to view the whole books:

Syllabus:
Groups and important examples (quaternion group, Klein group, dihedral groups), Isomorphism, Subgroups, Cosets, Lagrange's theorem, Normal subgroups, Simple groups, Factor groups, Homomorphisms, The homomorphism theorems, Isomorphism theorems, Symmetric and alternating groups, Direct sums, Permutation groups, Cayley's theorem, Rings, Ideals, Prime and maximal ideals in commutative rings, Field of quotients.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Calculus 2 (22-016) (Spring 2015, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Amirhossein Ghodrati, Mohammadali Nematollahi and Mohammad Sabokdast
  
Textbook:
Vector Calculus by Jerrold E. Marsden and Anthony J. Tromba (5th Edition)

Syllabus:
The geometry of Euclidean space, Differentiation, Higher-order derivatives: maxima and minima, Vector-valued functions, Double and triple integrals, The change of variables formula and applications of integration, Integrals over paths and surfaces, The integral theorems of vector analysis.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Instructor:
M. R. Pournaki

TA:
Zahra Ghafouri
  
Textbook:
Calculus by S. Shahshahani, Vol. 1

Syllabus:
Real numbers, Completeness, Complex numbers, Topics include limits, continuity, derivatives, and linear approximation, Differential and elementary integral calculus of functions of one variable, The fundamental theorem of calculus, Elementary techniques of integration, Applications of integration, Differential equations, Sequences and series of numbers, Convergent sequences and series, Cauchy sequences, Taylor series.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Calculus 2 (22-016) (Spring 2014, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Niloofar Farajzadeh and Mehdi Shaeiri
  
Textbook:
Calculus by S. Shahshahani, Vol. 2

Syllabus:
Real n-dimensional space, Affine subspaces, Inner product and Euclidean geometry, Linear mappings, Volume and determinant, Smooth curves, Functions of several variables, Level sets, Continuity and limit, Linear approximation and the derivative, The gradient vector field, The chain rule, Higher order derivatives, Taylor polynomials, Critical points, Implicit functions, Inverse functions, Coordinate change, Optimization, Multiple integrals, Computing multiple integrals, Change of variables in multiple integrals, Smooth surfaces, Line and surface integrals, Vector analysis.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Instructor:
M. R. Pournaki

TAs:
Khashayar Filom and Ahmad Reza Haj-Saeedi Sadegh
  
Textbook:
Topology by James R. Munkres

Syllabus:
Topological spaces, Basis for a topology, The order topology, The product topology on X x Y, The subspace topology, Closed sets and limit points, Continuous functions, The product topology, The metric topology, The quotient topology, Connected spaces, Connected subspaces of the real line, Components and local connectedness, Compact spaces, Compact subspaces of the real line, Limit point compactness, Local compactness, The Tychonoff theorem, The one-point Alexandroff compactification, The countability axioms, The separation axioms, Normal spaces, The Urysohn lemma, The Urysohn metrization theorem, The Tietze extension theorem.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking.
You may view the Midterm Exam 2 by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki
 
Textbook (Under Preparation):
Homological Methods in Commutative Algebra by S. Yassemi and M. R. Pournaki



Syllabus:
Rings and modules of fractions, Support of a module, Primary decomposition, Associated primes, Krull dimension theory, System of parameters, Regular local rings, Regular sequences, Grade, Depth, Cohen-Macaulay rings, Complexes and homology, Free resolutions, Projective resolutions, Injective resolutions, Derived functors, Ext and Tor, Homological dimensions theory, Depth revisited, Auslander-Buchsbaum formula, Bass formula, Classification of regular rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki
 
