**Math 111: Fundamentals of Mathematics**

**Prerequisites: **None

**Frequency: **Annually, Fall Term

**Credit: **(3-0)3

**Content: **Propositions. Connectives. Truth tables. Logical equivalencies. Logical implications. Methods of proofs of implications and equivalences. Quantifiers. The rules of inference for quantified propositions. Sets, subsets. Set operations. The laws of set theory. Cartesian product. Relations. Inverse relations. Composition of relations. Functions. Injective, surjective and bijective functions. Composition of functions. Equipollent sets. Countability of sets. More About Relations: Equivalence relations. Equivalence classes and Partitions. Quotient set. Order relations: Partial order, total order, well order. Mathematical Induction and Recursive definitions of functions.

**Goals: **To enable the student to comprehend and construct mathematical arguments, to develop the mathematical maturity of the student, to provide basic definitions, facts and tools necessary for his further studies in mathematics.

**Course Outline:**

(3 Weeks) Fundamentals of Logic

(1 Week) Sets

(1 Week) Relations

(2 Weeks) Functions

(2 Weeks) More about relations

(2 Weeks) Equivalent sets and countability

(1 Week) Order relations

(1 Week) Mathematical Induction

(1 Week) Recursive definitions

**Suggested textbook: **GRIMALDI, Ralph P.: Discrete and Combinatorial Mathematics

**Math 112: Introductory Discrete Mathematics**

**Prerequisites: **None

**Frequency: **Annually, Spring Term

**Credit: **(3-0)3

**Content: **Basic Counting: The sum and product rules, the pigeonhole principle, permutations and combinations. The binomial theorem. Discrete probability. Inclusion-Exclusion. Recurrence relations. Introduction to graphs and trees.

**Goals: **To introduce the beginning students to the topics and techniques of discrete methods and combinatorial reasoning by means of well chosen problems.

**Course Outline:**

(2 Weeks) Counting principles. The pigeonhole principle

(5 Weeks) Permutations and combinations. The binomial coefficients and the binomial theorem. Permutations and combinations with repetition. Discrete probability.

(4 Weeks) Advanced counting: Inclusion-Exclusion. Recurrence relations and their solutions.

(3 Weeks) Introduction to graphs: subgraphs, complements, graph isomorphisms, vertex degree. Planar graphs. Hamilton paths and cycles. Trees: definitions and examples.

**Suggested textbook:**GRIMALDI, Ralph P.: Discrete and Combinatorial Mathematics

**Math 115: Analytic Geometry**

**Prerequisites: **None

**Frequency: **Annually, Fall Term

**Credit: **(3-0)3

**Content: **Fundamental principles of Analytic Geometry. Cartesian coordinates in plane and space. Lines in the plane. Review of trigonometry and polar coordinates. Rotation and translation in the plane. Vectors and algebra of vectors; angles, parallelism and orthogonality. Lines and planes in 3-space. Intersection of three planes. Basics about conics whose axes are parallel to coordinate axes. Basic surfaces in space, cylinders, surface of revolutions, quadratic surfaces. Cylinderical and spherical coordinates.

**Goals: **Familiarize the student with the basic concepts of analytic geometry in the plane and space, and prepare him for later courses in advanced calculus and linear algebra.

**Course Outline:**

(2 Weeks) Fundamental Principle of Analytic Geometry. Cartesian coordinates in the plane and the space. Lines in the plane.

(2 Weeks) Review of Trigonometry and Polar coordinates. Rotation and translation in the plane.

(2 Weeks) Vectors in 2-3 space. Dot product, cross product

(2 Weeks) Lines and planes in 3-space

(3 Weeks) Basic about conics

(2 Weeks) Basic surfaces. Cylindrical and spherical coordinates.

