# Mathematics

Nakhle Asmar, Chair

College of Arts and Science

224 Math Sciences Building

(573) 882-6221

asmarn@missouri.edu

**Professor** I. M. Aberbach**, N. Asmar**, W. D. Banks**, P. Casazza**, T. Christiansen**, S. D. Cutkosky**, S. Dostoglou**, D. S. Edidin**, F. Gesztesy**, L. Grafakos**, A. D. Helfer**, S. Hofmann**, A. Koldobsky**, Y. Latushkin**, Y. Li**, K. A. Makarov**, D. I. Mitrea**, M. Mitrea**, S. Montgomery-Smith**, M. M. H. Pang**, Z. Qin**, H. Srinivasan**, A. Tsoi**, I. Verbitsky**, S. Wang**, Q. Zhang**

**Associate Professor** C. Chindris**, A. Harcharras**, C. Morpurgo**, J. Segert*, D. T. Weston**

**Assistant Professor** P. Pivovarov**, S. Walsh**, S. Takeda**, J. Tu**

**Business Mathematics Coordinator** S. Goldschmidt

**Calculus Coordinator** A. Clayton

**College Algebra Coordinator** T. E. Christiansen

**Professor Emeritus** M. S. Ashbaugh**, J. Beem*, C. C. Chicone**, R. Crownover, J. Lange*, I. J. Papick*, C. Petty*, D. H. Pettey*, J. Reeder, E. Saab*, P. Saab^{*,} D. Sentilles*, Z. Zhao*

* | Graduate Faculty Member - membership is required to teach graduate-level courses, chair master's thesis committees, and serve on doctoral examination and dissertation committees. |

** | Doctoral Faculty Member - membership is required to chair doctoral examination or dissertation committees. Graduate faculty membership is a prerequisite for Doctoral faculty membership. |

The Department of Mathematics offers a major with either a Bachelor of Arts or a Bachelor of Science degree. Within the BS degree, an emphasis in Actuarial Science and Financial Mathematics is available. Both the BA and BS degrees will prepare a student for a graduate program in Mathematics.

## Preparation for Graduate Study in Mathematics

Students satisfying the requirements for either the BA or the “traditional” BS will have the basic preparation for a graduate program in Mathematics. A student considering graduate work, however, should take additional coursework. Because of this, a BS degree would be considered preferable. Those students in the Actuarial Science area considering graduate work should take MATH 4720 as part of their program. Those students getting a dual degree in Mathematics and Mathematics Education considering graduate work in mathematics should choose to take both MATH 4700 and MATH 4720 as part of their program.

Courses recommended for students planning to pursue graduate studies in pure mathematics: MATH 4400, MATH 4500, MATH 4900, MATH 4920, and MATH 4940.

Courses recommended for students planning to pursue graduate studies in applied mathematics: MATH 4300, MATH 4310, MATH 4315, MATH 4320, MATH 4500, MATH 4540, MATH 4940.

## Departmental Honors

### Eligibility

To become a candidate for the BA or BS degree with a major in Mathematics with departmental honors, a student must have a cumulative grade point average that meets the Honors College standards. At present, students with a GPA of 3.30 or higher are automatically eligible to enter the departmental honors programs.

### Requirements

To graduate with departmental honors in mathematics, a student must satisfy the regular BA or BS degree requirements and must have a GPA of 3.5 or higher in all Mathematics Department courses. In addition, the student must have at least 26 credits in mathematics courses numbered 4000 or above. Furthermore, the student must complete one of the two options listed below.

### Option 1: Honors Thesis

The student must write an honors thesis in conjunction with a mentorship or in conjunction with MATH 4996. This option requires that the student enroll in MATH 4996.

### Option 2:

The student’s program of study must include MATH 4700, MATH 4900, MATH 4720 and MATH 4920.

College of Arts and Science

202 Mathematical Sciences Building

(573) 882-6221

https://www.math.missouri.edu/grad/index

**Director of Graduate Studies:** Stephen Montgomery-Smith

## About Mathematics

The Graduate Program in Mathematics is large enough to encompass research and courses in many areas, yet small enough to remain responsive to the needs of individual students. There are approximately 80 graduate students, 40 professors, and 15 postdoctoral and visiting researchers. The active areas of research include: algebraic geometry, analysis (real, complex, functional and harmonic), analytic functions, applied mathematics, ﬁnancial mathematics and mathematics of insurance, commutative rings, scattering theory, differential equations (ordinary and partial), differential geometry, dynamical systems, general relativity, mathematical physics, number theory, probabilistic analysis and topology.

The Mathematical Sciences Building houses a library with more than 34,000 volumes and 430 journal titles. MU students have access to an extensive array of computing resources.

## Admission Notice

Applicants for any graduate degree in mathematics should submit an application for graduate study. Admission to the graduate program does NOT guarantee admission to the Ph.D. program. International Applicants applying from outside North America who seek financial support from the Department will only be considered for the PhD program.

## Financial Aid from the Program

All domestic applications for admission are automatically considered for financial support, in most cases by Teaching Assistantships. Virtually all current students are supported financially. Scholarships, assistantships, fellowships and other sources of aid are available.

