For the Master of Arts degree in mathematics, students must meet the following requirements (but refer also to The Doctoral Program description):

1. A minimum of 30 hours of graduate credit must be completed. Math 6930 and 6940 do not count toward the 30 hours. The following courses are prerequisites and will not count toward the degree: Linear Algebra I (5300) and Advanced Calculus (5780).
2. The total graduate and undergraduate programs must include the following: Abstract Algebra I and II (5330, 5340), Topology I and II (5450, 5460), Real Analysis I and II (5820, 5830) and Complex Analysis (5880). Students may substitute the doctoral core courses in Algebra, Analysis or Topology for their 500-level counterparts, subject to the approval of the adviser. At most 20 credits from the above count toward the 30 hours requirement for the M.A. degree.
3. At least one two-semester sequence in a single discipline at the 6000 level must be taken from the following: Algebra I and II (6800, 6810), Topology I and II (6400, 6410), General Topology I and II (6420, 6430), Differential Geometry (6450,6460), Algebraic Topology I and II (6470, 6480), Real Analysis I and II (6800, 6810), Functional Analysis I and II (6820, 6830), Complex Analysis I and II (6840, 6850) or Ordinary Differential Equations I and II (6500, 6510).
4. The remaining course work must be chosen from the following electives: Topology I (5460), Introduction to Differential Geometry I and II (5540, 5550), Ordinary Differential Equations (5800), Partial Differential Equations (5810), Calculus of Variations I and II (5860, 5870) or any of the courses from (3).
5. The student must either pass comprehensive examinations or write a Master's thesis. The comprehensive examinations will be in Algebra, Analysis, and one of Topology, Differential Equations, or Probability and Statistics. If a thesis is elected the student must take an oral examination on the general areas of the thesis. As an alternative the student may take two of the three written doctoral examinations. In this case, if the student does not pass both at the Master's level it will not be considered as an attempt provided the regular M.A. exams are taken in the same semester.

1. A minimum of 30 graduate credits must be completed. Math 6930 and 6940 do not count toward the 30 hours.

2. Preparatory courses are Linear Algebra I (5300), Applied Linear Algebra (5350), Introduction to the Theory of Probability (5680), Mathematical Statistics (5690), and Advanced Calculus (5780). These may be taken for credit by applied master's students, but cannot be used to fulfill the course requirements for the Master of Science degree in the Applied Mathematics option.

3. The following courses or their equivalents are required: Numerical Analysis I, II (5710, 5720), Introductory Real Analysis I, II (5820, 5830), Complex Analysis (5880), Introduction to Differential Equations I, II (6500, 6510). In the case where there are corresponding 6000 level courses, students may substitute (with consent of the adviser) such courses to fulfill these requirements.

4. Elective courses may be chosen from: Applied Functional Analysis (6150), Functional Analysis I, II (6820, 6830), Methods of Mathematical Physics (6720, 6730), Introduction to Differential Geometry I, II (5540, 5550), Differential Geometry I, II (6440, 6450), Calculus of Variations I, II (5860, 5870), Linear and Nonlinear Programming (6180), Infinite Dimensional Optimization (6190), Theory of Computation (5390) and Discrete Structures and Analysis of Algorithms (5380), Partial Differential Equations I, II (6540, 6550), Dynamical Systems (6520, 6530), Applied Probability (5660), Linear Statistical Models (5620), Operational Mathematics (5850) or (for students not in the Ph.D. track), a one-semester course in an applied or pure science as approved by the adviser.

5. To complete the program a student may either elect to write a master's thesis under the supervision of a faculty adviser or pass comprehensive exams in analysis and differential equations. For full-time students it is expected that either of these requirements will be completed by the end of the student's second year of study.