**Suggested textbook: **H. Ibrahim Karakas Analytic Geometry Matematik Vakfi Yayini 1994.

**Math 116: Basic Algebraic Structures**

**Prerequisites: **None

**Frequency:**Annually, Spring Term

**Credit: **(3-0)3

**Content: **Binary operatons. Groups. The symmetric group. Subgroups. The order of an element. Cyclic groups. Rings. Integral domains. Subrings. Ideals. Fields: **Q**, **R**, **C**, **Z**_{p}. The concept of an isomorphism. The ring of integers and the ring of polynomials over a field: Division and Euclidean algorithms. GCD and LCM. Prime factorization. Quotient structures.

**Goals: **The goal of this course is to introduce basic algebraic structures-groups,rings and fields-with a special emphasis on the fundamental examples: the symmetric group *S _{n}*; rings of integers, polynominals and matrices; prime fields, fields of real and complex numbers.

**Course Outline:**

(2 Weeks) Binary operations, Groups, Subgroups, order of an element.

(2 Weeks) The symmetric group.

(1,5 Weeks) Cyclic groups.

(1 Week) Rings, integral domains, subrings.

(1 Week) Ideals.

(1,5 Weeks) Fields; **Q**, **R**, **C**, **Z**_{p}; the concept of an isomorphism.

(1,5 Weeks) The ring of Integers.

(1,5 Weeks) The ring of polynomials.

(2 Weeks) Quotient structures.

**Suggested textbooks:**

S. Lang; Algebraic Structures

S.Alpay, H.I.Karakas; An Introduction to Modern Mathematics, METU, 1979

** **

**Math 153: Calculus for Mathematics Students I**

**Frequency: **Annually, Fall Term

**Credit: **(4-2) 5

**Content: **Functions, Limit and Derivative of a Function of a Single Variable, A Thorough Discussion of the Basic Theorems of Differential Calculus: Intermediate Value, Extreme Value, and the Mean Value Theorems, Applications: Graph Sketching and Problems of Extrema.

**Goals: **The goal of the course is to endow the student with a good grasp of the basic conceptual tools of Calculus-functions, limits, and derivatives-and familiarize the student with computational skills.

**Course Outline:**

(2 Weeks) Domain, range and various representations of functions.

(2 Weeks) The Function Concept:** **Interpolation, monotonicty, extreme values, compositions of functions, implicitly defined functions.

(4 Weeks)** **The Limit Concept: ε–δ definition, properties, one-sided limits, limits at infinity, the substitution and Squeeze Theorems, Continuity, Properties of continuous functions, Intermediate-Value and Extreme-Value Theorems, using the Bisection Method to find roots.

( 5 Weeks) The Derivative Concept: Definition of the Derivative. Properties of the Derivative, Implicit Differentiation. Rolle's and the Mean Value Theorems. Monotonicity Theorem. the First and Second Derivative Tests, the Inverse Function Theorem in one variable. Construction of the exponential and the logarithmic functions. Inverse trigonometric functions, L'Hospital's Theorem. Asymptotes and Graph Sketching. Problems of Maxima and Minima.

**Suggested textbooks: Michael Spivak, Calculus,**** 2008**

**Math 154: Calculus for Mathematics Students II**

**Prerequisites: **Math 153 or Math 151

**Frequency: **Annually, Spring Term

**Credit: **(4-2)5

**Content: **The Riemann Integral, Mean Value Theorem for Integrals, Fundamental Theorem of Calculus, Techniques to Evaluate Anti-Derivative, Families, Various Geometric and Physical Applications, Sequences, Improper Integrals, Infinite Series of Constants, Power Series and Taylor's Series with Applications.

**Goals:**The goal of the course is to familiarize the student with the conceptual as well as computational aspects of integrals, sequences and series, thereby emphasizing the difference between the 'integral' and the 'anti-derivative', 'finite' and 'infinite' sums-the latter being a limit and not a sum-etc.

**Course Outline:**

(2,5 Weeks) The Integral: The concept and definition of Area, Riemann Sums and the definiton of the (definite) integral using Upper and Lower Sums, numerical Integration Methods.

(1 Week) Anti-derivative Families. The Mean-Value Theorem for Integrals, the Fundamental Theorem of calculus.

(2 Weeks) Techniques of Integration: Substitution, Integration by Parts, Partial Fractions, Trigonometric Substitution, the tan x/2 Substitution.

