# Department of Mathematics

Director, Graduate Program

The Department of Mathematics offers numerous graduate degree programs and one advanced certificate program. The graduate programs lead to an MS in Mathematics, an MS in Applied Mathematics-Data Analytics, a 5 year BA-MS or BS-MS in Applied Mathematics-Data Analytics, or, in cooperation with the School of Education, a 5 year BS-MS in Adolescence Education Mathematics, Teacher's Option. There is also an Advanced Certificate in Applied Mathematics-Data Analytics. The degree requirements, as well as the entrance requirements, for each degree, are listed below.

## MS in Mathematics

This program is designed for in-service secondary Mathematics teachers who desire further study in the discipline and for individuals who hope to teach Mathematics in a Community College or to pursue a Ph.D. in Mathematics. Students in the program will expand their understanding of tertiary Mathematics in four key areas: Algebra, Analysis, Statistics/Data Analysis, and Number Theory.

Normally, an undergraduate concentration in Mathematics is required for admission to this program, although students with a different preparation may apply if they have a strong background in mathematics. This preparation should include courses in Linear Algebra, Abstract Algebra, Analysis, and Probability Theory.

The program requires 30 credits of graduate mathematics including a 3 credit Statistics/Data Analysis elective, a 3 credit Algebra elective, a 3 credit Analysis elective, 3 credits of Number Theory, and a 3 credit Research Seminar. Fifteen additional elective credits round out the program. The student must pass Masters Comprehensive exams with a B or better in 3 of the 4 content areas of statistics, algebra, analysis, and number theory. Final exams in these courses will comprise the comprehensive exam. A research project is required.

This program may be completed on either a full-time or part-time basis. Qualified undergraduates may begin graduate classes as upperclassmen, thereby enabling completion of the MS degree in a single postgraduate year.

### Required Graduate Courses for the MS in Mathematics:

- Algebra Elective - MATG 571 Advanced Linear Algebra with Applications or MATG 678 Algebra II
- Analysis Elective - MATG 688 Graduate Analysis II or MATG 690 Graduate Complex Analysis
- Statistics / Data Analysis Elective - MATG 630 Probability and Statistics for Analytics or MATG 633 Advanced Statistical Inference
- Number Theory - MATG 542 Number Theory
- Research Seminar - MATG 699 Research in Mathematics
- 15 additional elective credits of graduate mathematics (MATG 500-799).

## MS in Applied Mathematics-Data Analytics

#### Program Description

This program is designed for students with a strong background in mathematics and a major in a quantitative field who wish to prepare for careers in data science, actuarial science, business, industry, or government, or for further study at the doctoral level. Applicants should possess a degree in a STEM or quantitative discipline, some exposure to computer programming, and have the desire to learn sufficient techniques of mathematical analysis to apply them in the profession. Students entering the program should have at least 18 credits of mathematics or the equivalent, including multi-variable calculus, probability, and linear algebra.

Students will typically complete the program in 3 semesters plus 1 summer. Students may also pursue the program part-time.

The 30 credit hour program consists of a core of study in computational methods, statistics, machine learning, linear algebra, and operations research, augmented by electives. There are four Masters Comprehensive Exams. These are the final exams (or term projects) in MATG 511, 571, 630, and 635. Three of these must be passed with a B or better, in order to complete the program.

Students may choose 2 electives in accordance with their personal interests, either in Mathematics, Computer Science, Engineering or Business Administration, or they may elect to pursue an Internship or Research Project.

### Required Courses:

MATG 511 | Computational Methods for Analytics | 3 |

MATG 555 | Operations Research | 3 |

MATG 557 | Machine Learning | 3 |

MATG 571 | Advanced Linear Algebra with Applications | 3 |

MATG 630 | Probability and Statistics for Analytics | 3 |

MATG 633 | Advanced Statistical Inference | 3 |

MATG 635 | Probabilistic Methods | 3 |

MATG 659 | Data Base Methods for Analytics | 3 |

Plus two electives | 6 | |

Total Credits | 30 |

## BA or BS Mathematics -- MS Applied Mathematics-Data Analytics

The program is a seamless 5 year BS-MS program with a major in Mathematics and an MS in Applied Mathematics-Data Analytics.