   Options:

   1. The option of writing a master's thesis is recommended to students who enter the program with a good background in mathematics and who do not intend to pursue doctoral studies. If a student chooses to write a master's thesis he or she must be assigned a thesis adviser by the end of the student's first year of study.
   2. Students choosing to write comprehensive examinations may either take the M.S. comprehensive examinations offered at the end of the Fall and Spring semesters, or pass the corresponding doctoral qualifying exam at the master's level. If a student does not pass either doctoral qualifying exam and takes the M.S. level examinations in the same semester, the first attempt will not be counted.

Prerequisites to the Statistics program are Calculus, Linear Algebra, Introductory Probability and Statistics and some proficiency in Computer Programming.

1. A minimum of 35 graduate credits must be earned, including at least 9 at the 6000 level; 6930, 6940 may not be applied toward this minimum.
2. Preliminary courses 5300 or 5350, 5680 and 5780 must be taken immediately upon enrolling in the program if they or their equivalents have not been previously taken. At most one of these courses may be counted toward the 35 hours required.
3. Required statistics courses include 5600, 5610, 5620, 5630, 5640, 6600, 6610, 6620, 6630, 6650 and 6690. Also required is one elective course numbered 5600 or 6600.
4. A two-part comprehensive examination, one part in probability and statistical theory and one in applied statistics, must be passed. (See Appendix A3.)

1. A minimum of 32 hours of graduate credit must be completed. Math 6930 and 6940 do not count in the 32 hours. At least 18 hours must be in Mathematics and 9 in Education, with an additional 6 hours to be assigned in consultation with advisers. As part of the additional 6 hours the student may elect to write a paper in Mathematical Education or one of an expository character in mathematics.
2. The total undergraduate and graduate programs must include the following : at least 6 hours of abstract algebra and/or linear algebra, 6 hours in geometry, statistics, probability and/or computer programming, 3-6 hours of analysis (beyond calculus), 3 hours of complex analysis, and one course in logic or foundations.
3. The student must pass comprehensive examinations in 3 of the areas of study in mathematics. The exact areas are to be arranged with the advisers. (See Appendix A4.)
4. For specific education requirements, consult the College of Education and Allied Professions.

The Masters Comprehensive Examination (required of all Masters students except those M.A. and M.S. applied students who do not plan to continue for the Ph.D. at UT and who elect to write a thesis) is scheduled regularly in the Fall and in the Spring. Students are expected to take the examination within two years of their entrance into the program.

The Graduate Student Affairs Committee (G.S.A.C.) will appoint the examining committees and oversee the implementation of examination procedures and policies.

Copies of some previous examinations are on file in the department office and may be obtained for perusal or copying. Some may also be found in the online graduate exam archive.

Detailed descriptions and a list of the regulations governing the Masters Comprehensive Examinations appear in Appendices A through C of this booklet. Students should note that the rules vary somewhat from program to program.

Included with each set of regulations is a syllabus and list of relevant core courses for each examination. It is important to realize that the list of courses is provided only as a guide; it is the syllabus that defines the scope of the examination.

Students choosing to work in pure or applied mathematics will be examined in two of the four areas of algebra, real analysis, topology, or differential equations. Students choosing to work in statistics will be examined in real analysis and probability and statistics. Doctoral students in pure or applied mathematics who have successfully completed the doctoral core courses in Algebra, Topology, and Real Analysis, and meet the university's credit requirement for the M.A. degree and who have passed the Ph.D. qualifying examinations at the master's level, may obtain a M.A. degree in pure mathematics by applying to the M.A. program. Doctoral students who have completed the course requirements for the M.S. degree in Applied Mathematics and have passed the Real Analysis and Differential Equations qualifying examinations at the master's level may obtain an M.S. degree in Applied Mathematics by applying to the M.S. program. For more details see the graduate adviser.

The student must pass an oral examination in the general area of the intended dissertation work. The intention of the oral exam is to demonstrate the student's ability to engage in mathematical research.

Candidates must satisfy the doctoral requirements of the College of Arts and Sciences.