(1,5 Week) Applications of Integrals: Calculating Areas. Volumes and Surface Areas of Rotational Surfaces, Length of Arcs, etc.

(1 Week) Sequences: Convergence with Emphasis on ε–δ Definition, Subsequences, Monotone, Bounded Sequences, and related theorems.

(1,5 Weeks) Improper Integrals, Comparison and Limit-Comparison Theorems.

(2 Weeks) Series of Numbers: Tests for Convergence of Series, Absolute vs. Conditional Convergence.

(1,5 Weeks) Power Series: Radius and Interval of Convergence, Statement of the Theorem on Termwise Differentation and Integration, Taylor's Theorem and Taylor's Series, Expansions of Various Functions (including the Binomial Function), Approximate Calculation of Special Constants and Various Definite Integrals Using Taylor's Series, and Other Applications.

(1 Week) Partial Differentiation: A Brief Introduction to Partial Differentiation to Enable the Students to Follow Beginnings of Math 254

**Suggested textbooks:**

E.Dubinsky, K.E. Schwingerdorf, and D.M.Mathews; Calculus, Concepts, and Computers, Mc Graw Hill, 1995 2nd Edition.

H.Arikan, A.Aytuna, M.Dabbagh, and Z. Nurlu; Calculus I for Mathematics Students, METU Department of Mathematics, 1994

H.Arikan, A.Baki, M.Dabbagh, and Z.Nurlu; Calculus II for Mathematics Students,METU Department of Mathematics, 1994

**Math 251: Advanced Calculus I**

**Prerequisites: **Math 154

**Credit: **(4-0)4

**Content: **Introductory Topology of IR, IR^{2} and IR ^{3}: Domains and regions. Functions of several variables. Limits and continuity. Partial derivatives. Directional derivatives. Gradients. Differentials and the tangent plane: the fundamental lemma, approximations. The mean Value, the implicite and the inverse function theorems. Extreme values. Introduction to vector differential calculus: the gradient, divergence and curl. Curvilinear coordinates.

**Goals: **A continuation of Math 153, Math 154. Differential calculus in several variables is rigorously developed.

**Course Outline:**

(2 Weeks)** **Introductory Topology of IR^{n}-1,2,3: Balls, Accumulation Points, Open, Closed Sets with Geometric Examples, Concepts of Boundary, Interior, and Closure, Domains, Regions in Plane.

(2 Weeks) Functions of Several Variables: Examples of Functions ƒ : **R**^{n} → **R** and **R**^{n} → **R**^{n}with n=2,3, Sketching Domains and Graphs of such Functions. Definition of the Limit and Continuity of Functions of 2,3 Variables, Examples on Approaching to a Point P from various Directions, Geometric interpretation of Continuous Surfaces.

(1 Week) Partial Derivatives: Definition and Geometric Meaning, Relation to Existence of Limit, Continuity, Higher Order Partial Derivatives, Directional Derivatives, The Gradient.

(4 Weeks) The Differential and the Tangent Plane: Definition of the Differential and its Relation to Partial Derivatives and Continuity, Examples of Functions w/o Differential, The Geometric Meaning of the Differential: Existence of the Tangent Plane. The Fundamental Lemma, Total derivative Formula and the Chain Rule, Jacobian, General Differential, Differential Approximation for Functions ƒ : **R**^{n} → **R**^{k}.

(1 Week)The Mean Value Theorem, Implicit Function and Inverse Function Theorems.

(2 Weeks)Extrema of Functions of Several variables: Hessian, Lagrange Multipliers with Applications.

(1 Week) Introduction to Vector Differential Calculus: Gradient, Divergence, Curl, orthogonal Curvilinear coordinates, Laplacian in Cylindrical and Spherical coordinates.