The Department Research Fellowship, the Blumenthal Scholarship and the McFarlan Fellowship are administered by the department, while the Huggins Scholarship, Gregory Fellowship and Ridgel Fellowship are administered by the university.

International applications with TOEFL of 85 or higher (or equivalent) will also be automatically considered for departmental financial support.

**MATH _0110: Intermediate Algebra**

MATH _0110 is a preparatory course for college algebra that carries no credit towards any baccalaureate degree. However, the grade received in MATH _0110 does count towards a student's overall GPA. The course covers operations with real numbers, graphs of functions, domain and range of functions, linear equations and inequalities, quadratic equations; operations with polynomials, rational expressions, exponents and radicals; equations of lines. Emphasis is also put on problem-solving.**Credit Hour****s**: 3

**MATH 1050: Quantitative Reasoning**

Promotes mathematical literacy among students. This course will cover important mathematical ideas and problem solving skills in the context of science, technology, and/or society. Topics may include: logic and critical thinking, Venn Diagrams, problem solving, sets, units of measure, percentages and ratios, counting and probability, exponential growth and decay, linear and exponential models. Quantitative Reasoning is designed to stimulate interest in and appreciation of mathematics and quantitative reasoning as valuable tools for comprehending the world in which we live. This course does not satisfy the prerequisite of any other MATH course.**Credit Hour****s**: 3**Prerequisites: **C- or higher in MATH _0110 or a sufficient score on the ALEKS (or MMT) exam

MATH 1050 - MOTR MATH 120: Mathematical Reasoning & Modeling

**MATH 1100: College Algebra**

A review of exponents, order of operations, factoring, and simplifying polynomial, rational, and radical expressions. Topics include: linear, quadratic, polynomial, rational, inverse, exponential, and logarithmic functions and their applications. Students will solve equations involving these functions, and systems of linear equations in two variables, as well as inequalities. See the Math website for specific requirements. A student may receive at most 5.0 credit hours among MATH 1100, MATH 1120, MATH 1140, MATH 1160.**Credit Hour****s**: 3**Prerequisites: **C- or higher in MATH _0110 or a sufficient score on the ALEKS exam or MyMathTest Intermediate Algebra score of 70% or higher

MATH 1100 - MOTR MATH 130: Pre-Calculus Algebra

**MATH 1140: Trigonometry**

A student may receive only 5 credits from among MATH 1100, MATH 1140, and MATH 1160. A Student may receive at most 5.0 credit hours from MATH 1100, MATH 1120, MATH 1140, and MATH 1160.**Credit Hour****s**: 2**Prerequisites: **C- or higher in MATH 1100 or sufficient ALEKS score or MyMathTest College Algebra score of 70% or higher

**MATH 1160: Precalculus Mathematics**

Review of elementary algebra. Background material for MATH 1500, including algebraic, trigonometric, logarithmic, exponential functions. A student may receive at most 5 credits from among MATH 1100, MATH 1140, and MATH 1160.**Credit Hour****s**: 5**Prerequisites: **B+ or higher in MATH _0110 (at MU), or C- or higher in MATH 1100, or sufficient ALEKS score or MyMathTest College Algebra score of 60% or higer

MATH 1160 - MOTR MATH 150: Pre-Calculus

**MATH 1300: Finite Mathematics**

A selections of topics in finite mathematics such as: basic financial mathematics, counting methods and basic probability and statistics, systems of linear equations and matrices. Warning: without a College Algebra exemption, a sufficient ALEKS score will not suffice unless it is a proctored exam (for MATH 1100 credit).**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 1100, or MATH 1160, or both a College Algebra exemption and sufficient ALEKS score

**MATH 1360: Geometric Concepts**

This course is primarily for education majors. This course covers topics of Euclidean geometry such as the study of points, lines, angles, polygons, circles, congruence, similarity, transformations, symmetry, area, surface area, arc length, and volume. Polyhedra, spheres, cones, and other solids are discussed. The course includes constructions and proofs, and uses inductive and deductive reasoning throughout. Math Reasoning Proficiency Course.**Credit Hour****s**: 3**Prerequisites: **C- or higher in MATH 1100 or sufficient ALEKS exam score or MATH 1160 or equivalent

**MATH 1400: Calculus for Social and Life Sciences I**

The real number system, functions, analytic geometry, derivatives, integrals, maximum-minimum problems. No credit for students who have completed a calculus course. A student may receive credit for MATH 1320 or MATH 1400 but not both. A student may receive at most 5 units of credit among the MATH 1320 or MATH 1400 and MATH 1500. Math Reasoning Proficiency Course.**Credit Hour****s**: 3**Prerequisites: **grade of C- or higher in MATH 1100, or MATH 1160, or sufficient ALEKS score

**MATH 1500: Analytic Geometry and Calculus I**

Elementary analytic geometry, functions, limits, continuity, derivatives, antiderivatives, definite integrals. A student may receive at most 5 units of credit among the Mathematics courses MATH 1320 or MATH 1400 and MATH 1500. Math Reasoning Proficiency Course.**Credit Hour****s**: 5**Prerequisites: **grade of C- or higher in MATH 1160 or C - or higher in both MATH 1100 and MATH 1140 or sufficient ALEKS score or MyMathTest PreCalculus score of 70% or higher