The departmental requirements for the Ph.D. are as follows:

1. A minimum of 90 hours of graduate credit must be completed. Math 6930 and 6940 do not count in the 90 hours. Of the 90 hours, at least 18 but no more than 36 shall be allotted for the dissertation.
2. As a general rule, in the first year of study for the Ph.D. students must enroll in Abstract Algebra (6300, 6310), Real Analysis (6800, 6810), and Topology (6400, 6410). (In any event, successful completion of these or equivalent courses is required for the degree.)
3. The student must successfully complete the year-long sequence in Complex Analysis (6840, 6850), and two other regularly scheduled year-long 6000/8000 sequences excluding (6720, 6730).
4. Students must pass the qualifying and oral examination. Full-time students must pass the qualifying examination within the first two years of study and pass the oral exam within a year of passing the qualifying examination.
5. The student must demonstrate the ability to read mathematical literature in one foreign language, ordinarily chosen from among French, German and Russian. Another language, e.g., Polish, Spanish, Italian or Japanese, may be substituted if it is necessary for the student's specific program. The language requirement must be met before dissertation research is begun.
6. All doctoral students are expected to spend two consecutive semesters in supervised teaching. This requirement should be met before dissertation research is begun.
7. The student must write a Ph.D. dissertation under the direction of a faculty member. Before completing the dissertation the student must report on it in an open seminar. A completed dissertation must be approved by an outside examiner, who should be an authority in the field, and the student must defend it before a faculty committee. This committee is to be chosen by the adviser in consultation with the student and must consist of at least three faculty members including the adviser. The choice of the committee must be approved by the Graduate Student Affairs Committee. Normally this committee will be formed when the student's efforts in research have begun to bear fruit.
8. The student will present a bound copy of the dissertation to the department.

A1: M.A. Comprehensive Examination

The M.A. Comprehensive Examination is a written examination consisting of three parts. Each part is based on topics normally treated in the courses indicated.

1. Real and Complex Analysis (5820, 5830, 5880)
2. Abstract and Linear Algebra (5330, 5340, 5300, 5310)
3. One of:
   1. Topology (5450, 5460)
   2. Differential Equations (5800, 5810)
   3. Probability and Statistics (5680, 5690)

Parts 1 and 2 are each three hours in length and Part 3 is a two hour exam. The parts will normally be written on three consecutive Saturdays in the Fall or in the Spring.

The following rules apply to this examination:

1. If a student's score on any part of the written exam is only marginally below the passing level, the examining committee for that part may defer its final decision pending the outcome of a one-hour follow-up oral examination of the student, to be administered within one month of the date of the written exam. This option will not be exercised by the committee if the margin of failure on the entire written exam is deemed to be decisive.
2. A student who passes two of the three parts of the exam is credited with those parts and is required to retake only the part failed; a student who fails two or more parts will receive no credit for any part.
3. A student is allowed two attempts on the full three-part examination. A third (and final) attempt on one part of the exam will be permitted if credit has previously been earned for the other two parts.
4. A student's first attempt at the examination must occur at one of the two regularly scheduled periods (Fall or Spring). Following an unsuccessful attempt made during one of these periods, a student who is eligible under rule (3) may elect to reattempt the exam (or part thereof as governed by rule (2)) within two months, at a time agreeable to both the student and the exam committee. Otherwise, the retake must take place at the next regularly scheduled examination.
5. A student who intends the M.A. degree to be her or his terminal degree in this department may elect to satisfy the degree requirements by submitting and defending a thesis instead of writing the M.S. Comprehensive Examination, but a student may not switch from the exam option to the thesis option after an unsuccessful attempt at the examination.

