# Mathematics (MATH)

**MATH-100 Precalculus**

**MATH-100QR Precalculus: 'Problem Solving and Quantitative Reasoning'**

*Fall. **Credits: 4*

This course is intended for students who, based on the results of their mathematics assessment and the agreement of the instructor, need to strengthen their quantitative and algebraic skills in order to be ready to progress to further mathematics, science, and economics courses. In this class students learn to translate real problems into mathematics, to solve complex multi-step problems, and to gain confidence in using logarithms, exponents, and trigonometry in different contexts.

*Applies to requirement(s): Math Sciences**T. Day**Instructor permission required.**Advisory: Permission of instructor. Send score from math online self-assessment and background information to Dylan Shepardson, dshepard@mtholyoke.edu.*

**MATH-101 Calculus I**

*Fall and Spring. **Credits: 4*

This course is for students who have not studied calculus and who have the necessary precalculus background. It presents rates of change and their applications, integrals, the fundamental theorem, and modeling of phenomena in the natural and social sciences. All students are required to complete the online self assessment of precalculus skills before the course begins.

*Applies to requirement(s): Math Sciences**C. Bozeman, R. Quarles*

**MATH-102 Calculus II**

*Fall and Spring. **Credits: 4*

Topics include techniques of integration, applications of integration, differential equations, sequences, series, and Taylor series.

*Applies to requirement(s): Math Sciences**C. Cox, V. Ferlini*

**MATH-131 Explorations in Mathematics**

**MATH-131GM Explorations in Mathematics: 'Games, Systems, and Strategic Thinking'**

*Spring. **Credits: 4*

Board games have a long history of use as both entertainment and as a training ground for higher-level reasoning and analysis. Recent innovations in board game design have produced games (so-called euro-style games) that are mathematically sophisticated and embody systems that model different aspects of reality. In this course we will use board games to explore and analyze different mathematical systems and structures as well as to develop and apply skills in strategic thinking. Topics will include probability, modeling, and network theory.

*Applies to requirement(s): Math Sciences**K. Mulder*

**MATH-139 Cryptography: The Mathematics of Sending Secret Messages**

*Fall. **Credits: 4*

Cryptography is the study of secret communication between different groups of people. From 4,000 years ago in ancient Egypt when secret hieroglyphs were used to communicate the messages of royalty to today when credit card numbers are encrypted to be transmitted over the internet, cryptography has been a necessary part of human life. In this class we will discuss classical cryptography and some historical ciphers along with the mathematical concepts of the modern field. We will study public key cryptography, prime numbers, the discrete logarithm problem, the Diffie-Hellman key exchange, and RSA encryption. If time permits we will also discuss elliptic curve encryption. In particular, we will use the Python programming language and Jupyter notebooks to implement the encryption schemes that we study.

*Applies to requirement(s): Math Sciences**Other Attribute(s): Speaking-Intensive, Writing-Intensive**M. Robinson**Notes: Students who have taken a 100-level Mathematics, Statistics, or Computer Science course can take this at the 200-level with permission of the professor.*

**MATH-203 Calculus III**

*Fall and Spring. **Credits: 4*

Topics include differential and integral calculus of functions of several variables.

*Applies to requirement(s): Math Sciences**H. Wang**Prereq: MATH-102 or its equivalent. *

**MATH-206 Introduction to Proofs Through Analysis**

*Fall and Spring. **Credits: 4*

An introduction to abstract reasoning in the context of real analysis. Topics will be drawn from the real numbers, mathematical induction, functions, sequences, and continuity. The emphasis is on formal mathematical reasoning and writing through proofs.

*Applies to requirement(s): Math Sciences**L. Mrad, T. Chumley**Prereq: MATH-102 or above. **Advisory: Students may not take this course after completing MATH-301.*

**MATH-211 Linear Algebra**

*Fall and Spring. **Credits: 4*

Topics include elements of the theory of matrices and vector spaces.

*Applies to requirement(s): Math Sciences**C. Bozeman, S. Hart, L. Mrad, D. Young**Prereq: MATH-102 or above. *

**MATH-232 Discrete Mathematics**

*Fall and Spring. **Credits: 4*

Studies some aspects of discrete mathematics. Topics include sets, functions, elementary probability, induction proofs, and recurrence relations.

*Applies to requirement(s): Math Sciences**C. Bozeman, C. Cox, R. Quarles, D. Young**Prereq: MATH-102 or above or COMSC-150/151. *

**MATH-251 Mathematical Experimentation: An Introduction to Research in the Mathematical Sciences**

*Not Scheduled for This Year. **Credits: 4*

A selection of projects with a goal of discovery of properties and patterns in mathematical structures. The choice of projects varies from year to year and is drawn from algebra, analysis, discrete mathematics, geometry, applied mathematics, and statistics.

*Applies to requirement(s): Math Sciences**Other Attribute(s): Writing-Intensive**The department**Prereq: MATH-102 or above. **Advisory: MATH-232 recommended*

**MATH-272 Numerical Calculus**

*Not Scheduled for This Year. **Credits: 4*

This course is an introduction to computation and computing from a mathematical perspective, covering topics such as numerical algorithms for differentiation, integration, root finding, curve fitting, and error analysis. These tools are very powerful when one finds a mathematical or an applied problem that cannot be solved using the types of analytical functions one learns in calculus. This course is for students with little or no programming knowledge and an interest in learning skills for mathematical computations. The course will cover the basics of programming: types of variables, lists, arrays, for and while loops, if statements, file handling, plotting, pseudo-code and documentation.