**MATH 1500H: Analytic Geometry and Calculus I - Honors**

Elementary analytic geometry, functions, limits, continuity, derivatives, antiderivatives, definite integrals. Honors eligibility required. A student may receive at most 5 units of credit among MATH 1320 or MATH 1400 and MATH 1500. Math Reasoning Proficiency course.**Credit Hour****s**: 5**Prerequisites: **C- or higher in MATH 1160 or C- in both MATH 1100 and MATH 1140 and sufficient ALEKS score. Honors Eligibility required

**MATH 1601: Selected Topics in Mathematics-General**

The special topics covered may vary from term to term. This course may be repeated.**Credit Hour**: 1-3**Prerequisites: **instructor's consent

**MATH 1602: Selected Topics in Mathematics-Biological/Physical/Math**

The special topics covered may vary from term to term. This course may be repeated.**Credit Hour**: 1-3**Prerequisites: **instructor's consent

**MATH 1700: Calculus II**

Definite integrals, applications and techniques of integration, elementary transcendental functions, infinite series. Math Proficiency Reasoning course.**Credit Hour****s**: 5**Prerequisites: **a grade of C- or better in MATH 1500

**MATH 1700H: Calculus II - Honors**

Definite integrals, applications and techniques of integration, elementary transcendental functions, infinite series. Math Reasoning Proficiency course.**Credit Hour****s**: 5**Prerequisites: **a grade of C- or better in MATH 1500. Honors eligibility required

**MATH 2100: Calculus for Social and Life Sciences II**

Riemann integral, transcendental functions, techniques of integration, improper integrals and functions of several variables. No credit for students who have completed two calculus courses. Math Reasoning Proficiency course.**Credit Hour****s**: 3**Prerequisites: **C- or higher in MATH 1320 or MATH 1400 or MATH 1500

**MATH 2300: Calculus III**

Vectors, solid analytic geometry, calculus of several variables. Math Reasoning Proficiency course.**Credit Hour****s**: 3**Prerequisites: **grade of C- or better in MATH 1700

**MATH 2300H: Calculus III - Honors**

Vectors, solid analytic geometry, calculus of several variables. Math Reasoning Proficiency course.**Credit Hour****s**: 3**Prerequisites: **grade of C or better in MATH 1700. Honors eligibility required

**MATH 2320: Discrete Mathematical Structures**

Sets, functions, logic, relations, induction, recursion, counting techniques, graphs, trees, algorithms. Math Reasoning Proficiency course.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 1700

**MATH 3000: Introduction to Advanced Mathematics**

Gateway to theoretical math courses. Focus on reading and writing math proofs/rigorously developing background needed in Adv Calc/Abstract Alg. Topics include logic, set theory, properties of functions and integers, the real number system, completeness of the real numbers, sequences of real numbers.**Credit Hour****s**: 3**Prerequisites: **Consent of Department required. Recommended MATH 1700

**MATH 3000W: Introduction to Advanced Mathematics - Writing Intensive**

Gateway to theoretical math courses. Focus on reading and writing math proofs/rigorously developing background needed in Adv Calc/Abstract Alg. Topics include logic, set theory, properties of functions and integers, the real number system, completeness of the real numbers, sequences of real numbers.**Credit Hour****s**: 3**Prerequisites: **Consent of Department required. Recommended MATH 1700

**MATH 4002: Topics in Mathematics-Biological/Physical/Math**

Organized study of selected topics. Subjects and earnable credit may vary from semester to semester. May repeat for credit with Departmental consent.**Credit Hour**: 1-99**Prerequisites: **MATH 2300 and instructor's consent

**MATH 4060: Connecting Geometry to Middle and Secondary Schools**

(cross-leveled with MATH 7060). Euclidean foundations, logic, Euler Characteristic, congruence, area, Pick's Theorem, volume, Cavalieri's Principle, surface area, similarity, symmetry, transformations, matricies, introduction to spherical geometry.**Credit Hour****s**: 3**Prerequisites: **Consent of Department required**Recommended: **MATH 1360 or MATH 1500

**MATH 4070: Connecting Algebra to Middle and Secondary Schools**

A detailed study of integer and rational arithmetic and algebra. Topics include: Bionomial Theorem, induction, division algorithm, Euclid's Algorithm, Fundamental Theorem of Arithmetic, Pythogorian triples, modular arithmetic and generalizations to polynomials, matrices and other axiomatic structures.**Credit Hour****s**: 3**Prerequisites: **MATH 1320, enrollment is restricted to Math Education majors

**MATH 4080: Calculus Connections**

Course topics include: sequences, series, functions, limits, continuity, differentiation, optimization, curve sketching, antidifferentiation, areas of plane regions, lengths of plane curves, areas of surfaces of revolution, and volumes of solids.**Credit Hour****s**: 3**Prerequisites: **MATH 1160, enrollment is restricted to Math Education majors