This program is designed for strong students of mathematics who wish to prepare for careers in business, industry, or government, or for further study at the doctoral level. In addition to the core undergraduate courses in the discipline, at the graduate level students will master probabilistic and statistical methods, machine learning, and optimization. Students also have the opportunity to complete minors in cognate disciplines.

Students will typically complete all requirements for the BS in 4 years. They will apply to the MS program during the spring of their sophomore year or during the junior year. If accepted, they take graduate classes during the 3rd and 4th year of study and will complete the requirements for the MS degree in a fifth year.

## Advanced Certificate in Applied Mathematics - Data Analytics

To complete an Advanced Certificate in Applied Mathematics-Data Analytics, a student must complete 18 credits, to be chosen in consultation with the Graduate Director from the MATG courses eligible for credit towards the MS in Applied Mathematics-Data Analytics.

## BS in Adolescent Education Mathematics, MS in Mathematics, Teacher's Option

This program is a seamless 5 year program designed for the undergraduate student seeking certification for teaching in grades 7-12. After a major in Adolescence Education Mathematics, the advanced student will complete an MS in Mathematics, Teacher's Option. Initial certification requirements are completed during the first four years. The MS is one required step towards conversion of initial certification to professional certification in Adolescence Education Mathematics, the second step being the completion of three years of teaching experience. The 5-year program is grounded in a deep knowledge of college mathematics and its connection to secondary mathematics and to the common core standards. Qualified students may use undergraduate electives to begin graduate course work, earning nine graduate credits during the first four years. Upon satisfactory completion of specific program requirements for the bachelor’s degree, and successful completion of LAST, ATS-W, and CST (Multi-subject and Students with Disabilities), students will be recommended for initial certification in each area.

After completion of the requirements for initial certification, students pursue a 5^{th} year of graduate mathematics. Required courses include a two-term sequence in the roots of high school Geometry and Trigonometry and a course on Contemporary Issues in Teaching Mathematics. Students also participate in departmental seminars and colloquia. An additional elective rounds out the program.

Graduates of the Manhattan College masters program in Mathematics, Teachers Option, will gain deep knowledge in five content areas: Algebra, Number Theory, Calculus, Probability and Statistics, and Geometry and make connections from these content areas to the Secondary Curriculum and to the common core standards. They will understand how to use software and technology effectively in the teaching of mathematics and communicate mathematics effectively both orally and in writing.

### Admission

Undergraduates will apply after completion of the fall term of junior year. A minimum cumulative GPA of 3.3 in the mathematics classes MATH 185 Calculus I, MATH 186 Calculus II, MATH 285 Calculus III, MATH 243 Foundations for Higher Mathematics,MATH 272 Linear Algebra I , and MATH 377 Algebra is normally required.

### Degree Program

Students complete the required sequence of undergraduate courses during freshman, sophomore, junior and senior year. In the third year, they enroll in one graduate class. In the fourth year, they are enrolled in two 3 credit graduate courses and complete the requirements for initial certification. In the fifth year, students complete an additional 21 graduate credits, for a total of 30 graduate credits. Graduates will demonstrate mastery of the tertiary mathematics curriculum, its connections to the secondary curriculum, the common core standards, and in particular, the foundations for Calculus, Algebra, Number Theory, Geometry and Statistical Inference. Students will take Masters Comprehensives in three content areas to be chosen by the student in consultation with the director.

