# Department of Mathematics

Director, Graduate Program

The Department of Mathematics offers two graduate degree programs and one advanced certificate program. The graduate programs lead to an MS in Applied Mathematics-Data Analytics, a 5 year BA-MS or BS-MS in Applied Mathematics-Data Analytics and an MS in Mathematics. There is also an Advanced Certificate in Applied Mathematics-Data Analytics. The degree requirements, as well as the admission requirements, for each degree, are listed below.

## MS in Applied Mathematics-Data Analytics

#### Program Overview

The 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 industry, business, government, or for further study at the doctoral level. It is a particularly good fit for students who want to transition into data analytics and data science careers. The curriculum emphasizes the application of mathematics and programming with open ended course projects. The courses combine theory and application striving to give students practical tools with the understanding to make them useful.

Students will typically complete the program in 3 semesters plus 1 internship or research project during a summer. Students may pursue the program part-time and courses are scheduled to accommodate students who work full time.

#### Admission Requirements

Applicants should possess a degree in a STEM or quantitative discipline, some exposure to computer programming, and have the desire to learn mathematical and computational methods to apply them to real world problems. The prerequisites for the program are multi-variable calculus, probability or calculus-based statistics, linear algebra and a programming class.

#### Degree Requirements

The 30 credit hour program consists of a core of study in computational methods, statistics, machine learning, linear algebra, and operations research, complemented by electives. Manhattan College undergraduate students from any major can count up to six graduate credits toward both their undergraduate and graduate degrees in mathematics which may allow them to complete the master's program in one additional year.

There are four master's 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 for the MS in Applied Mathematics-Data Analytics:

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 graduate electives (e.g. in Mathematics, Business, Computer Science or Engineering) | 6 | |

Total Credits | 30 |

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

The program is a seamless 5 year BS-MS or BA-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 BA or BS in 4 years. They will apply to the MS program during their junior or senior year. If accepted, they take graduate classes during the 3^{rd} and 4^{th} years of study and will complete the requirements for the MS degree in a fifth year. Manhattan College mathematics students can count up to six graduate credits toward both their undergraduate and graduate degrees.

## Advanced Certificate in Applied Mathematics-Data Analytics

To complete the 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.

## MS in Mathematics

#### Program Overview

This program is for individuals who hope to pursue the PhD in Mathematics or a related discipline, or who wish to teach mathematics in a community college. Students in the program will complete course work in foundational areas of pure mathematics: linear and abstract algebra, real and complex analysis, topology and probability-statistics. Electives may be chosen to deepen the applicant’s knowledge in preparation for study at the PhD level, for breath including applications, and may include research. A thesis option is available for those who wish to do research. This program may be completed on either a full-time or a part-time basis. Qualified undergraduates may begin graduate classes as upperclassmen, thereby enabling completion of the MS degree in a single postgraduate year plus two summers.

#### Admission Requirements

Entering students should have seen, at a minimum, Calculus I-II-III, a Proof-theoretic Linear Algebra, a Probability or Statistics class, Algebra and Analysis. A major in mathematics is desirable. A course in programming is recommended.

#### Degree Requirements

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. Manhattan College students can count up to six graduate credits toward both their undergraduate and graduate degrees.

The student must pass master's 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.

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

Algebra Elective (MATG 571 Advanced Linear Algebra with Applications or MATG 678 Algebra II) | 3 | |

Analysis Elective (MATG 688 Graduate Analysis or MATG 690 Graduate Complex Analysis) | 3 | |

Statistics / Data Analysis Elective (MATG 630 Probability and Statistics for Analytics or MATG 633 Advanced Statistical Inference) | 3 | |

MATG 542 | Number Theory | 3 |

MATG 699 | Research in Mathematics | 3 |

15 Additional Elective Credits of Graduate Mathematics (MATG 500-799) | 15 | |

Total Credits | 30 |

### Courses

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

This course is a survey of programming tools used in solving problems in applied mathematics and data analytics. The course material comprises the following broad areas: programming techniques in scientific languages such as MATLAB, Python, and R; and an overview of selected topics in data handling, data visualization, and introductory predictive analytics. Additional topics may include analytics-related topics in linear algebra and numerical analysis. A final project is required for this course.
Prerequisites: CMPT 101 and MATH 285 or MATH 287 and MATH 272 or MATH 351.

**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 or MATH 351.

**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 or MATH 351.

**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 the 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 or MATH 351.

**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. Prerequisite: Graduate status or permission of the 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 or MATH 351) and MATH 285 and (MATH 331 or MATH 336).

**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. Prerequisite: MATG 511.

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

A survey of the fundamental concepts in point set topology: open and closed sets, topological spaces, homeomorphisms, metric spaces, connectedness and compactness, illustrated by examples from applications in other disciplines. After a survey of the basics, the course will cover selected topics such as homotopy theory, chaos, fixed point theory, knots, manifolds and cosmology.
Prerequisites: MATH 243 and 387, with a grade of B or better, or equivalent.

**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. Prerequisite: MATH 272 or equivalent.

**MATG 577. Foundations of Abstract Algebra. 3 Credits.**

An introduction to algebraic structures with an emphasis on the theory of groups, subgroups, isomorphism, normal subgroups, cosets. Lagrange's theorem and the fundamental homomorphism theorems. This is a prerequisite course for graduate study in mathematics for students who lack the undergraduate background.

**MATG 587. Foundations of Mathematical Analysis. 3 Credits.**

A rigorous treatment of differential calculus of one variable: sequences, limits, continuity, the derivative, the Riemann Integral. This is a prerequisite course for graduate study in mathematics for students who lack the undergraduate background.

**MATG 588. Principles of Mathematical Analysis. 3 Credits.**

Review of Riemann integral and the major theorems of integration in calculus. Review of infinite Series. Sequences and series of functions and their convergence properties, focusing on uniform convergence as the central notion for developing the properties of function spaces of real valued maps, such as equicontinuity and the Stone-Weierstrass theorem. Power series, Fourier series and the Gamma function. A rigorous definition of multivariable calculus using linear maps and differential forms over R n . Implicit and Inverse Function theorems and the general Stokes theorem. Introduction to Measure Theory.
Prerequisites: MATH 387 or MATG 587.

**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. Prerequisite: MATH 331.

**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 and nonparametric modeling. Students will utilize appropriate software for data analytics. Prerequisite: MATH 432 or MATG 630.

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

An introduction to probability models including random variables, conditional probability and expectation, Markov chains and time series. Additional topics may include Poisson processes, continuous time Markov processes, queueing theory, spatial, text and network models.
Prerequisite: 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. Prerequisite: MATG 630.

**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. Prerequisite: CMPT 101 or equivalent.

**MATG 678. Abstract Algebra. 3 Credits.**

Study of algebraic structures, such as rings, fields and integral domains, further study in group theory, applications of abstract theory. The course requires a written project.
Prerequisite: MATH 377 or MATG 577.

**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. Prerequisite: MATH 387.

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

Topics in Applied Mathematics. Offered in response to interests and needs of faculty and students. Can be repeated for credit. Prerequisite: Permission of the graduate director.

**MATG 692. Topics in Mathematics. 3 Credits.**

Topics in Mathematics. Offered in response to interests and needs of faculty and students. Can be repeated for credit. Prerequisite: Permission of the graduate 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. Prerequisite: 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.

**MATG 700. Thesis. 3 Credits.**

A sequel to MATG 699, research in Mathematics. Continuation of research culminating in a
Master’s Thesis.

**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, Geometer's Sketchpad or an equivalent software product. Prerequisite: MATG 761.