Textbook (Under Preparation):
Real Analysis by M. Hesaaraki and M. R. Pournaki

Syllabus:
Lebesgue measure on the line, Lebesgue outer measure, Measurable sets, Nonmeasurable sets, Regularity properties, Measurable functions, Lusin's theorem, Egoroff's theorem, Littlewood's three principles, Convergence in measure, Lebesgue integration on the line, Bounded convergence theorem, Fatou's lemma, Monotone convergence theorem, Dominated convergence theorem, Differentiation, Vitali's covering lemma, Lebesgue-Young theorem, Functions of bounded variations, Derivatives of integrals, Absolute continuity, The fundamental theorems of calculus, Normed linear spaces, Hahn-Banach theorem, Banach spaces, Open mapping theorem, Closed graph theorem, Classical Banach spaces, L^p spaces, Hilbert spaces, Orthonormal sequences and bases, Fourier series, Riesz representation theorem, Sigma algebras, Borel sets, Measurable spaces, Measure spaces, Complete measure spaces, Sigma finite measure spaces, Outer measures, General measure and integration theory, Dynkin systems, Product measures, Fubini and Tonelli theorems, Signed measures, Hahn decomposition theorem, Jordan decomposition theorem, Lebesgue decomposition theorem, Radon-Nikodym theorem.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Syllabus:
Topics include limits, continuity, derivatives, and linear approximation, Differential and elementary integral calculus of functions of one variable, The fundamental theorem of calculus, Elementary techniques of integration, Applications of integration, Sequences and series of numbers, Convergent sequences and series, Cauchy sequences, Taylor series.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki
 
Textbook (Under Preparation):
Homological Methods in Commutative Algebra by S. Yassemi and M. R. Pournaki

Syllabus:
Rings and modules of fractions, Support of a module, Primary decomposition, Associated primes, Krull dimension theory, System of parameters, Regular local rings, Regular sequences, Grade, Depth, Cohen-Macaulay rings, Complexes and homology, Free resolutions, Projective resolutions, Injective resolutions, Derived functors, Ext and Tor, Homological dimensions theory, Auslander-Buchsbaum formula, Classification of regular rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Instructor:
M. R. Pournaki

Instructor:
M. R. Pournaki
 
Textbook:
Introduction to Module Theory by S. Yassemi and M. R. Pournaki

Syllabus:
Modules and submodules, Homomorphisms and exact sequences, Products and sums of modules, Hom, Tensor, Category and functor, Free modules, Projective modules, Injective modules, Flat modules, Jacobson radical of rings, Chain conditions on modules, Noetherian and Artinian rings, Simple and semisimple modules, Semisimple and primitive rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki
  
Textbook:
Fields and Galois Theory by John M. Howie

Click on the cover to view the whole book:

Syllabus:
Rings and fields, The field of fractions of an integral domain, The characteristic of a field, Prime fields, Integral domains and polynomials, ED's, PID's, UFD's, Polynomial rings, Finite and infinite extensions, Simple and finitely generated extensions, Algebraic and transcendental extensions, Constructions with straight edge and compass, Splitting fields, Finite fields, Normal extensions, Separable extensions, Galois extensions, Galois groups, Solvability of algebraic equations.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Instructor:
M. R. Pournaki

TAs:
Khashayar Filom, Nima Moeini and Peyman Morteza
  
Textbook:
Real Mathematical Analysis by Charles C. Pugh

Click on the cover to view the whole book:

Syllabus:
Metric space concepts, Convergent sequences and subsequences, Continuity, Homeomorphism, Closed sets, Open sets, Open subsets of R, Topological description of continuity, Closure, Interior, Boundary, Inheritance, Clustering, Condensing, Product metrics, Continuity of arithmetic in R, Cauchy sequences, Boundedness, Compactness, Nests of compacts, Continuity and compactness, Homeomorphisms and compactness, Absolute closedness, Uniform continuity and compactness, Connectedness, Coverings, Total boundedness, Perfect metric spaces, Cantor sets, Completion, Differentiation, The rules of differentiation, Higher derivatives, Smoothness classes, Analytic functions, Taylor approximation, Inverse functions, Riemann integration, Darboux integrability, Uniform convergence and function spaces, Nowhere differentiable continuous functions, Power series, Compactness and equicontinuity in the function spaces, Uniform approximation in the function spaces.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking and its solutions by clicking .

