Undergraduate Courses
Graduate Courses
Undergraduate Courses
You may view more
detailed information by clicking the following buttons.
2016
Calculus 1, Fall 2016
2015
Algebra 1, Fall 2015
Calculus 2, Spring 2015
2014
Introductory Algebraic Topology, Fall 2014
Calculus 1, Spring 2014
Calculus 2, Spring 2014
2013
Topology 1, Fall 2013
2012
Calculus 1, Fall 2012
Analysis 2, Spring 2012
2011
Algebra 3, Fall 2011
Analysis 1, Spring 2011
2010
Engineering Mathematics, Fall 2010
Topology 1, Spring 2010
2009
Differential Equations, Fall 2009
Elementary Theory of Ordinary Differential Equations, Fall 2009
Algebra 2, Spring 2009
2008
Engineering Mathematics,
Fall 2008
Algebra 1, Fall 2008
Complex Functions 1, Spring 2008
2007
Linear Algebra 1, Fall 2007
2006
Differential Equations, Fall 2006
Elementary Functional Analysis, Fall 2006
Differential Equations, Spring 2006
2005
Engineering Mathematics, Fall 2005
Calculus 2, Spring 2005
2004
Calculus 1, Fall 2004
Analysis 1, Fall 2004
2000
Analysis 1, Spring 2000
1999
Algebra 1, Fall 1999
1998
Linear Algebra 2, Fall 1998
Algebra 2, Fall 1998
Algebra 1, Spring 1998
1997
Number Theory, Fall 1997
1996
Analysis 2, Fall 1996
1995
Number Theory, Fall 1995
Graduate Courses
You may view more detailed information by
clicking the following buttons.
2016
Topics in Algebra (Gröbner Bases Theory),
Fall 2016
Homological Methods in Commutative Algebra,
Spring 2016
2015
Advanced Algebra, Fall 2015
2014
Real Analysis, Fall 2014
2013
Homological Methods in Commutative Algebra,
Spring 2013
2012
Real Analysis, Fall 2012
Homological Methods in Commutative Algebra,
Spring 2012
2011
Advanced Algebra, Fall 2011
2010
Commutative Algebra 2, Fall 2010
Homological Algebra 2, Spring 2010
2009
Homological Algebra, Fall 2009
Commutative Algebra, Spring 2009
2007
Linear Groups, Fall 2007
2006
Commutative Algebra, Fall 2006
Homological Algebra, Spring 2006
2005
Algebra 3, Fall 2005
Finite Groups, Spring 2005
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Topics in Algebra (Gröbner Bases Theory)
(22045+) (Fall 2016,
Sharif University of Technology)
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, Kalgebra
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Calculus 1 (22015) (Fall 2016,
Sharif University of Technology)
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here
to switch to the Sharif Calculus home page!
Instructor:
M. R.
Pournaki
TA:
To be announced.
Textbook:
Calculus: A Complete Course by Robert A. Adams (5th Edition)
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 .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Homological Methods in Commutative Algebra
(22252+) (Spring 2016,
Sharif University of Technology)
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, CohenMacaulay rings,
Complexes and homology, Free resolutions, Projective resolutions, Injective
resolutions, Derived functors, Ext and Tor, Homological dimensions theory, Depth revisited,
AuslanderBuchsbaum 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Advanced Algebra
(22226+) (Fall 2015,
Sharif University of Technology)
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 (22217) (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 (22016) (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, Higherorder derivatives:
maxima and minima, Vectorvalued 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Real Analysis
(22412+) (Fall 2014,
Sharif University of Technology)
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,
LebesgueYoung theorem, Functions of bounded variations, Derivatives of integrals, Absolute continuity, The fundamental theorems of
calculus, Normed linear spaces, HahnBanach 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, RadonNikodym 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Introductory Algebraic Topology
(22565) (Fall
2014,
Sharif University of Technology)
Instructor:
M. R. Pournaki
Textbook:
Introduction to Topology
by Theodore W. Gamelin and Robert E. Greene
Syllabus:
Homotopic paths, The fundamental group, Induced homomorphisms,
Covering spaces, Seifertvan Kampen theorem, Some applications of the index, Brouwer fixed point theorem, BorsukUlam theorem,
Ham sandwich theorem, Homotopic maps, Maps into the punctured plane, The fundamental theorem of algebra, Vector fields, Hairy ball
theorem, Jordan curve theorem, Higher homotopy groups, The EckmannHilton argument, Noncontractibility of S^n.
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 1 (22015) (Spring 2014,
Sharif University of Technology)
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 (22016) (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 ndimensional 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Topology 1
(22556) (Fall
2013,
Sharif University of Technology)
Instructor:
M. R. Pournaki
TAs:
Khashayar Filom and Ahmad Reza HajSaeedi 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 onepoint
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Homological Methods in Commutative Algebra
(22252+) (Spring 2013,
Sharif University of Technology)
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, CohenMacaulay rings,
Complexes and homology, Free resolutions, Projective resolutions, Injective
resolutions, Derived functors, Ext and Tor, Homological dimensions theory, Depth revisited,
AuslanderBuchsbaum 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Real Analysis
(22412+) (Fall 2012,
Sharif University of Technology)
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,
LebesgueYoung theorem, Functions of bounded variations, Derivatives of integrals, Absolute continuity, The fundamental theorems of
calculus, Normed linear spaces, HahnBanach 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, RadonNikodym 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Calculus 1 (22015) (Fall 2012,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
TA:
Khashayar Filom
Textbook:
Calculus by J. Stewart
Click on the cover to view
the whole book:
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Homological Methods in Commutative Algebra
(22252+) (Spring 2012,
Sharif University of Technology)
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, CohenMacaulay rings,
Complexes and homology, Free resolutions, Projective resolutions, Injective
resolutions, Derived functors, Ext and Tor, Homological dimensions theory,
AuslanderBuchsbaum 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Analysis 2 (22326) (Spring
2012,
Sharif University of Technology)
Instructor:
M. R. Pournaki
TAs:
Khashayar Filom and Ahmad Reza HajSaeedi Sadegh
Textbooks:
1) Calculus on Manifolds
by Michael Spivak
2) Principles of
Mathematical Analysis by Walter Rudin
Syllabus:
Norm and inner product, Subsets of Euclidean space, Functions and
continuity, Differentiation, Partial derivatives, Inverse functions,
Implicit functions, Integration, Measure zero and content zero, Integrable
functions, Fubini's theorem, Partitions of unity, Change of variable,
Algebraic preliminaries, Fields and forms, Geometric preliminaries,
Integration on chains, The fundamental theorem of Calculus, Stokes'
theorem, Manifolds, Some measure 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 .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Advanced Algebra
(22226+) (Fall 2011,
Sharif University of Technology)
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 3 (22209)
(Fall
2011,
Sharif University of Technology)
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Analysis 1 (22325) (Spring
2011,
Sharif University of Technology)