Syllabus for M.A. Exam

1. Real and Complex Analysis

   Real Analysis:

   The real number system

   Elementary metric space theory

   Sequences and series

   Differential calculus

   Integral calculus (the Riemann integral)

   Sequences and sums of functions (Weierstrass Approximation theorem, uniform convergence, Arzela-Ascoli theorem)

   The Lebesgue integral (on the real line)

   Complex Analysis:

   Complex functions: limits, continuity, properties of elementary functions including branches

   Differentiability: derivatives Cauchy-Riemann equations, analyticity, harmonic functions

   Integration: Cauchy theorems, Residue theorem, Morera's theorem, Maximum-Modulus theorem

   Series: Taylor's theorem, Laurent series expansions

   Mappings: elementary functions, properties of conformal maps

   Further properties of analytic functions: singular points, zeros, analytic continuation, residues, evaluation of real and complex integrals using residues

   Some references:

   Real Analysis:

   Rudin, Principals of Mathematical Analysis, McGraw-Hill

   Goldberg, Methods of Real Analysis, Blaisdell

   Complex Analysis:

   Churchill, Complex Variables and Applications, McGraw-Hill

   Saff & Smider, Fundamentals of Complex Analysis, Prentice-Hall

2. Abstract and Linear Algebra

   Abstract Algebra:

   Basic concepts of groups and rings including Lagrange's theorem, normal subgroups, factor groups, homomorphisms and isomorphisms, permutations, the field of quotients of an integral domain, polynomial rings, factorization in integral domains.

   Linear Algebra:

   Vector spaces, linear transformations and matrices, determinants, canonical forms.

   Some references:

   Abstract Algebra:

   Herstein, Topics in Algebra, Wiley

   Fraleigh, A First Course in Abstract Algebra, Addison-Wesley

   Linear Algebra:

   Hoffman and Kunze, Linear Algebra, Prentice-Hall

   Curtis, Linear Algebra, Springer-Verlag

   Friedberg, Insel, Spence, Linear Algebra, Prentice-Hall

3. One of:

   1. Topology

      Axiomatization of topological spaces

      Different ways of introducing a topological structure.

      Continuous maps

      Characterizations of continuity, initial sources and final sinks, discrete and indiscrete spaces.

      Fundamental constructions

      Basis for open (or closed) sets, subbase, subspaces, products, quotients, sums. Lattice of topologies on a set.

      Convergence

      Sequences; filters and ultrafilters.

      Countability

      First and second axiom. Lindelof spaces.

      Separation

      Hausdorff, regular, completely regular, normal spaces; T_i-spaces, i=0,1,2,3,3.5,4. Urysohn's Lemma.

      Compactness

      In Euclidean spaces. Tychonoff Theorem, Stone-Cech compactification. Local compactness in T-spaces, Alexandroff compactification.

      Connectedness

      Components, local connectedness, path connectedness.

      Metric spaces

      Cauchy sequences, completeness. Uniform continuity. Baire's theorem.

      Metrization theorems and paracompactness

      The classical theorems of Urysohn and of Nagata-Smirnov-Bing. Stone's theorem.

      Function spaces

      Pointwise and compact convergence. The compact-open topology, Ascoli's theorem. Uniform convergence for metric spaces.

      Approximation

      Stone-Weierstrass theorem.

      Some References

      J. Dugundji, Topology, Allyn and Bacon, Inc., 1966

      R. Engelking, General Topology, PWN-Polish Scientific Publishers, 1977

      James R. Munkres, Topology, Prentice-Hall, Inc., 1975

      Stephen Willard, General Topology, Addison-Wesley, 1968

   2. Differential Equations

      Ordinary Differential Equations:

      Generalities and soluble classes of first order ODE's.

      Second order linear ODE's (General theory and explicit solutions in case of constant coefficients).

      Systems of linear ODE's with constant coefficients.

      Series solutions of second order linear ODE's with analytic coefficients near an ordinary point and near a regular singularity.

      Non-linear autonomous ODE's of second order. Phase plane analysis; in particular, equilibrium solutions, their classifications and their stability.

      Existence and uniqueness theorems. Nature of dependence on initial conditions.

      Partial Differential Equations:

      First order linear (and quasi-linear) equations.

      Classification of second order equations and their canonical forms.

      Method of separation of variables to solve boundary value problems, in particular the heat equation, the wave equation and the Laplace equation. In this connection: the Convergence theorem for Fourier series. Also Fourier integrals.