Instructor:
M. R. Pournaki

Textbooks:
1) Commutative Ring Theory by H. Matsumura
2) Cohen-Macaulay Rings by W. Bruns and J. Herzog

Click on the left cover to view the whole book:

Syllabus:
Graded rings and modules, Artin-Rees lemma, Inverse systems, Inverse limits, Completion, Krull intersection theorem, The Hilbert function, The Hilbert-Samuel function, Dimension theory revisited, Regular sequences, Grade, Depth, Auslander-Buchsbaum formula, Cohen-Macaulay rings, Minimal injective resolutions, Bass numbers, Gorenstein rings, Complete intersection rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Engineering Mathematics (22-035) (Fall 2010, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Mohsen Karmozdi, Kayvan Yaghmaei and Mohsen Yousefnezhad

Textbooks:
1) Complex Functions by M. Hesaaraki and M. R. Pournaki
2) Differential Equations and Their Applications by M. Braun

Click on the right cover to view the whole book:

Syllabus:
Complex Functions: Structure of complex numbers, Sequences of complex numbers, Topology of the complex plane, Compactness, Path-connectedness, Domains, Complex functions, Limits, Continuity, Sequences of complex functions, Cauchy-Riemann equations, Analytic functions, Harmonic functions, Exponential functions, Logarithmic functions, Trigonometric functions, Roots, Branches, Conformal maps, Linear fractional transformations, Line integrals, Cauchy-Goursat theorem, Cauchy integral formula, Morera's theorem, Maximum modulus principle, Fundamental theorem of algebra, Series of complex numbers and functions, Uniform convergence, Taylor's series, Laurent's series, Isolated singularities, Residues, Evaluation of integrals of real-valued functions.

Partial Differential Equations: Two point boundary value problems, Introduction to partial differential equations, The heat equation, Separation of variables, Periodic functions, Fourier series, Fourier coefficients, Fourier cosine and sine series, Parseval formula, The heat equation revisited, The wave equation, Laplace's equation, Fourier integrals, Partial differential equations on unbounded domains, Sturm-Liouville theory.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking and its solutions by clicking .

Instructor:
M. R. Pournaki

Textbooks:
1) Local Cohomology: An Algebraic Introduction ... by M. P. Brodmann and R. Y. Sharp
2) An Introduction to Homological Algebra by Joseph J. Rotman

Click on the right cover to view the whole book:

Syllabus:
Local cohomology functors, Local cohomology modules, Direct and inverse systems, Direct limits, Local cohomology modules as direct limits of Ext modules, Torsion modules, Ideal transforms, Some vanishing theorems, Mayer-Vietoris sequence and its applications, Arithmetic rank, Some acyclic modules, The independence theorem, A non-vanishing theorem, The flat base change theorem, Use of Čech complexes, Use of Koszul complexes, Grothendieck's vanishing theorem, Introduction to spectral sequences.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Instructor:
M. R. Pournaki

TAs:
Khashayar Filom and Peyman Morteza
  
Textbook:
A Taste of Topology by Volker Runde

Click on the cover to view the whole book:

Syllabus:
Metric spaces, Examples of metric spaces, Open and closed sets, Convergence and continuity, Completeness, Cantor's intersection theorem, Completion, Bourbaki's Mittag-Leffler theorem, Baire's theorem, Compactness for metric spaces, Bolzano-Weierstraß property, Heine-Borel theorem, Topological spaces, Examples of topological spaces, Zariski topology, Continuity and convergence of nets, Compactness, Tychonoff's theorem, Locally compact spaces, Alexandroff one-point compactification, Connectedness and path connectedness, Intermediate value theorem, Locally connected spaces, Locally path connected spaces, Totally disconnected spaces, Separation properties, Normal spaces, Completely regular spaces, Urysohn's lemma, Urysohn's metrization theorem, Tietze's extension theorem.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Instructor:
M. R. Pournaki

Textbook:
An Elementary Approach to Homological Algebra by L. R. Vermani

Syllabus:
Homology of complexes, Ker-Coker sequence, Connecting homomorphism: the general case, Homotopy, Free resolutions, Projective resolutions, Injective resolutions, Derived functors, Torsion functors, Extension functors, Some further properties of Tor and Ext, Hereditary rings, Dedekind domains and rings, Prüfer rings, Universal coefficient theorem for homology, Universal coefficient theorem for cohomology, The Künneth formula, Homological dimensions theory, Dimensions of modules and rings, Global dimension of rings, Global dimension of Noetherian and Artinian rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Differential Equations (22-034) (Fall 2009, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Farzaneh Jannat, Ahmad Mousavi and Farnaz Salehi
  