**MATH 4100: Differential Equations**

(cross-leveled with MATH 7100). Traditional introductory course in ordinary differential equations. Includes 1st and 2nd order linear differential equations with numerous applications; Laplace transforms; power series solutions; numerical methods, linear systems.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or above in MATH 2300

**MATH 4140: Matrix Theory**

(cross-leveled with MATH 7140). Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or better in MATH 2300 or MATH 2320. Writing intensive sections require ENGLSH 1000

**MATH 4140W: Matrix Theory - Writing Intensive**

Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or better in MATH 2300 or MATH 2320. Writing intensive sections require ENGLSH 1000

**MATH 4150: History of Mathematics**

This is a history course with mathematics as its subject. Includes topics in the history of mathematics from early civilizations onwards. The growth of mathematics, both as an abstract discipline and as a subject which interacts with others and with practical concerns, is explored. Prerequisites or Cirequisite: MATH 2300 or MATH 2340. Writing intensive sections require ENGLSH 1000.**Credit Hour****s**: 3

**MATH 4150W: History of Mathematics - Writing Intensive**

This is a history course with mathematics as its subject. Includes topics in the history of mathematics from early civilizations onwards. The growth of mathematics, both as an abstract discipline and as a subject which interacts with others and with practical concerns, is explored. Prerequisites or Cirequisite: MATH 2300 or MATH 2340. Writing intensive sections require ENGLSH 1000.**Credit Hour****s**: 3

**MATH 4300: Numerical Analysis**

Machine arithmetic, approximation and interpolation, numerical differentiation and integration, nonlinear equations, linear systems, differential equations, error analysis. Selected algorithms will be programmed for solution on computers.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or better in MATH 2300 and MATH 4100

**MATH 4310: Numerical Linear Algebra**

Solution of linear systems of equations by direct and iterative methods. Calculation of eigenvalues and eigenvectors of matrices. Selected algorithms programmed for solution on computers.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 and familiarity with software such as Mathematica, MatLab, Maple, etc

**MATH 4315: Introduction to Mathematical Statistics**

(same as STAT 4710). Introduction to theory of probability and statistics using concepts and methods of calculus.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 or instructor's consent

**MATH 4320: Introduction to Probability Theory**

(same as STAT 4750). Probability spaces; random variables and their distributions; repeated trials; probability limit theorems.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 or instructor's consent

**MATH 4330: Theory of Numbers**

Divisibility, factorization, arithmetic functions, means value theorems, distribution of prime numbers, congruences, primitive roots, character theory, Riemann zeta function, and Dirichlet L-functions.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 2300. Recommended MATH 2320

**MATH 4350: Introduction to Non-Euclidean Geometry**

(cross-leveled with MATH 4350). Account of rise, development of non-Euclidean geometries. Intensive study of plane hyperbolic geometry.**Credit Hour****s**: 3**Prerequisites: **MATH 2300

**MATH 4355: Mathematics of Financial Derivatives I**

(cross-leveled with MATH 7355). Long and short positions, forward contracts, exchange traded index futures, European and American call and put options, put-call parity, trading and hedging strategies, synthetic transactions, arbitrage, currency options, fixed income portfolio management, duration, convexity, interest rate and currency swaps, embedded options.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 2300 and either STAT 2500 or STAT 4710 or MATH 4315

**MATH 4370: Interest Theory**

(cross-leveled with MATH 7370). This course covers the concepts underlying the theory of interest and their applications to valuation of various cash flows, annuities certain, bonds, and loan repayment. This course is designed to help students prepare for Society of Actuaries exam FM (Financial Mathematics). It is oriented towards problem solving techniques applied to real-life situations and illustrated with previous exam problems.**Credit Hour****s**: 3**Prerequisites: **grade of C-or better in MATH 2300

**MATH 4371: Models for Life Contingencies I**

(cross-leveled with MATH 7371). The sequence MATH 4371 - MATH 4372 is designed to help students prepare for the Society of Actuaries exam LTAM (Long-Term Actuarial Mathematics). This course teaches the basic theory of life contingent actuarial models and the applications of those models to insurance and other financial risks. Covered topics include: survival models, life tables and selection, insurance benefits, life annuities, premium calculation and reserves.**Credit Hour****s**: 3**Prerequisites: **MATH 4320 or STAT 4750, and MATH 4370

**MATH 4372: Models for Life Contingencies II**

(cross-leveled with MATH 7372). The sequence MATH 4371 - MATH 4372 is designed to help students prepare for the Society of Actuaries exam LTAM (Long-Term Actuarial Mathematics). This course extends the life-death contingency models of MATH 4371 to more general multiple-state and multiple-life models applied to problems involving a wide range of insurance and pension benefits. Covered topics include: Markov chains, multiple decrement models, joint life and last survivor benefits, pension mathematics, profit testing.**Credit Hour****s**: 3**Prerequisites: **A grade of C- or better in MATH 4371

**MATH 4400: Introduction to Topology**

Topics from topology of Euclidean spaces, generalizations to metric spaces and topological spaces. Fundamentals of point set topology.**Credit Hour****s**: 3**Prerequisites: **MATH 2300