      Wave equation (or other hyperbolic equations). The Initial (Boundary) value problem. D'Alembert's Principle. (Huygens' principle.)

      Heat equation (or other parabolic equations). The Initial (Boundary) value problem. Existence and uniqueness theorem in one-dimensional heat equation.

      Laplace equation (or other elliptic equation). Basic properties of harmonic functions, in particular, the Maximum Principle. Boundary value problems. The Dirichlet problem, Green's function and Poisson's formula.

      Some References:

      ODE's:

      Boyce and DePrima, Elementary Differential Equations, Wiley.

      Simmons, Ordinary Differential Equations, McGraw-Hill.

      Birkhoff and Rota, Ordinary Differential Equations.

      PDE's:

      Zachmano glu and Thoe, Introduction to Partial Differential Equations, William & Willkins Comp., Baltimore.

      Colton, Partial Differential Equations: An Introduction, Random House Birkhaeuser Math Series.

      Berg and McGregor, Elementary Partial Differential Equations, Holden-Day, San Francisco.

   3. Probability and Statistics

      This will be based upon a knowledge of the material in Mathematics 5680 and 5690.

A2: M.S. (Applied Math) Comprehensive Exam

The comprehensive examination for the M.S. in Applied Mathematics will be a written examination consisting of two parts. Each part is based on topics normally treated in the courses indicated.

1. Real and Complex Analysis (5820, 5830, 5880)

   The examination will test the student's knowledge of elementary real and complex analysis.

2. Differential Equations (6500, 6510)

   The examination will test students' knowledge of both ordinary and partial differential equations. The exam will be based upon the more computational aspects of the material in the above courses.

The two parts of the exam will each be three hours in length and will normally be taken on consecutive Saturdays in the Fall or in the Spring.

The following rules apply to this examination:

1. The entire (two part) examination can be taken only twice.
2. If a student fails only one part of the examination, only that part need be retaken. A third (and final) attempt on one part only will be allowed.
3. If a student needs to retake one or both parts, this may be done either within two months or at the next regularly scheduled examination session.
4. If a student's score on one or both parts of the examination is only marginally below the passing level, in the examining committee's judgment, the committee may defer its final decision pending the outcome of a one-hour oral examination on that part. Such oral examination(s) are to be given within one month of the date of the written examination and at the mutual convenience of the student and the examining committee.

Syllabus for M.S. (Applied Math) Exam

1. Real and Complex Analysis

   Real Analysis:

   Completeness of Rⁿ, Sequences and Series.

   Compactness of Rⁿ, Connectedness.

   Uniform Convergence.

   Riemann integral, existence of the integral, uniform convergence and the integral.

   Improper integrals.

   Complex Analysis:

   Analytic functions and the Cauchy-Riemann Equations.

   Elementary conformal mappings.

   Cauchy-Goursat theorem, Cauchy integral formula, residue calculus.

   Taylor and Laurent series.

2. Differential Equations

   Ordinary Differential Equations:

   Linear systems, calculation of fundamental matrices.

   Variation of parameters for systems.

   Boundary value problems, eigenvalue problems, Sturm-Liouville theory.

   Plane autonomous systems, Liapunov stability.

   Partial Differential Equations:

   Method of characteristic for first order equations.

   Boundary value problems for the Laplace, Wave and Heat equations, separation of variables.

   Greens functions for the Laplace, Wave and Heat equations, Poisson's kernel, Dirichlet problem, method of images.

A3: M.S. (Statistics) Comprehensive Exam

The purpose of this examination is to ensure that the M.S. graduate has acquired statistical knowledge and skills adequate for a practicing statistician in engineering, science, management, pharmacy, medicine, and other areas.

The examination will be a written examination, including a take-home project. It will consist of two parts:

1. Probability and Statistical Theory.

   This part will normally be based upon a detailed knowledge of the material in Mathematics 5680, 5690 and 6680. In addition, on a more general level, and with some choice given between questions, material from Mathematics 5660, 5700, 6630, 6690, and 6710 will also be tested. If material from any other courses will be tested, you will be notified of that fact in writing at least one month in advance of the examination.