Textbook:
Differential Equations and Their Applications by Martin Braun

Click on the cover to view the whole book:

Syllabus:
Ordinary differential equations and mathematical models, First-order linear equations, Separable equations, Exact equations, Picard's existence and uniqueness theorem, Second-order linear differential equations, The method of variation of parameters, The method of judicious guessing, Series solutions, The method of Laplace transforms and its applications, Higher-order equations, Systems of differential equations, Stability of linear systems, Stability of equilibrium solutions, The phase-plane.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Syllabus:
Linear differential equations, The need for qualitative analysis, Existence and uniqueness, Dependence on initial data and parameters, Maximal interval of existence, Fixed point method, General nonhomogeneous linear equations, Linear equations with constant coefficients, Periodic coefficients and Floquet theory, Linear autonomous equations in the plane, Perturbations on linear equations in the plane.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki
  
Textbook:
Steps in Commutative Algebra by Rodney Y. Sharp

Click on the cover to view the whole book:

Syllabus:
Commutative rings and subrings, Ideals, Maximal ideals, Prime ideals, Minimal prime ideals, Rings and modules of fractions, Support of a module, Primary decomposition, Associated primes, Integral extensions, Krull dimension theory.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki

TA:
Amin Fakhari
  
Textbooks:
1) Algebra by Thomas W. Hungerford
2) A First Course in Rings and Ideals by David M. Burton

Click on the cover to view the whole book:

Syllabus:
The action of a group on a set, The Sylow theorems, Classification of finite groups, Introductory concepts in rings, Ideals and their operations, The classical isomorphism theorems, Integral domains and fields, Maximal, prime and primary ideals, Divisibility theory in integral domains, Polynomial rings, PID's, UFD's, ED's.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Engineering Mathematics (22-035) (Fall 2008, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Negin Bagherpour, Amin Fakhari, Asmaa Hasannejad and Akram Sheikhalishahi

Textbooks:
1) Partial Differential Equations of Mathematical Physics by Tyn Myint-U
2) Complex Variables and Applications by R. V. Churchill, J. W. Brown and R. F. Verhey

Syllabus:
Partial Differential Equations: Periodic functions, Fourier series, Fourier coefficients, Fourier cosine and sine series, Parseval formula, Fourier integral, Fourier transforms, What is a PDE? Initial and boundary conditions, Separable partial differential equations, Classical equations and boundary value problems, Wave equation, Heat equation, Laplace's equation.

Complex Functions: Structure of complex numbers, Sequences of complex numbers, Topology of the complex plane, Compactness, Path-connectedness, Domains, Complex functions, Limits, Continuity, Sequences of complex functions, Cauchy-Riemann equations, Analytic functions, Harmonic functions, Exponential functions, Logarithmic functions, Trigonometric functions, Roots, Branches, Conformal maps, Linear fractional transformations, Line integrals, Cauchy-Goursat theorem, Cauchy integral formula, Morera's theorem, Maximum modulus principle, Series of complex numbers and functions, Uniform convergence, Taylor's series, Laurent's series, Isolated singularities, Residues, Evaluation of integrals of real-valued functions.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking and its solutions by clicking .

Algebra 1 (22-217) (Fall 2008, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Amin Fakhari
  
Textbook:
Contemporary Abstract Algebra by Joseph A. Gallian

Click on the cover to view the whole book:

Syllabus:
Groups and important examples (quaternion group, Klein group, dihedral groups), Subgroups, Center, Cosets, Lagrange's theorem, Normal subgroups, Factor groups, Symmetric and alternating groups, Conjugacy classes, Homomorphisms, The homomorphism theorems, Isomorphism, Isomorphism theorems, Cayley's theorem, Direct sums, Fundamental theorem on finite abelian groups, Automorphism groups, Inner and outer automorphisms, Commutator subgroup, Rings, Ideals, Prime and maximal ideals in commutative rings, Field of quotients.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki

TA:
Mohammad Safdari
  
Textbook:
Complex Functions by M. Hesaaraki and M. R. Pournaki

Syllabus:
Structure of complex numbers, Sequences of complex numbers, Topology of the complex plane, Compactness, Path-connectedness, Domains, Complex functions, Limits, Continuity, Sequences of complex functions, Cauchy-Riemann equations, Analytic functions, Harmonic functions, Exponential functions, Logarithmic functions, Trigonometric functions, Roots, Branches, Conformal maps, Linear fractional transformations, Line integrals, Cauchy-Goursat theorem, Cauchy integral formula, Morera's theorem, Maximum modulus principle, Series of complex numbers and functions, Uniform convergence, Taylor's series, Laurent's series, Isolated singularities, Residues, Evaluation of integrals of real-valued functions.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking and its solutions by clicking .

Instructor:
M. R. Pournaki
  
Textbook:
Linear Groups by M. R. Darafsheh

Syllabus:
Group actions, Finite fields, Projective geometry, General linear groups, Projective line, Transvections, Permutation groups, Simplicity of PSL_n(F), Symplectic groups, Witt's theorem, Symplectic transvections, Simplicity of projective symplectic groups, Quasi-bilinear and quadratic forms, Finite unitary groups, Finite orthogonal groups.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki

TA:
Asmaa Hasannejad
  
Textbook:
Linear Algebra by S. H. Friedberg, A. J. Insel and L. E. Spence

Syllabus:
Vector spaces, Subspaces, Linear combinations and systems of linear equations, Linear dependence and linear independence, Bases and dimension, Maximal linearly independent subsets, Linear transformations, Null spaces, Ranges, The matrix representation of a linear transformation, Composition of linear transformations and matrix multiplication, Invertibility and isomorphisms, The change of coordinate matrix, Dual spaces, Elementary matrix operations and elementary matrices, The rank of a matrix and matrix inverses, Systems of linear equations, Determinants, Properties of determinants, Eigenvalues, Eigenvectors, Diagonalizability, Invariant subspaces, The Cayley-Hamilton theorem, The minimal polynomial, Triangular forms, Jordan canonical forms, Inner products and norms, The Gram-Schmidt orthogonalization process.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking .
You may view the Final Exam by clicking and its solutions by clicking .

Instructor:
M. R. Pournaki
  
Textbook:
Steps in Commutative Algebra by Rodney Y. Sharp

Click on the cover to view the whole book:

Syllabus:
Commutative rings and subrings, Ideals, Maximal ideals, Prime ideals, Minimal prime ideals, Rings and modules of fractions, Support of a module, Primary decomposition, Associated primes, Integral extensions, Krull dimension theory.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki

Syllabus:
Ordinary differential equations, Mathematical models, First-order equations, Picard's existence theorem, Second-order linear equations, Oscillation, Separation theorems, Solution in series, Legendre and Bessel functions, Systems of equations, Stability theory, The Laplace transform and its applications.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Instructor:
M. R. Pournaki
 
Textbook:
Linear Functional Analysis by B. P. Rynne and M. A. Youngson

Click on the cover to view the whole book:

Syllabus:
Normed spaces, Banach spaces, Inner product spaces, Hilbert spaces, Linear operators, Linear operators on Hilbert spaces, Compact operators, Integral and differential equations.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki

Textbook:
An Elementary Approach to Homological Algebra by L. R. Vermani

Syllabus:
Homology of complexes, Ker-Coker sequence, Connecting homomorphism: the general case, Homotopy, Free resolutions, Projective resolutions, Injective resolutions, Derived functors, Torsion functors, Extension functors, Some further properties of Tor and Ext, Homological dimensions theory, Dimensions of modules and rings, Global dimension of rings, Global dimension of Noetherian and Artinian rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking.
You may view the Final Exam by clicking.