**Suggested textbooks:**

1.** **W.Kaplan; Advanced Calculus,** **Addison-Wesley, 1973 2nd Edition

2. Buck; Advanced Calculus, McGraw Hill, 1965

3. W. Fulks; Advanced Calculus: An Introduction to Analysis, John Wiley 1961

** Math 252: Advanced Calculus II **

**Prerequisites: **Math 251

**Frequency:**

**Credit: **(3-2)4

**Content: **Double Integral as Iterated integrals, polar coordinates, Applications to Volumes, Areas, Improper Double Integrals. General Change of Variables in Double Integrals. Triple Integrals: Cylindrical and Spherical coordinates in Triple Integrals, Various Applications. Line Integrals: Parametrisation of Curves, Green's Theorem, Independence of Path, Exact Differentials. Line Integrals of Multiply Connected Regions. Proof of the General Change of Variables Formula in Double Integrals. Surface Integrals: Parametrisation of Surfaces. Orientation of Surfaces. Surface Integrals. Divergence and Stokes' Theorems, Applications.

**Goals: **The last course in the calculus sequence where multiple and vector integrals are developed, and treated rigorously.

**Course Outline:**

(4 Weeks) Double Integrals: Motivation through Brief Mentioning of 'content', Iterated integrals, polar coordinates, Applications to Volumes, Areas, Improper Double Integrals, General Change of Variables in Double Integrals (Statement and Applications).

(2 Weeks) Triple Integrals: Cylindrical and Spherical coordinates in Triple Integrals, Various Applications.

(3 Weeks) Line Integrals: Parametrisation of Curves, Green's Theorem, Independence of Path, Exact Differentials, Line Integrals of Multiply Connected Regions, Proof of the General Change of Variables Formula in Double Integrals.

(4 Weeks) Surface Integrals: Parametrisation of Surfaces, Orientation of Surfaces, Surface Integrals, Divergence and Stokes' Theorems, Applications.

**Suggested textbooks:**

1.** **W.Kaplan; Advanced Calculus,** **Addison-Wesley, 1973 2nd Edition

2. Buck; Advanced Calculus, McGraw Hill, 1965

3. W. Fulks; Advanced Calculus: An Introduction to Analysis, John Wiley 1961

**Math 254 Introduction to Differential Equations**

**Prerequisites: **Math 152 or Math 154

**Frequency: **Fall and Spring Term

**Credit: **(4-0)4

**Content:**

First order equations: separable, linear, homogeneous exact equations, orthogonal and oblique trajectories, applications. Higher order linear differential equations: Reduction of order, method of undetermine coefficients, method of variation of parameters, Cauchy-Euler equations, operator methods, applications. Power series solutions: ordinary points, regular singular points. The Laplace Transform: basic properties, solution of initial value problems, convolution integral, solution of various equations. Systems of linear differential epuations: Brief discussion of theory of linear systems, solving linear systems; by operator method, by Laplace transform. Introduction to Partial Differential equations: Separation of variables.

**Goals: **This is one of the basic calculus course of the sequence (Math 153, 154,251,252) which serves as the foundation of all advanced subjects in applied and theoretical mathematics.

**Course Outline:**

(1st Week) Preliminaries. Solutions. Existence-Uniqueness Theorem. Separable Equations. Linear Equations. Homogeneous Equations.

(2nd Week)Exact Equations and Integrating Factors.Substitutions. Approximate solutions (2.6.1 and 2.6.2). Applications (2.7.2).

(3th Week) Basic Theory of Higher Order Linear Equations.

(4th Week) Reduction of Order. Homogeneous Constant Coefficient Equations.

(5th Week) Undetermined Coefficients. Variation of Parameters.

(6th Week) The Cauchy-Euler equation. Operator Method.

(7th Week) Power Series Solutions (ordinary points).

(8th Week) Power Series Solutions (regular singular points).

(9th Week) The Laplace Transform. Basic Properties. Convolution.

(10th Week) Solution of Differential Equations by the Laplace Transform.

(11th Week) Solutions of Systems of Linear Differential Equations by the Laplace Transform.

(12th Week) Solutions of Systems of Linear Differential Equations by Elimination: simple elimination and operator method.

(13 and 14th Weeks) Introduction to Partial Differential Equations

**Suggested textbook: **Safak Alpay, Ersan Akyildiz, Albert Erkip; Lectures on Differential Equations. Matematik Vakfi, 1995.