2. Applied Statistics.

   This part will normally be based upon a detailed knowledge of the material in Mathematics 5620, 6630, 6640, and 6690. In addition, on a more general level, and with some choice given between questions, material from Mathematics 5610, 5640, 5660, 5700, 6610, and 6710 will also be tested. If material from any other courses will be tested, you will be notified of that fact in writing at least one month in advance of the examination.

The two parts of the exam will each be three hours in length and will normally be given on consecutive Saturdays in the Fall and/or in the Spring.

The take-home project will be given at least one week before the Applied Statistics portion of the exam, and will be due at the time that the Applied Statistics portion is given. In addition to material from the courses listed above in the description for the Applied Statistics exam, material from Mathematics 5670 may also be covered on the take-home portion of the exam.

The rules applying to this examination are the same as rules (1) through (4) of the M.S. Applied Mathematics exam as delineated in Appendix A2 of this Guide with one additional rule:

5. Books and notes (as approved by the examining committee) and calculators may be used during this examination. Your course notes and course textbooks will be approved. Please obtain approval in advance for the use of other written materials during the examination.

Any questions regarding the content of this examination should be directed to Professor Donald White, Head of the Statistics Group.

A4: M.S. ED. Comprehensive Exam

The M.S. Ed. Comprehensive Examination consists of three parts, each part in an area of mathematics studied by the student. The exact areas are to be arranged with the advisers, but will usually include complex variables and algebra.

B1: Ph.D. Qualifying Exam

The Ph.D. qualifying examination is a preliminary examination for the Ph.D. program. It consists of two three-hour parts, each on a separate topic. The two topics on the examination are to be chosen from among the following general areas: algebra, real analysis,topology, differential equations (for students intending to write a dissertation in an applied area) or statistics (for students intending to write a dissertation in the area of statistics). The content of each part will be based on the material presented in the respective required first year-long sequences in algebra (6300, 6310), real analysis (6800, 6810), topology (6400, 6410), differential equations (6500, 6510). See below for more details on the exams syllabus. (Students intending to take the Statistics exam should consult with the statistics program adviser.)

The following rules apply:

1. The Ph.D. preliminary exam will generally be offered twice a year, in the Fall and Spring semesters, and will be administered over a period of two weeks.
2. Each part of the examination will be graded on a pass/fail basis. Failure in one topic will result in a failing grade on the exam. Passing in just one topic does not relieve the student of the need to retake and pass both topics on any subsequent attempt.
3. In order to continue in the program both exams must be passed by the end of the student's second year. There will be two opportunities to pass the exam. However, should the student elect to take the exams at the end of their first year this will not count as one of the two opportunities.

B2: Ph.D. Oral Examination

A student shall take the Ph.D. oral examination upon successful completion of the Ph.D. qualifying examination, or at the request of a faculty member whom the student has asked to be his/her thesis supervisor and who has not yet accepted the student as an advisee. The oral examination will be administered by a committee of three faculty members. The exam will be concerned with the general area of specialization of the student. It is the student's responsibility to ask a faculty member to agree to serve as the chair of the examination committee. The examination will consist of two parts. Part I shall be a talk by the student on a topic at a level sufficient to demonstrate the student's ability to engage in mathematical research. The topic will be chosen in consultation with and approval of the committee chair. Part II shall be an examination of the student by the committee to ascertain the level of the student's understanding of the topic and related background material. Parts I and II will be administered at the same session.

The following rules apply:

1. The oral exam will be graded on a pass/fail basis.
2. The student must pass the oral examination within one year of passing the Ph.D. qualifying exam or by the end of the second year, which ever is later.