Instructor:
M. R. Pournaki

TAs:
Roozbeh Tabrizian, Alireza Vahid and Ehsan Varyani
  
Textbook:
Differential Equations by D. Lomen and J. Mark

Syllabus:
Basic concepts of differential equations, First-order differential equations, Linear first-order differential equations, Higher order linear differential equations, Systems of linear differential equations, Series solutions, Laplace transforms.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Instructor:
M. R. Pournaki

TA:
Majid Hadian
  
Textbook:
Introduction to Module Theory by S. Yassemi and M. R. Pournaki

Syllabus:
Modules and submodules, Homomorphisms and exact sequences, Products and sums of modules, Hom, Tensor, Category and functor, Free modules, Projective modules, Injective modules, Flat modules, Jacobson radical of rings, Chain conditions on modules, Noetherian and Artinian rings, Simple and semisimple modules, Semisimple and primitive rings.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Take Home Exam by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Instructor:
M. R. Pournaki

TA:
Hassan Dadashi
  
Textbooks:
1) Partial Differential Equations of Mathematical Physics by Tyn Myint-U
2) Complex Variables and Applications by R. V. Churchill, J. W. Brown and R. F. Verhey

Syllabus:
Partial Differential Equations: Periodic functions, Fourier series, Fourier coefficients, Fourier cosine and sine series, Parseval formula, Fourier integral, Fourier transforms, What is a PDE? Initial and boundary conditions, Separable partial differential equations, Classical equations and boundary value problems, Wave equation, Heat equation, Laplace's equation.

Complex Functions: Structure of complex numbers, Sequences of complex numbers, Topology of the complex plane, Compactness, Path-connectedness, Domains, Complex functions, Limits, Continuity, Sequences of complex functions, Cauchy-Riemann equations, Analytic functions, Harmonic functions, Exponential functions, Logarithmic functions, Trigonometric functions, Roots, Branches, Conformal maps, Linear fractional transformations, Line integrals, Cauchy-Goursat theorem, Cauchy integral formula, Morera's theorem, Maximum modulus principle, Series of complex numbers and functions, Uniform convergence, Taylor's series, Laurent's series, Isolated singularities, Residues, Evaluation of integrals of real-valued functions.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

Instructor:
M. R. Pournaki
  
Textbook:
The Theory of Finite Groups, An Introduction by H. Kurzweil and B. Stellmacher

Syllabus:
Groups, Subgroups, Homomorphisms, Normal subgroups, Automorphisms, Cyclic groups, Commutators, Products of groups, Minimal normal subgroups, Composition series, The structure of abelian groups, Automorphisms of cyclic groups, Group actions, Sylow’s theorem, Complements of normal subgroups, Permutation groups, Transitive groups, Frobenius groups, Primitive action, The symmetric group, Imprimitive groups, Wreath products.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Calculus 2 (22-016) (Spring 2005, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Ali Mohammadian and Ramin Okhrati
  
Textbook:
Calculus by J. Stewart

Click on the cover to view the whole book:

Lecture notes (taken by S. Shahshahani):

Linear Algebra:

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

Smooth Curves:

12)

13)

14)

Functions of Several Variables:

15)

16)

17)

Continuity and Limit:

18)

19)

Differentiation:

20)

21)

22)

23)

24)

25)

26)

27)

28)

29)

30)

31)

Multiple Integrals:

32)

33)

34)

Line and Surface Integrals:

35)

36)

37)

38)

Vector Analysis:

39)

40)

41)

Syllabus:
Real n-dimensional space, Affine subspaces, Inner product and Euclidean geometry, Linear mappings, Volume and determinant, Smooth curves, Functions of several variables, Level sets, Continuity and limit, Linear approximation and the derivative, The gradient vector field, The chain rule, Higher order derivatives, Taylor polynomials, Critical points, Implicit functions, Inverse functions, Coordinate change, Optimization, Multiple integrals, Computing multiple integrals, Change of variables in multiple integrals, Smooth surfaces, Line and surface integrals, Vector analysis.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Calculus 1 (22-015) (Fall 2004, Sharif University of Technology)

Instructor:
M. R. Pournaki

TAs:
Asghar Bahmani and Azam Heydari
  
Textbook:
Calculus by J. Stewart

Click on the cover to view the whole book:

Lecture notes (taken by S. Shahshahani):

Real and Complex Numbers:

1)

2)

3)

4)

5)

Sequences and Series of Numbers:

6)

7)

8)