**Math 261: Linear Algebra I**

**Prerequisites: **None

**Frequency: **Annually, Fall Term

**Credit: **(4-0)4

**Content: **Matrices and Systems of linear equations. Vector spaces; subspaces, sums and direct sums of subspaces. Linear dependence, Bases, dimension, quotient spaces. Linear transformations, Kernel, range, isomorphism. Spaces of linear transformations, Hom (V,W),V*, V** transpose. Representations of linear transformations by matrices, similarity.

**Goals: **This course introduces fundamental concepts of linear algebra which are indispensible in all branches of basic sciences.

**Course Outline:**

(2 Weeks) Matrices and Systems of linear equations.

(2 Weeks) Vector spaces subspaces, sums and direct sums of subspaces.

(3 Weeks) Linear dependence, Bases, dimension, quotient spaces.

(2 Weeks) Linear transformations, Kernel, range, isomorphism.

(2 Weeks) Spaces of linear transformations, Hom (V,W),V*, V** transpose.

(2 Weeks) Representations of linear tnasformations by matrices, similarity

**Suggested textbook: **C. Koç; Linear Algebra I, METU, Department of Mathematics, 1996

**Math 262: Linear Algebra II**

**Prerequisites: **Math 261

**Frequency: **Annually, Spring Term

**Credit: **(4-0)4

**Content: **Characteristic and minimal polynomials of an operator, eigenvalues, diagonalizability, Canonical forms, Smith normal form, Jordan and rational forms of matrices. Inner product spaces, norm and orthogonality, projections. Linear operators on inner product spaces, adjoint of an operator, normal, self adjoint, unitary and positive operators. Bilinear and quadratic forms.

**Goals: **This course is a continuation of Math 261 and introduces further concepts of linear algebra used in all branches of mathematics.

**Course Outline:**

(3 Weeks) Inner product Spaces, norm and orthogonality, projections.

(2 Weeks) Linear operators on inner product spaces, adjoint of an operator, normal, self adjoint, unitary and positive operators.

(4 Weeks) Characteristic and minimal polynomials of an operator, eigenvalues, diagonalizability, diagonalization of special operators on inner product spaces.

(1 Week) Bilinear and quadratic forms.

(3 Week) Canonical forms, Smith normal form, primary decomposition, cyclic decomposition.

**Suggested textbook:**C. Koç; Topics in Linear Algebra (Revised Edition), METU, Department of Mathematics, 1996

** Math 349: Introduction to Mathematical Analysis **

**Prerequisites:**Math 252

**Frequency: **Annually, Fall Term

**Credit: **(4-0)4

**Content: **LUB Property of Real Numbers, Metric Spaces, Review of Sequences and Series at scalors with Emphasis on rigorous proofs, Continuity, Sequences and Series of Functions, Uniform Convergence, Applications.

**Goals: **This course serves as a bridge between calculus courses and more advanced courses. Primarily the course is intended to develop analysis required for higher mathematics and applications. The main topics are uniform convergence of sequences and series of functions.

**Course Outline:**

Week 1 Real number system, least upper bound property and its consequences.

Week 2 Sequences of real numbers.

Week 3 Bolzano-Weierstrass Theorem.

Week 4 Cantor intersection property.

Week 5 Limit superior and limit inferior.

Week 6 Introduction to metric spaces open and closed sets.

Week 7 Sequences in metric spaces continuous and uniformly continuous functions. Series of functions.

Week 8 Products of metric spaces.

Week 9 Completion of a metric space.

Week 10 Compactness in metric spaces.

Week 11 Connectedness and connected components.

Week 12 Ascoli-Arzela Theorem.

Week 13 Baire's Theorem and Applications.