**MATH 4500: Applied Analysis**

Solution of the standard partial differential equations (wave, heat, Laplace's eq.) by separation of variables and transform methods; including eigenfunction expansions, Fourier and Laplace transform. Boundary value problems, Sturm-Liouville theory, orthogonality, Fourier, Bessel, and Legendre series, spherical harmonics.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 4100

**MATH 4510: Higher Algebra**

Introduction to rings, integral domains, fields, groups.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 2300 or MATH 2320

**MATH 4520: Statistical Inference I**

(same as STAT 4760). Sampling; point estimation; sampling distribution; tests of hypotheses; regression and linear hypotheses.**Credit Hour****s**: 3**Prerequisites: **MATH 4320

**MATH 4540: Mathematical Modeling I**

Solution of problems from industry, physical, social and life sciences, economics, and engineering using mathematical models.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 2300 and MATH 4100**Recommended: **Familiarity with software such as MATHEMATICA, MATLAB, or MAPLE

**MATH 4560: Nonlinear Dynamics, Fractals and Chaos**

Conceptual introduction to nonlinear dynamics, bifurcation and stability of steady states, chaos in nonlinear differential equations and maps, fractal dimension, strange attractors, and applications to physical science.**Credit Hour****s**: 3**Prerequisites: **MATH 4100 or MATH 7100, MATH 4140 or MATH 7140, and familiarity with software such as MATHEMATICA, MATLAB, or MAPLE

**MATH 4590: Mathematics of Financial Derivatives II**

(cross-leveled with MATH 7590). Binomial and Black-Scholes pricing models.Option Greeks, delta and gamma hedging, market maker profit theory. Asian, barrier, compound gap and exchange options. Lognormal and Monte Carlo price simulation. Geometric Brownian Motion and Ito's Lemma. Interest rate models and volatility.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 and either STAT 2500 or STAT 4710 or MATH 4315**Recommended: **MATH 4355

**MATH 4700: Advanced Calculus of One Real Variable I**

(cross-leveled with MATH 7700). Basic topology of the real line, numerical sequences and series, continuity, differentiability, Riemann integration, uniform convergence, power series.**Credit Hour****s**: 3**Prerequisites: **Grade of C - or higher in MATH 3000

**MATH 4720: Introduction to Abstract Algebra I**

Basic properties of integers, fundamental theorem of arithmetic, introduction to groups, rings and fields.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 3000

**MATH 4900: Advanced Multivariable Calculus**

(cross-leveled with MATH 7900). This is a course in calculus in several variables. The following is core material: Basic topology of n-dimensional Euclidian space; limits and continuity of functions; the derivative as a linear transformation; Taylor's formula with remainder; the Inverse and Implicit Function Theorems, change of coordinates; integration (including transformation of integrals under changes of coordinates); Green's Theorem. Additional material from the calculus of several variables may be included, such as Lagrange multipliers, differential forms, etc.**Credit Hour****s**: 3**Prerequisites: **MATH 4700

**MATH 4920: Introduction to Abstract Linear Algebra**

(cross-leveled with MATH 7920). Study of vector spaces over arbitrary fields: topics include linear maps on finite dimensional vector spaces, bilinear and multi-linear forms, invariant subspaces and canonical forms.**Credit Hour****s**: 3**Prerequisites: **Grade of C- or higher in MATH 4720

**MATH 4940: Introduction to Complex Variables**

Complex functions, contour integration, power series, residues and poles, conformal mapping.**Credit Hour****s**: 3**Prerequisites: **MATH 4110 or MATH 4700

**MATH 4960: Special Readings in Mathematics**

**Credit Hour**: 1-3**Prerequisites: **Consent of Department required

**MATH 4996: Honors in Mathematics**

Special work for senior B.A. Honors and B.S. Honors candidates.**Credit Hour****s**: 2**Prerequisites: **Consent of Department required

**MATH 7100: Differential Equations**

(cross-leveled with MATH 4100). Traditional introductory course in ordinary differential equations. Includes 1st and 2nd order linear differential equations with numerous applications; Laplace transforms; power series solutions; numerical methods, linear systems.**Credit Hour****s**: 3**Prerequisites: **graduate standing and MATH 2300

**MATH 7140: Matrix Theory**

(cross-leveled with MATH 4140). Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs.**Credit Hour****s**: 3**Prerequisites: **graduate standing and one of MATH 2300, MATH 2320, MATH 2120 or MATH 2340

**MATH 7150: History of Mathematics**

This is a history course with mathematics as its subject. Includes topics in the history of mathematics from early civilizations onwards. The growth of mathematics, both as an abstract discipline and as a subject which interacts with others and with practical concerns, is explored. Pre- or Co-requisite: MATH 2300 or MATH 2340 and graduate standing.**Credit Hour****s**: 3

**MATH 7300: Numerical Analysis**

Machine arithmetic, approximation and interpolation, numerical differentiation and integration, nonlinear equations, linear systems, differential equations, error analysis. Selected algorithms will be programmed for solution on computers.**Credit Hour****s**: 3**Prerequisites: **graduate standing and MATH 2300 and MATH 4100 or equivalent