### Required Graduate Courses for the BS-MS in Adolescence Education Mathematics, Teacher's Option:

MATG 542 | Number Theory | 3 |

Algebra Elective | 3 | |

Analysis Elective | 3 | |

Statistic Elective | 3 | |

MATG 622 | Graduate Seminar for Mathematics Education | 3 |

MATG 724 | Cont. Issues in Teaching Math | 4 |

MATG 761 | Classical Methods in Plane Euclidean Geometry | 4 |

MATG 762 | Modern Methods in Plane Euclidean Geometry | 4 |

Additional elective credits of Graduate mathematics (MATG 500-799) | 3 | |

Total Credits | 30 |

### Courses

**MATG 511. Computational Methods for Analytics. 3 Credits.**

A survey of programming tools of use in solving problems in predictive analytics and applied mathematics: Possible languages include Python, Matlab, R. Introduction to methods for data visualization.

**MATG 532. Statistical Inference. 3 Credits.**

Topics covered in this course include sampling distributions, point estimation, interval estimation, testing statistical hypotheses, regression and correlation. Requires a project. Prerequisite: MATH 331. Not available to students with credit for MATH 432 or MATG 630.

**MATG 542. Number Theory. 3 Credits.**

An introduction to number theory with connections to the Middle and High school curriculum. Divisibility, prime numbers and their distribution, congruences, quadratic residues and nonresidues, Diophantine equations, elliptic curves, primality testing, applications to cryptology. Recent progress. The course requires a written project connecting the course content to the 6-12 curriculum. Prerequisite: MATH 272.

**MATG 548. Combinatorics and Graph Theory. 3 Credits.**

Fundamental concepts in combinatorics include binomial coefficients, inclusion-exclusion, and generating functions. Topics in graph theory include connectivity, planarity, colorings and chromatic polynomials, and max-flow-min-cut in networks. This course will require a written project and an oral presentation on some particular application of Graph Theory or Combinatorics. The project will consist of a case study that will require researching a particular area of application, and then formulating, solving, and analyzing an appropriate mathematical model. Findings will be presented at the end of the term. Not open to students with credit for MATH 448 or Cmpt. 335. Prerequisites: MATH 243, MATH 272.

**MATG 550. Financial Models. 3 Credits.**

The course covers the following topics: the growth of money, equations of value and yield rates, annuities certain, annuities with different payment and conversion periods, loan repayment, bonds, stocks and financial markets, arbitrage, term structure of interest rates, derivatives, and interest rate sensitivity. Prerequisite: Graduate status or permission of Graduate director.

**MATG 551. Financial Engineering. 3 Credits.**

Basic mathematical foundations and numerical techniques required to understand quantitative finance. Interest rates, hedging, Black Scholes formula, bootstrapping, finite differences and PDE's, Bond and portfolio optimization. Prerequisite: MATG 550.

**MATG 555. Operations Research. 3 Credits.**

Optimization, linear programming, simplex method, duality theory, transportation problems, scheduling problems, queuing theory. Students will be required to complete an independent project. The project will consist of a case study that will require researching a particular area of application, and then formulating, solving, and analyzing an appropriate mathematical model. Findings will be presented at the end of the term. Not open to students with credit for MATH 455. Prerequisite: MATH 272.

**MATG 556. Non-Linear Optimization. 3 Credits.**

Methods for solving non-linear optimization problems. Topics include unconstrained optimization, convex sets, approximation methods, method of least squares, convex programming, penalty methods and mixed constraints. Pre-requisite: Graduate status or permission of Graduate director.

**MATG 557. Machine Learning. 3 Credits.**

An introduction to the field of machine learning and its real-world applications, emphasizing the coding of machine learning algorithms to iteratively learn from data and to automate analytical model building. Topics include supervised & unsupervised learning, Bayesian decision theory, non-parametric methods, linear discriminant functions, multi-layer neural networks, stochastic methods and cluster analysis. Programming experience, preferably in Matlab, will be useful. A project is required. Prerequisites: MATH 272 and MATH 285 and (MATH331 or MATH 336) or permission of the Graduate Director. 3 credits.