Continuity:

9)

10)

11)

12)

13)

Differentiation:

14)

15)

16)

17)

18)

19)

20)

Integration:

21)

22)

23)

24)

25)

26)

27)

Taylor Series:

28)

29)

30)

31)

Syllabus:
Real numbers, Completeness, Complex numbers, Topics include limits, continuity, derivatives, and linear approximation, Differential and elementary integral calculus of functions of one variable, The fundamental theorem of calculus, Elementary techniques of integration, Applications of integration, Sequences and series of numbers, Convergent sequences and series, Cauchy sequences, Taylor series.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking and its solutions by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analysis 1 (22-325) (Fall 2004, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Hassan Dadashi
  
Textbooks:
1) Methods of Real Analysis by Richard R. Goldberg
2) Principles of Mathematical Analysis by Walter Rudin

Syllabus:
Introduction to metric spaces and important examples, Normed linear spaces and important examples, Brief excursion into topology, Open and closed sets, Convergence in metric spaces, Limits and metric spaces, Continuous functions on metric spaces, Uniform continuity, Connectedness, Completeness, Compactness, Heine-Borel theorem, Bolzano-Weierstrass theorem, Differentiation of functions of one real variable, Theory of Riemann integration, Sequences and series of functions, Uniform convergence, Ascoli's theorem, The Stone-Weierstrass theorem.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analysis 1 (22-325) (Spring 2000, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Mohsen Bahramgiri

Textbooks:
1) Methods of Real Analysis by Richard R. Goldberg
2) Principles of Mathematical Analysis by Walter Rudin

Syllabus:
Sequences of real numbers, Convergent sequences, Cauchy sequences, Completeness, Lim-sup, Lim-inf, Series of real numbers, Introduction to metric spaces and important examples, Normed linear spaces and important examples, Brief excursion into topology, Open and closed sets, Convergence in metric spaces, Limits and metric spaces, Continuous functions on metric spaces, Uniform continuity, Connectedness, Completeness, Compactness, Heine-Borel theorem, Bolzano-Weierstrass theorem, Differentiation of functions of one real variable.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Algebra 1 (22-217) (Fall 1999, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Mohsen Bahramgiri

Textbook:
Contemporary Abstract Algebra by Joseph A. Gallian

Click on the cover to view the whole book:

Syllabus:
Groups and important examples (quaternion group, Klein group, dihedral groups), Subgroups, Center, Cosets, Lagrange's theorem, Normal subgroups, Factor groups, Symmetric and alternating groups, Conjugacy classes, Homomorphisms, The homomorphism theorems, Isomorphism, Isomorphism theorems, Cayley's theorem, Direct sums, Fundamental theorem on finitely generated abelian groups, Automorphism groups, Inner and outer automorphisms, Commutator subgroup, Rings, Ideals, Prime and maximal ideals in commutative rings, Field of quotients, Polynomial rings, PID's, UFD's, ED's.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Linear Algebra 2 (6103-193) (Fall 1998, University of Tehran)

Instructor:
M. R. Pournaki

TA:
Hadi Jorati

Textbook:
Linear Algebra by K. Hoffman and R. Kunze

Syllabus:
Eigenvectors, Eigenvalues, Characteristic polynomial, Minimal polynomial, The Cayley-Hamilton theorem, Direct sums of subspaces, Primary decomposition theorem, Invariant subspaces, Elementary divisors, Nilpotent operators, Reduction to the triangular forms, Reduction to the Jordan canonical forms, Reduction to the rational canonical forms, Reduction to the classical forms, Inner product spaces, Orthogonal direct sums, Orthonormal basis, Gram-Schmidt theorem, The Riesz representation theorem, The spectral theorem, Bilinear and quadratic forms, Real Normality.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Algebra 2 (22-218) (Fall 1998, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Keivan Mallahi-Karai
  
Textbooks:
1) Algebra by Thomas W. Hungerford
2) Basic Abstract Algebra by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul

Click on the covers to view the whole books:

       