**MATH 7310: Numerical Linear Algebra**

Solution of linear systems of equations by direct and iterative methods. Calculation of eigenvalues and eigenvectors of matrices. Selected algorithms programmed for solution on computers.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 and prior experience writing programs in Mathematica and /or in a computer language such as Fortran, Pascal, or C**Recommended: **MATH 4140

**MATH 7330: Theory of Numbers**

Divisibility, factorization, arithmetic functions, means value theorems, distribution of prime numbers, congruences, primitive roots, character theory, Riemann zeta function, and Dirichlet L-functions.**Credit Hour****s**: 3**Prerequisites: **MATH 2300; recommended MATH 2320 or MATH 2340, and MATH 4940 or MATH 7940

**MATH 7355: Mathematics of Financial Derivatives I**

(cross-leveled with MATH 4355). Long and short positions, forward contracts, exchange traded index futures, European and American call and put options, put-call parity, trading and hedging strategies, synthetic transactions, arbitrage, currency options, fixed income portfolio management, duration, convexity, interest rate and currency swaps, embedded options.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 and STAT 2500 or STAT 4710/ MATH 4315, or instructor's consent

**MATH 7370: Interest Theory**

(cross-leveled with MATH 4370). This course covers the main probability tools applied to financial risk modeling, and the financial mathematics concepts used in calculating present and accumulated values for various cash flows. It is a helpful tool in preparing for the Society of Actuaries exams P (Probability) and FM (Financial Mathematics), and it is oriented towards problem solving techniques illustrated with previous exam problems. Students are encouraged to take MATH 4355 prior to this course.**Credit Hour****s**: 3**Prerequisites: **MATH 2300, MATH 4320/ STAT 4750

**MATH 7371: Models for Life Contingencies I**

(cross-leveled with MATH 4371). The sequence MATH 4371 - MATH 4372 is designed to help students prepare for the Society of Actuaries exam LTAM (Long-Term Actuarial Mathematics). This course teaches the basic theory of life contingent actuarial models and the applications of those models to insurance and other financial risks. Covered topics include: survival models, life tables and selection, insurance benefits, life annuities, premium calculation and reserves.**Credit Hour****s**: 3

**MATH 7400: Introduction to Topology**

Topics from topology of Euclidean spaces, generalizations to metric spaces and topological spaces. Fundamentals of point set topology.**Credit Hour****s**: 3**Prerequisites: **MATH 2300

**MATH 7500: Applied Analysis**

Solution of the standard partial differential equations (wave, heat, Laplace's eq.) by separation of variables and transform methods; including eigenfunction expansions, Fourier and Laplace transform. Boundary value problems, Sturm-Liouville theory, orthogonality, Fourier, Bessel, and Legendre series, spherical harmonics.**Credit Hour****s**: 3**Prerequisites: **MATH 4100/ MATH 7100

**MATH 7510: Higher Algebra**

Introduction to rings, integral domains, fields, groups.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 OR MATH 2320

**MATH 7540: Mathematical Modeling I**

Solution of problems from industry, physical, social and life sciences, economics, and engineering using mathematical models.**Credit Hour****s**: 3**Prerequisites: **graduate standing and 3 semesters of calculus and some exposure to ordinary differential equations or instructor's consent

**MATH 7560: Nonlinear Dynamics, Chaos and Fractals**

Conceptual introduction to nonlinear dynamics, bifurcation and stability of steady states, chaos in nonlinear differential equations and maps, fractal dimension, strange attractors, and applications to physical science.**Credit Hour****s**: 3**Prerequisites: **graduate standing and MATH 4100/ MATH 7100, MATH 4140/ MATH 7140, and familiarity with software such as MATHEMATICA, MATLAB, or MAPLE

**MATH 7590: Mathematics of Financial Derivatives II**

(cross-leveled with MATH 4590). Binomial and Black-Scholes pricing models.Option Greeks, delta and gamma hedging, market maker profit theory. Asian, barrier, compound, gap and exchange options. Lognormal and Monte Carlo price simulation. Geometric Brownian Motion and Ito's Lemma. Interest rate models and volatility. Prerequisites: MATH 2300 and either STAT 2500 or STAT 4710 or MATH 4315.**Credit Hour****s**: 3**Prerequisites: **MATH 2300 and either STAT 2500 or STAT 4710 or MATH 4315**Recommended: **MATH 4355

**MATH 7700: Advanced Calculus of One Real Variable I**

(cross-leveled with MATH 4700). Series of real numbers, limits of functions, continuity and uniform continuity, differentiability, and Riemann integration.**Credit Hour****s**: 3**Prerequisites: **MATH 2300**Recommended: **MATH 4140 and one other mathematics course number above MATH 2300

**MATH 7720: Introduction to Abstract Algebra I**

Basic properties of integers, fundamental theorem of arithmetic, introduction to groups, rings and fields.**Credit Hour****s**: 3**Prerequisites: **MATH 2300**Recommended: **MATH 4140 and one other Mathematics course numbered above 2300