**MATG 558. Data Mining. 3 Credits.**

Basic concepts of data mining. Fundamental aspects and techniques of analyzing large, complex data-sets. Topics include data objects and attributes, measuring data similarity, data visualization, data processing, apriori algorithm, classification methods, cluster analysis, outlier identification. Pre-requisite: MATG 511.

**MATG 564. Topology. 3 Credits.**

An Introduction to Topology, beginning with the concept of topological equivalence, and topological invariants. Knots and Links, colorings, knot polynomials, Euler characteristic, cut and paste techniques, classification of surfaces, 3-manifolds, and the fundamental group, the Poincare conjecture. This course will require a written project and an oral presentation on some particular application of, or historical development in Topology. Not open to students with credit for MATH 464. (Cr. 3) Prerequisite: MATH 243.

**MATG 571. Advanced Linear Algebra with Applications. 3 Credits.**

A continuation of topics introduced in Linear Algebra, (MATH 272), covering factorization of matrices. eigenvalues and eigenvectors, orthogonality, optimization problems, ill-conditioned matrices, applications to topics such as least-squares approximation, difference and differential equations, linear programming, networks, game theory. (Cr. 3) Prerequisite: MATH 272 or equivalent.

**MATG 622. Graduate Seminar for Mathematics Education. 3 Credits.**

This course is intended for prospective mathematics teachers. Topics in high school mathematics are examined from an advanced perspective. Topics include the real and complex numbers, functions, and trigonometry. The course requires a written project and an oral presentation. The use of appropriate technology will incorporated throughout the course. (Cr. 3) Prerequisites: MATH 243 and MATH 272.

**MATG 630. Probability and Statistics for Analytics. 3 Credits.**

Basic theorems in probability, random variables, distribution functions, expected values; binomial, Poisson and normal distributions. Sampling distributions, point estimation, interval estimation, testing statistical hypotheses. Not open to students who have taken MATH 331-432.

**MATG 633. Advanced Statistical Inference. 3 Credits.**

This is a data intensive course on statistical inference. Topics covered in this course include regression analysis, hypothesis testing, analysis of variance, nonparametic modeling, and sequential tests of hypotheses. Students will utilize appropriate software for data analytics. Not open to students with credit for MATH 433. Prerequisite: MATH 432 or MATG 630 or permission of the Graduate Director.

**MATG 635. Probabilistic Methods. 3 Credits.**

An introduction to probability models including random variables, conditional probability and expectation, Markov chains, Poisson processes, continuous time Markov processes and queueing theory. Prerequisites: MATH 331 or MATG 630 or equivalent.

**MATG 639. Statistical Learning. 3 Credits.**

u A course on the statistical foundations of machine learning, this course develops the fundamental ideas of statistical learning for drawing conclusions from multivariate data sets using statistical theory and applied linear algebra. The course combines a theoretical presentation with computation of the resulting machine-learning algorithms on real data sets to develop intuition of both how the methods work and how they perform in practice. It will cover the major techniques and concepts for both supervised and unsupervised learning. Topics will include regression, classification, resampling methods, model selection, regularization, principal components and clustering. Optional selected topics include tree-based methods, support vector machines and neural networks. Prerequisites: MATG 633, MATG 571.

**MATG 659. Data Base Methods for Analytics. 3 Credits.**

Provides students with an in-depth understanding of the design, implementation and management of SQL, transactional database design, normalizing tables, functional dependencies, entity-relationship and relational database models; use of object-oriented design and event programming. Additional topics may include data warehouse modeling, analytics database systems, administration, security and other topics as time and interest permit.

**MATG 678. Algebra II. 3 Credits.**

This is the second part of a two-semester sequence. We undertake further study of algebraic structures, such as rings, fields and integral domains. Significant results include the Fundamental Homomorphism Theorem and Unique Factorization. The course requires a written project. Not open to students with credit for MATH 478. Prerequisite: MATH 377.