**Suggested textbook:** An Introduction to Real Analysis Matematik Vakfi Yayini, 2000.

**Math 353: Complex Variables and Applications**

**Prerequisites: **Math 252

**Frequency: **Annually, Fall Term

**Credit: **(4-0)4

**Content: **Algebra of complex numbers. Polar representation. Analyticity. Cauchy-Riemann equations. Power series. Elementary functions. Mapping by elementary functions. Linear fractional trasformations. Line integral. Cauchy-Theorem. Cauchy integral formula. Taylor's Series. Laurent series. Residues, Residue theorem. Improper integrals.

**Goals: **This is an introductory course in complex analysis, giving the basics of the theory, with an emphasis on complex integration.

**Course Outline:**

(1 Week)** **Complex Numbers (Definition, Algebraic operations, polar form) nth roots of a complex number and their geometric meaning.

(1 Week) More geometry in the complex plane, Loci, Stereographic projection, point at infinity.

(1 Week) Regions in the Complex plane (points, sets, neighbounhoods, bounded, connected set, etc..) Functions of a complex variable, limit, Continuity, Uniform continuity.

(1 Week) Derivative, Cauchy-Riemann equations (also in polar form), Analytic functions, Harmonic functions.

( Elemantary functions (exponential, trigonometric, hyperbolic, logarithmic (branch point, branch cut), inverse trignometric, inverse hyperbolic)

(1 Week) Definite Integrals, Contours, Line Integrals, Cauchy theorem, Path independence, Indefinite Integrals.

(1 Week) Cauchy Integral formula and its extension to multiply-connected domains. Derivatives of analytic functions. Morera's th., Liouville's th. Maximum Modulus th.

(1 Week) Sequence, Series, Power series, Taylor's series, Laurent Series.

(1 Week) Further properties of series, Uniform Convergence. Integration and Differentiation of Power Series. Uniqueness of representation. Multiplication and Division of power series. Examples.

(1 Week) Zeros of Analytic functions, Types of singularities, Residues, Residue theorem. (in case of pole some practical formulas for finding residues). Residue at ∞

(1 Week) Improper Real Integrals, Improper integrals involving trigonometric functions. Definite integral of rational functions of sin θ, cos θ).

(1 Week) Integraton around branch point. Mapping by linear entire functions, and by 1/*z*

(1 Week) Linear fractional Transformation, Some special cases. Conformal mapping, basic properties of conformal mapping.

(1 Week) Mappings by other elementary functions (exp, log, *z*^{2}, √*z*, sin*z*)

**Suggested textbooks:**

R.V.Churcill, J.W.Brown; Complex Variables and Applications, McGraw-Hill, 1984

S.D.Fisher; Complex Variables, Wadsworth&Brodes/cole, 1990

J.E.Marsden; Basic complex analysis, W.H.Freeman and Co. 1973

**Math 358: Partial Differential Equations**

**Prerequisites: **Math 252 and Math 254

**Frequency: **Annually, Spring Term

**Credit: **(4-0)4

**Content: **Equations formation. Linear first-order and quasilinear equations, Lagrange's method, Cauchy problem. Non-Linear equations of first order; compatible systems, Charpit's method, Cauchy problem for nonlinear first order equations. Linear second-order equations in two independent variables, normal forms. Reduction to hyperbolic, parabolic and elliptic equations. Cauchy problem for linear second-order equations, adjoint operator. Laplace's and Poisson's equation. Properties of harmonic functions. Sturm-Liouville problems and generalized Fourier Series. Seperation of variables. Vibrating string. Boundary and initial value problems for heat equation.

**Goals:**To present an introduction to the theory of partial differential equations in a systematic way and to present some methods of solutions of the classical problems of mathematical physics.

**Course Outline:**

(1 Week) Introduction, classical equations formation of eqquations, geometric examples. (1 Week) Linear first-order equations.

(1 Week) First-order quasilinear equations, Lagrange's method, Cauchy problem.

(1 Week) Non-Linear equations of first order; compatible systems, Charpit's method, Cauchy problem for nonlinear first order equations.

(1 Week) Linear second-order equations in two independent variables, normal forms.

(1 Week) Reduction to hyperbolic, parabolic and elliptic equations.

(1 Week) Cauchy problem for linear second-order equations, adjoint operator.