**MATH 7900: Advanced Multivariable Calculus**

(cross-leveled with MATH 4900). This is a course in calculus in several variables. The following is core material: Basic topology of n-dimensional Euclidian space; limits and continuity of functions; the derivative as a linear transformation; Taylor's formula with remainder; the Inverse and Implicit Function Theorems, change of coordinates; integration (including transformation of integrals under changes of coordinates); Green's Theorem. Additional material from the calculus of several variables may be included, such as Lagrange multipliers, differential forms, etc.**Credit Hour****s**: 3**Prerequisites: **MATH 4700/ MATH 7700

**MATH 7920: Introduction to Abstract Linear Algebra**

(cross-leveled with MATH 4920). Study of vector spaces over arbitrary fields: topics include linear maps on finite dimensional vector spaces, bilinear and multi-linear forms, invariant subspaces and canonical forms.**Credit Hour****s**: 3**Prerequisites: **MATH 2300

**MATH 7940: Introduction to Complex Variables**

Complex functions, contour integration, power series, residues and poles, conformal mapping.**Credit Hour****s**: 3**Prerequisites: **MATH 4110/ MATH 7110 OR MATH 4700/ MATH 7700

**MATH 7960: Special Readings in Mathematics**

**Credit Hour**: 1-3**Prerequisites: **MATH 2300 and instructor's consent

**MATH 7980: Mathematics Problem Solving**

Creative advanced problem solving bringing together methods such as integration, probability and Euclidean geometry.**Credit Hour****s**: 3**Prerequisites: **MATH 4140 and another 4000 level Mathematics course, or instructor's consent

**MATH 8085: Problems in Mathematics**

Problems in Mathematics**Credit Hour****s**: 3

**MATH 8090: Master's Thesis Research in Mathematics**

Students will be required to complete an independent project. Topics are chosen in consultation with a faculty advisor and are subject to departmental consent. Graded on S/U basis only.**Credit Hour****s**: 3

**MATH 8190: Masters Project in Mathematics**

Masters Project in Mathematics**Credit Hour****s**: 3

**MATH 8202: Topics in Functional Analysis**

Topics in Functional Analysis**Credit Hour****s**: 3

**MATH 8210: Basic Algebra**

Accelerated problem solving course in linear and abstract algebra. Will prepare students for the algebra qualifying exam.**Credit Hour****s**: 3**Prerequisites: **MATH 4720, MATH 4920, or instructor's consent, or equivalent**Corequisites: **MATH 8220 and MATH 8250

**MATH 8220: Basic Analysis**

Accelerated problem-solving course in advanced calculus and complex analysis. Will prepare students for the analysis qualifying exam.**Credit Hour****s**: 6**Prerequisites: **MATH 4700, MATH 4900, MATH 4940, instructor's consent or equivalent

**MATH 8250: Basic Topology and Geometry**

Topological spaces, differential manifolds, differential forms, integration of vector fields.**Credit Hour****s**: 3**Prerequisites: **MATH 4700, MATH 4900, MATH 4140, or instructor's consent, or equivalent**Corequisites: **MATH 8210 and MATH 8220

**MATH 8302: Topics in Harmonic Analysis**

Topics in Harmonic Analysis**Credit Hour****s**: 3

**MATH 8402: Topics in Mathematical Physics**

Topics in Mathematical Physics**Credit Hour****s**: 3

**MATH 8410: Algebra I**

Theory of algebraic structures--groups, rings, fields, algebraic and transcendental extensions of fields.**Credit Hour****s**: 3**Prerequisites: **MATH 4720 and MATH 4920, or equivalent

**MATH 8411: Algebra II**

Theory of modules, Galois theory and additional topics to be selected by the instructor.**Credit Hour****s**: 3**Prerequisites: **MATH 8410 or equivalent

**MATH 8420: Theory of Functions of Real Variables I**

Properties of functions of one real variable. Lebesgue measure and integration on the line.**Credit Hour****s**: 3**Prerequisites: **MATH 4700 or MATH 7700 and MATH 4900 or MATH 7900, or equivalent

**MATH 8421: Theory of Functions of Real Variables II**

Continuation of MATH 8420. General measure and integration theory. Elements of the theory of Hilbert and Banach spaces, linear functions and linear operators.**Credit Hour****s**: 3**Prerequisites: **MATH 8420

**MATH 8425: Complex Analysis I**

Rigorous introduction to the theory of functions of a complex variable.**Credit Hour****s**: 3**Prerequisites: **MATH 4940 or MATH 7940 or equivalent

**MATH 8426: Complex Analysis II**

Analytic continuation, Riemann surfaces, entire and meromorphic functions, selected topics.**Credit Hour****s**: 3**Prerequisites: **MATH 8425

**MATH 8440: Advanced Ordinary Differential Equations I**

Topics from existence and uniqueness theorems, plane autonomous systems, periodicity and boundedness of solutions of second order nonlinear equations, perturbation theory, Sturm-Liouville systems, behavior of solutions at singularities.**Credit Hour****s**: 3**Prerequisites: **MATH 4700 or MATH 7700 or equivalent

**MATH 8445: Partial Differential Equations I**

Fourier and integral transforms, first and second order partial differential equations, methods of characteristics, Laplace's equation, Direchlet and Neumann problems, Green's functions and maximum principles.**Credit Hour****s**: 3**Prerequisites: **MATH 4700 or MATH 7700 or instructor's consent required