Syllabus:
Rings, Ideals, Prime and maximal ideals in commutative rings, Field of quotients, Polynomial rings, Matrix rings, Boolean rings, PID's, UFD's, ED's, Rings with chain conditions, Hilbert basis theorem, Prime fields, Finite and infinite extensions, Simple and finitely generated extensions, Algebraic and transcendental extensions, Algebraically closed fields, Normal extensions, Finite fields, Galois extensions, Cyclic extensions, Galois groups, Solvability of algebraic equations, Constructions with straight edge and compass.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Algebra 1 (22-217) (Spring 1998, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Keivan Mallahi-Karai

Textbook:
Algebra by Thomas W. Hungerford

Click on the cover to view the whole book:

 

Syllabus:
Groups and important examples (quaternion group, Klein group, dihedral groups), Isomorphism, Subgroups, Cosets, Lagrange's theorem, Normal subgroups, Simple groups, Factor groups, Homomorphisms, The homomorphism theorems, Isomorphism theorems, Symmetric and alternating groups, Direct sums, Free abelian groups, Fundamental theorem on finitely generated abelian groups, Group actions, Orbits, Stabilizers, Permutation groups, Conjugacy classes, Center, Cayley's theorem, Automorphism groups, Inner and outer automorphisms, Sylow theorems, Commutator subgroup,  Schreier's refinement theorem, Jordan-Hِlder theorem, Solvable groups, Nilpotent groups. 

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Number Theory (22-215) (Fall 1997, Sharif University of Technology)  

Instructor:
M. R. Pournaki

Textbooks:
1) Introduction to Number Theory by W. W. Adams and L. J. Goldstein
2) Introduction to Analytic Number Theory by Tom M. Apostol

Click on the cover to view the whole book:

Syllabus:
Basic properties of the integers, The greatest common divisor, The least common multiple, The Euclidean algorithm, Primes, The fundamental theorem of arithmetic, Linear Diophantine equations, Modular arithmetic, Fermat's little theorem, The Chinese remainder theorem, Roots of polynomial congruences, Primitive roots, Power residues, Legendre's symbols, The law of quadratic reciprocity, Jacobi's symbol, Number theoretic functions, Distribution of primes, The prime number theorem, Pythagorean triples and Fermat's last theorem, Sums of two squares, Pell's equation.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analysis 2 (22-326) (Fall 1996, Sharif University of Technology)

Instructor:
M. R. Pournaki

TA:
Kamran Reihani

Textbooks:
1) Principles of Mathematical Analysis by Walter Rudin
2) Mathematical Analysis by Tom M. Apostol
3) A Second Course in Mathematical Analysis by J. C. Burkill and H. Burkill

Click on the cover to view the whole book:

Syllabus:
Theory of Riemann integration, Functions of bounded variation, Theory of Riemann-Stieltjes integration, Sequences and series of functions, Uniform convergence, Ascoli's theorem, The Stone-Weierstrass theorem, Taylor series, Power Series, Fundamental theorem of algebra, Fourier series and Fourier integrals, The L_2-space as a complete inner product space, Basic Hilbert space theory, Orthonormal sets, Bessel, Gram-Schmidt, Riesz-Fischer, Riemann-Lebesgue and Parseval's results, Periodic functions, Dirichlet and Fejer's kernels, Cesaro summability, Denseness of trigonometric polynomials in L_2 (I) and in L_1(I).

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Number Theory (22-215) (Fall 1995, Sharif University of Technology)

Instructor:
M. R. Pournaki

Textbook:
Introduction to Number Theory by W. W. Adams and L. J. Goldstein

Syllabus:
Basic properties of the integers, The greatest common divisor, The least common multiple, The Euclidean algorithm, Primes, The fundamental theorem of arithmetic, Linear Diophantine equations, Modular arithmetic, Fermat's little theorem, The Chinese remainder theorem, Roots of polynomial congruences, Primitive roots, Power residues, The law of quadratic reciprocity, Number theoretic functions, Pythagorean triples and Fermat's last theorem, Sums of two squares, Pell's equation.

Course Schedule and Exams:
You may view the Time Table by clicking.
You may view the Midterm Exam 1 by clicking and its solutions by clicking.
You may view the Midterm Exam 2 by clicking and its solutions by clicking.
You may view the Final Exam by clicking.