(1 Week) Laplace's equation and Poisson's equation. Properties of harmonic functions.

(1 Week) Sturm-Liouville problems and generalized Fourier Series.

(1 Week) Seperation of variables in Laplace's equation (Continued)

(1 Week) Vibrating string, seperation of variables.

(1 Week) Initial value problem for heat equation.

(1 Week) Boundary and initial value problems for heat equation.

**Suggested textbooks:**

R. De Myer; An introduction to PDE and BVP, I.N.

Sneddon; Elements of PDE

**Math 367: Abstract Algebra**

**Prerequisites:**Math 116

**Frequency: **Annually, Fall Term

**Credit: **(3-2)4

**Content: **Groups, Lagrange's theorem. Factor groups, homomorphisms. Isomorphism theorems, direct products. Groups acting on sets, Cayley's theorem, Class equation. Statements of Sylow theorems and the fundamental theorem on finite abelian groups. Rings, quotient rings, homomorphisms, Isomorphisms theorems. Prime and maximal ideals. Integral domains, field of fractions. Euclidean domains, PIDs, UFDs. Polynomials, polynomials in several variables. Field extensions, algebraic and transcendental elements. Finite extensions, algebraically closed fields. Impossibility of certain geometric constructions. Finite fields.

**Goals:**The goal of the course is to give fundamental concepts of abstract algebra and thus to exhibit a typical model of mathematics as much as possible.

**Course Outline:**

(1 Week) Groups, subgroups, cosets, Lagrange's theorem.

(1 Week) Normal subgroups, Factor groups, homomorphisms.

(1 Week) Isomorphism, isomorphism theorems, direct products.

(1 Week) Groups acting on sets, Cayley's theorem, Class equation.

(1 Week) Statements of Sylow theorems and the fundamental theorem on finite abelian groups.

(1 Week) Rings, ideals, quotient rings, homomorphisms.

(1 Week) Isomorphisms, isomorphism theorems, prime and maximal ideals.

(1 Week) Integral domains, field of fractions.

(1 Week) Euclidean domains, PIDs, UFDs.

(1 Week) Polynomials, polynomials in several variables.

(1 Week) Field extensions, algebraic and transcendental elements.

(1 Week) Finite extensions, algebraically closed fields.

(1 Week) Impossibility of certain geometric constructions.

(1 Week) Finite fields.

**Suggested textbook: **Herstein; Abstract Algebra

**Math 371: Differential Geometry**

**Prerequisites: **Math 251 and Math 261

**Frequency: **Annually, Spring Term

**Credit: **(4-0)4

**Content: **Curves in 3 space: Local Theory of curves. Frenet formulas and Fundamental Theorem. Regular surfaces, definition and examples. Inverse image of regular values. Change of parameters, differentiable functions on surfaces. The tangent plane; The differential of a map, vector fields, the first fundamental form. Gauss map, second fundamental form, normal curvature, principal curvature and principal directions, asymptotic directions. Gauss map in local coordinates. Covariant derivative, geodesics.

**Goals: **To give the student essentials of curves and surfaces in 3-dimension, to reinforce their advanced calculus and linear algebra knowledge giving a good opportunitly to exhibit their interplay through application to geometry.

**Course Outline:**

(4 Weeks) Review of smooth functions, curves in IR^{3}, tangent, normal, binormal vectors, curvature, torsion, Fundamental Theorem of Curves, Plamar curves.

(4 Weeks) Coordinate patches and Smooth Surfaces, Examples of smooth surfaces, tangent and normal vectors the first fundamental form, directional derivative, covariant derivatives, vector fields, length and area, isometries.

(4 Weeks) Gauss map, Weingarter map, second fundamental form, Gaussian curvature, mean curvature, principal curvatures, principal direction, conputations of curvature using coordinates, Theorem a Egregium and the Fundamental Theorem of surfaces.

(2 Weeks) Geodesics, Statement of Gauss-Bonnet Theorem.

**Suggested textbook: **Ethan D. Block; A first course in Geometric Topology and Differential Geometry. Chapters 4-6 and Section 72-73.