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 .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Commutative Algebra
2
(22245+) (Fall 2010,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
Textbooks:
1) Commutative Ring Theory by H. Matsumura
2) CohenMacaulay Rings
by W. Bruns and J. Herzog
Click on the left cover to view
the whole book:
Syllabus:
Graded rings and modules, ArtinRees lemma, Inverse systems, Inverse limits, Completion, Krull intersection theorem, The Hilbert function, The
HilbertSamuel function, Dimension theory
revisited, Regular sequences, Grade, Depth, AuslanderBuchsbaum formula, CohenMacaulay 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
(22035) (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, Pathconnectedness, Domains,
Complex functions, Limits, Continuity, Sequences of complex
functions, CauchyRiemann equations, Analytic functions, Harmonic
functions, Exponential functions, Logarithmic functions,
Trigonometric functions, Roots, Branches, Conformal maps, Linear
fractional transformations, Line integrals, CauchyGoursat 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 realvalued
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, SturmLiouville 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 .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Homological Algebra
2
(22247+) (Spring 2010,
Sharif University of Technology)
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, MayerVietoris sequence and its applications, Arithmetic rank, Some acyclic modules,
The independence theorem, A nonvanishing 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Topology 1 (22556) (Spring
2010,
Sharif University of Technology)
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 MittagLeffler
theorem, Baire's theorem, Compactness for metric spaces, BolzanoWeierstraß property, HeineBorel theorem,
Topological spaces, Examples of topological spaces, Zariski topology, Continuity and convergence of nets,
Compactness, Tychonoff's theorem, Locally compact spaces, Alexandroff onepoint 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Homological Algebra
(22246+) (Fall 2009,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
Textbook:
An Elementary Approach to Homological Algebra
by L. R. Vermani
Syllabus:
Homology of complexes, KerCoker 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 (22034) (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, Firstorder linear
equations, Separable equations, Exact equations, Picard's existence and
uniqueness theorem, Secondorder linear differential equations, The method
of variation of parameters, The method of judicious guessing, Series
solutions, The method of Laplace transforms and its applications,
Higherorder equations, Systems of differential equations, Stability of
linear systems, Stability of equilibrium solutions, The phaseplane.
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Elementary Theory of Ordinary Differential
Equations (22384) (Fall 2009,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
Textbook:
A First Course in the Qualitative Theory of Differential Equations
by James H. Liu
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Commutative Algebra
(22243+) (Spring 2009,
Sharif University of
Technology)
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Algebra 2 (22218) (Spring 2009,
Sharif University of Technology)
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
(22035) (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 MyintU
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, Pathconnectedness, Domains,
Complex functions, Limits, Continuity, Sequences of complex
functions, CauchyRiemann equations, Analytic functions, Harmonic
functions, Exponential functions, Logarithmic functions,
Trigonometric functions, Roots, Branches, Conformal maps, Linear
fractional transformations, Line integrals, CauchyGoursat 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 realvalued
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
(22217) (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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Complex Functions 1
(22335) (Spring 2008,
Sharif University of Technology)
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, Pathconnectedness, Domains,
Complex functions, Limits, Continuity, Sequences of complex
functions, CauchyRiemann equations, Analytic functions, Harmonic
functions, Exponential functions, Logarithmic functions,
Trigonometric functions, Roots, Branches, Conformal maps, Linear
fractional transformations, Line integrals, CauchyGoursat 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 realvalued
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 .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Linear Groups
(22285+) (Fall 2007,
Sharif University of Technology)
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, Quasibilinear 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Linear Algebra 1
(22255) (Fall 2007,
Sharif University of Technology)
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 CayleyHamilton theorem,
The minimal polynomial, Triangular forms, Jordan canonical forms, Inner products and norms, The GramSchmidt 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 .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Commutative Algebra
(22243+) (Fall 2006,
Sharif University of Technology)
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Differential
Equations (22034) (Fall 2006,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
TAs:
Alireza Mohammadi and Shokoofeh Shafiei Ebrahimi
Textbook:
Ordinary Differential Equations
by Tyn MyintU
Syllabus:
Ordinary differential equations, Mathematical models, Firstorder
equations, Picard's existence theorem, Secondorder 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Elementary Functional Analysis
(22475)
(Fall 2006,
Sharif University of Technology)
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Homological Algebra
(22246+) (Spring 2006,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
Textbook:
An Elementary Approach to Homological Algebra
by L. R. Vermani
Syllabus:
Homology of
complexes, KerCoker 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Differential
Equations
(22034) (Spring 2006,
Sharif University of Technology)
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, Firstorder differential
equations, Linear firstorder 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Algebra 3
(22219+) (Fall 2005,
Sharif University of Technology)
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Engineering Mathematics
(22035) (Fall 2005,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
TA:
Hassan Dadashi
Textbooks:
1) Partial Differential Equations of Mathematical Physics
by Tyn MyintU
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, Pathconnectedness, Domains, Complex functions, Limits, Continuity, Sequences of complex functions, CauchyRiemann equations, Analytic functions, Harmonic functions, Exponential functions, Logarithmic functions, Trigonometric functions, Roots, Branches, Conformal maps, Linear fractional transformations, Line integrals, CauchyGoursat 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 realvalued 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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Finite Groups
(22282+) (Spring 2005,
Sharif University of Technology)
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
(22016) (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:
Smooth Curves:
Functions of Several Variables:
Continuity and Limit:
18)