**MATG 688. Graduate Analysis II. 3 Credits.**

This course is a successor to Analysis I. The approach followed here is a rigorous treatment of the material found in Calculus I and II leading to an introduction to measure theory and the modern definition of the integral. The first part of the course covers the Riemann Integral, infinite series, sequences and series of functions with an emphasis on uniform convergence and its consequences. This leads to the need to extend the definition of the integral to allow for the treatment of more complicated functions. The approach of Lebesgue leads to a new integral with vastly improved convergence properties. Not open to students with credit for MATH 488. Prerequisite: MATH 387.

**MATG 690. Graduate Complex Analysis. 3 Credits.**

This course focuses on the complex plane, complex functions, limits and continuity, as well as analytic functions, the Cauchy-Riemann equations, the Cauchy Integral Theorem, and consequences. Additional topics may include: power series, Taylor and Laurent Series, classification of singularities, the Residue Theorem and its applications, conformal mapping, and selected applications. This course will require a written project and an oral presentation on some particular application of, or historical development in complex analysis. Not open to students with credit for MATH 490 or 407. (Cr. 3) Prerequisites: MATH 387 (formerly 313).

**MATG 691. Topics in Applied Mathematics. 3 Credits.**

Special Topics in Applied Mathematics-Data Analytics. Offered in response to interests and needs of faculty and students. Pre-reqs: MAT G611 or MATG 630 or
Permission of Director.

**MATG 698. Internship. 3 Credits.**

Students will receive guidance in securing an appropriate internship and must obtain faculty sponsorship. Faculty supervisors will define appropriate academic activities in parallel to the work requirement in order to provide a complete internship experience. Pre-requisite: Permission of the Graduate Director.

**MATG 699. Research in Mathematics. 3 Credits.**

Investigation of a research topic in mathematics culminating in a written paper and oral presentation. Prerequisite: Permission of the Graduate Director. (Cr. 3).

**MATG 724. Cont. Issues in Teaching Math. 4 Credits.**

Discussion of issues related to mathematics instruction at the secondary and early college level: how to develop student competence in effective communication, cooperative learning, use of technology, quantitative literacy, knowledge of content, professional responsibilities. Students will gain experience running review sessions and assist in labs, and evaluate lesson plans incorporating the use of technology as appropriate. Selected readings and evaluation of lesson plan portfolios generated in Algebra and Number Theory. (Cr. 4).

**MATG 761. Classical Methods in Plane Euclidean Geometry. 4 Credits.**

This is a the first part of a two-semester introduction to classical and modern plane Euclidean geometry. The sequence fills a critical gap for prospective secondary school mathematics teachers, who may have to teach courses in Euclidean geometry with an undergraduate background that may have no geometry or may include geometry courses whose nature is not well-adapted to the demands of most secondary geometry curricula. The first part of the course includes a recapitulation of the classical content of high-school Euclidean geometry, based on the SMSG axioms, with additional topics and most exercises well beyond the high-school curriculum. For example, the classical results about the concurrence of various special lines in a triangle are obtained as consequences of a single overarching result, Ceva's theorem. Ptolemy's theorem for cyclic quadrilaterals is proved, and then is used in the second semester to obtain the formula for sin(x). Heron's formula for the area of a triangle in terms of the sides and semi-perimeter is derived, and the equidecomposability of plane polygons of equal area is proved. Finally, a geometric treatment of the conic sections and their reaction properties is presented. The course will use Geogebra, Geometers SketchPad or an equivalent software product. (Cr. 4) Prerequisite: MATH 243.

**MATG 762. Modern Methods in Plane Euclidean Geometry. 4 Credits.**

This is a the second part of a two-semester introduction to classical and modern plane Euclidean geometry. The course continues with the introduction of modern methods. Topics include trigonometry, coordinate methods and the algebra associated with the conic sections, complex numbers, vector methods, transformations, and inversion with respect to a circle. Many of the results of the first semester are revisited from new perspectives (for example Heron's formula is found by complex number methods), and a host of more modern results are obtained. The course will use Geogebra, Geometers SketchPad or an equivalent software product. (Cr. 4) Prerequisite: MATG 761.