B3: Syllabus for Ph.D. Qualifying Exams

1. Differential Equations

   Ordinary Differential Equations

   General theory for first order equations

   Existence and uniqueness of solutions

   Continuous dependence on parameter and initial conditions

   Infinite series solutions and method of majorants

   Linear Systems

   General theory of linear systems

   Linear periodic systems

   Second Order linear equations

   Boundary value problems, Green's functions, Sturm-Liouville theory

   Comparison theorems

   Qualitative theory of ordinary differential equations in the plane

   Limit cycles, Poincaré-Bendixson Theorem

   Stability, Liapunov's method

   Partial Differential Equations

   Cauchy-Kowalevski Theorem

   Hyperbolic systems

   Existence and uniqueness of solutions

   Method of characteristics

   Energy estimates

   Fourier transforms, Green's functions

   Second order elliptic equations

   Application of the Maximum Principle

   Elementary Sobolev space theory:

   H¹(Rⁿ) and H²(Rⁿ)

   Existence and uniqueness of solutions

   Dirichelet Principle

   Perron's Method

   Eigenvalues of the Laplace operator

   Heat equation

   Existence and uniqueness of solutions

   Fundamental solutions

   Energy estimates

   Maximum principle

2. Algebra

   Background material

   Linear Algebra

   Vector spaces and linear transformations

   Determinants

   Canonical forms

   Diagonalization of quadratic forms

   General

   Homomorphism theorems

   Jordan-Hölder Theorem

   Fittings's Lemma

   Krull-Schmidt Theorem

   Groups

   Group actions

   Fundamental counting theorem

   Permutation groups

   Transitivity and primitivity

   Simplicity of A_n for n at least 5

   Class equation

   Frattini Argument

   Sylow's theorems and p-groups

   Group constructions

   Direct products

   Semidirect products

   Structure of finitely generated Abelian groups

   Derived series and central series

   Solvable groups and nilpotent groups

   Fields

   Simple extensions (algebraic and transcendental)

   Galois Group of an extension

   Algebraic closure

   Separable and inseparable extensions

   Normal extensions

   Fundamental theorem of Galois Theory

   Finite fields

   Rings

   Projective and Injective modules

   Commutative Rings

   Factorization

   Localization with respect to multiplicatively closed sets

   Simple modules and primitive rings

   Jacobson radical

   Jacobson Density Theorem

   Artinian Rings

   Wedderburn-Artin Theorems

3. Real Analysis

   Background material

   Infinite series and products

   Power series

   Elementary theory of the derivative

   Monotone function and functions of bounded variation

   Metric spaces

   Topology

   Completeness

   Connectedness

   Compactness and totally boundedness

   Uniform convergence

   Baire Category Theorem

   Ascoli-Arzela Theorem

   Stone-Weierstrass Theorem

   Integration

   Measure Theory on the real line

   Outer measure

   Relation between measure and content

   Measurable function, Egoroff and Lusin Theorems

   Lebesgue integral on the real line

   Monotone and bounded convergence theorems, Fatou's Lemma

   Riemann integral and relation to Lebesgue integral

   Improper Riemann and Lebesgue integrals

   Some References:

   T. Apostol, Mathematical Analysis

   R. Goldberg, Methods of Real Analysis

   H.L. Royden, Real Analysis

   W. Rudin, Principles of Mathematical Analysis

4. Topology

   Background material

   Cardinality and countability

   Axiom of Choice, Well Ordering, Maximum Principle

   Basic facts about the ordinals

   General topology

   Open set, closed set, closure, interior, and neighborhood systems

   Convergence of filters and nets

   Separation and countability axioms

   Continuous functions

   Connectedness and path connectedness

   Compactness

   Product spaces and Tychonov Theorem

   Urysohn Lemma and Tietze Extension Theorem

   Local compactness and paracompactness

   Quotient spaces

   Algebraic topology

   Homotopy of maps and homotopy equivalence

   Fundamental group

   Covering spaces and classification

   Seifert-van Kampen Theorem

   Some references:

   W.S. Massey, A Basic Course in Algebraic Topology

   J.R. Munkres, Topology, A First Course

   I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry

   S. Willard, General Topology