19)

Differentiation:
20)

21)

22)

23)

24)

25)

26)

Multiple Integrals:
Line and Surface Integrals:
Vector Analysis:
Syllabus:
Real ndimensional 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
(22015) (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:
Sequences and Series of Numbers:
Continuity:
Differentiation:
14)

15)

16)

17)

18)

19)

20)

Integration:
21)

22)

23)

24)

25)

26)

27)

Taylor Series:
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
(22325) (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, HeineBorel
theorem, BolzanoWeierstrass theorem, Differentiation of functions of one
real variable, Theory of Riemann integration, Sequences and series of
functions, Uniform convergence, Ascoli's theorem, The StoneWeierstrass
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 (22325) (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, Limsup, Liminf,
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, HeineBorel
theorem, BolzanoWeierstrass 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
(22217)
(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 (6103193) (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 CayleyHamilton
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, GramSchmidt 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
(22218) (Fall
1998,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
TA:
Keivan MallahiKarai
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
(22217) (Spring
1998,
Sharif University of Technology)
Instructor:
M. R.
Pournaki
TA:
Keivan MallahiKarai
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, JordanHِlder 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
(22215) (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
(22326) (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
RiemannStieltjes integration, Sequences and series of functions,
Uniform convergence, Ascoli's theorem, The StoneWeierstrass theorem,
Taylor series, Power Series, Fundamental theorem of algebra,
Fourier series and Fourier
integrals, The
L_2space as a complete inner product space, Basic Hilbert space theory,
Orthonormal sets, Bessel, GramSchmidt, RieszFischer, RiemannLebesgue
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 (22215) (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.