**MATH 8446: Partial Differential Equations II**

The Cauchy-Kovalevski theorem, the Lewy example, the heat operator, the wave operator, Sobolev spaces, local regularity of elliptic boundary value problems.**Credit Hour****s**: 3**Prerequisites: **MATH 8445**Recommended: **MATH 8420

**MATH 8460: Mathematical Finance I**

(same as FINANC 8340). Financial instruments and derivative: stocks, bonds, futures option prices on interest rates, swaps, etc. Mathematical models of stock price fluctuations. Interest rates and options on interest rates. Swaps. Open markets and properties of stock option prices. Stochastic models. Bionomial trees. Continuous time stochastic modeling. No arbitrage modeling. European and American options. BlackScholes model and differential equation, for the price of European option. Exotic options. Interest rate models.**Credit Hour****s**: 3**Prerequisites: **graduate standing in Mathematics. Knowledge of elementary probability or instructor's consent

**MATH 8461: Mathematical Finance II**

Diffusion Processes as models for stock price fluctuations. Contingent claims and arbitrage. Mathematical analysis of risk neutral valuation of contingent claims. Self-financing portfolios and hedging. Hedging contingent claims. Partial differential equations for valuation of derivative securities. Completeness of the markets and hedging. Parity relations and delta hedging. Several underlying assets.**Credit Hour****s**: 3**Prerequisites: **knowledge of advance probability/stochastic processes or instructor's consent**Recommended: **MATH 8460

**MATH 8480: Advanced Probability**

(same as STAT 9810). Measure theoretic probability theory. Characteristic functions; conditional probability and expectation; sums of independent random variables including strong law of large numbers and central limit problem.**Credit Hour****s**: 3**Prerequisites: **MATH 4320 or MATH 8220; or instructor's consent

**MATH 8502: Topics of Geometry**

Topics of Geometry.**Credit Hour****s**: 3**Prerequisites: **instructor's consent

**MATH 8587: Topology Seminar**

Topology Seminar**Credit Hour****s**: 3

**MATH 8615: Algebraic Geometry I**

Affine and projective varieties and schemes; nullstellensatz; Zariski topology, morphisms and rational maps; divisors and linear systems; topics from curves, surfaces, Grassmann varieties; commutative algebra and homological algebra as needed.**Credit Hour****s**: 3**Prerequisites: **MATH 8410

**MATH 8618: Introduction to Algebraic Topology**

Development of singular homology theory; reference to other homology and cohomology theories. Introduction to homological algebra.**Credit Hour****s**: 3**Prerequisites: **MATH 8250

**MATH 8628: Functional Analysis I**

Linear topological spaces, Banach spaces, Hilbert spaces. Operator theory, including the Hahn-Banach, uniform boundedness and closed graph theorems.**Credit Hour****s**: 3**Prerequisites: **MATH 4900 and instructor's consent or MATH 8420

**MATH 8630: Harmonic Analysis I**

An introduction to Fourier Analysis in one and higher Dimensions. Topics include Fourier Series, conjugate functions, Fourier transforms, distributions, interpolation, and maximal functions.**Credit Hour****s**: 3**Prerequisites: **MATH 8420

**MATH 8631: Harmonic Analysis II**

Singular integrals, Littlewood-Paley theory, Hardy spaces, bounded mean oscillation, weighted norm inequalities, boundary value problems, and analysis on groups.**Credit Hour****s**: 3**Prerequisites: **MATH 8630

**MATH 8650: Differentiable Manifolds and Riemannian Geometry**

Tensor product spaces and tensor fields on manifolds. Differentiation and integration of differential forms. Riemannian geometry and applications.**Credit Hour****s**: 3**Prerequisites: **MATH 4700 or MATH 4400

**MATH 8655: General Topology I**

Introduction to axiomatic theory of general topology. Continuous functions and homeomorphisms. Convergence in abstract topological spaces. Compact and locally compact spaces. Connectedness. Metrizable spaces.**Credit Hour****s**: 3**Prerequisites: **MATH 4900, MATH 4400 or instructor's consent

**MATH 8702: Topics in Applied Mathematics**

Selected topics in applied mathematics drawn from variety of areas: partial differential equations, tensor analysis, calculus of variations, asymptotic methods, integral equations, advanced theory of transforms and distributions, numerical analysis.**Credit Hour****s**: 3

**MATH 9090: Doctoral Dissertation Research in Mathematics**

Doctoral Dissertation Research in Mathematics. Graded on a S/U basis only.**Credit Hour**: 1-9

**MATH 9387: Harmonic Analysis Seminar**

Harmonic Analysis Seminar**Credit Hour****s**: 3

**MATH 9487: Mathematical Physics Seminar**

Mathematical Physics Seminar**Credit Hour****s**: 3

**MATH 9502: Topics in Topology**

Advanced topics in topology or topological algebra.**Credit Hour****s**: 3

**MATH 9787: Applied Mathematics Seminar**

Applied Mathematics Seminar**Credit Hour****s**: 3

**MATH 9887: Analysis Seminar**

Analysis Seminar**Credit Hour****s**: 3