*Applies to requirement(s): Math Sciences**The department**Prereq: MATH-102. **Advisory: Students who have completed COMSC-150 or any version of COMSC-151 are not allowed to take this course.*

**MATH-295 Independent Study**

*Fall and Spring. **Credits: 1 - 4*

*The department**Instructor permission required.**Notes: The permission of The department is required for independent work to count towards the major or minor.*

**MATH-301 Real Analysis**

*Fall and Spring. **Credits: 4*

Topics include the real number system, convergence of sequences and series, power series, uniform convergence, compactness and connectedness, continuity, abstract treatment of differential and integral calculus, metric spaces, and point-set topology.

*Applies to requirement(s): Math Sciences**H. Wang**Prereq: MATH-102, and MATH-211, and either MATH-206 or MATH-232. *

**MATH-302 Complex Analysis**

*Not Scheduled for This Year. **Credits: 4*

Topics include differentiation and integration of functions of a complex variable, the Cauchy integral formula, residues, conformal mapping, and applications to physical science and number theory.

*Applies to requirement(s): Math Sciences**The department**Prereq: MATH-211 and either MATH-206 or MATH-232. **Notes: offered alternate years at Mount Holyoke and Smith Colleges*

**MATH-312 Abstract Algebra**

**MATH-312GT Abstract Algebra: 'Groups'**

*Spring. **Credits: 4*

Abstract algebra is the study of the common principles that govern computations with seemingly disparate objects. One way to begin is by studying groups, which are sets with a single operation under which each non-identity element is invertible. Examples include the integers with addition, invertible matrices of size n, permutations of a fixed set, and the symmetries of an object. Our goal is to study a definition of groups that unifies all of the important examples above and more.

*Applies to requirement(s): Math Sciences**D. Young**Prereq: MATH-211 and either MATH-206 or MATH-232. **Advisory: Students who have taken MATH-312GT Rings may only take MATH-311 Abstract Algebra: Groups and Rings with instructor permission.**Notes: This course will satisfy the MATH-311 requirement for the mathematics major.*

**MATH-312RT Abstract Algebra: 'Rings'**

*Fall. **Credits: 4*

Abstract algebra is the study of the common principles that govern computations with seemingly disparate objects. One way to begin is by studying rings, which are sets with two operations, typically addition and multiplication. Examples include the integers, the integers modulo n, and polynomials in n variables. Our goal is to study a definition of rings that unifies all of the important examples above and more.

*Applies to requirement(s): Math Sciences**M. Robinson**Prereq: MATH-211 and either MATH-206 or MATH-232. **Advisory: Students who have taken MATH-312RT Rings may only take MATH-311 Abstract Algebra: Groups and Rings with instructor permission.**Notes: This course will satisfy the MATH-311 requirement for the mathematics major.*

**MATH-319 Topics in Algebra**

**MATH-319GR Topics in Algebra: 'Graph Theory'**

*Not Scheduled for This Year. **Credits: 4*

Graph theory gives us both an easy way to pictorially represent many major mathematical results and insights into the deep theories behind them. Graphs seem simple -- they're just collections of dots connected by curves -- but are very rich structures that arise naturally in applications ranging from social networks to electric power grids. We will examine properties such as isomorphism, connectivity, planarity, and coloring using classic examples such as paths, cycles, trees, complete graphs, and polyhedral graphs. More advanced topics will be determined by student interest and course trajectory.

*Applies to requirement(s): Math Sciences**The department**Prereq: MATH-232. *

**MATH-319NT Topics in Algebra: 'Number Theory'**

*Not Scheduled for This Year. **Credits: 4*

This course will begin with an introduction to number theory, covering material on congruences, prime numbers, arithmetic functions, primitive roots, quadratic residues, and quadratic fields. We will then continue our study of number theory by picking special topics which might include some of the following: Finite Fields, Prime Factorization of Ideals, Fermat's Last Theorem, Elliptic curves, Dirichlet's Theorem on Arithmetic Progressions, the Prime Number Theorem, or the Riemann Zeta function.

*Applies to requirement(s): Math Sciences**Other Attribute(s): Speaking-Intensive, Writing-Intensive**The department**Prereq: MATH-211 and either MATH-206 or MATH-232. *

**MATH-329 Topics in Geometry**

**MATH-329TP Topics in Geometry and Topology: 'Topology'**

*Spring. **Credits: 4*

This course is an introduction to point-set topology, which is a fundamental language for much of modern mathematics. One of the goals of topology is to understand what it means for a function to be continuous, first in Euclidean space, and then to generalize the notion of continuity to other spaces. The core topics to be studied include: basic set theory, various interesting topologies, continuous functions, connectedness and compactness. Topics from algebraic topology will be covered if time permits.

*Applies to requirement(s): Math Sciences**C. Cox**Prereq: MATH-232 and any 300-level math class. *

**MATH-333 Differential Equations**

*Fall. **Credits: 4*

This is an introduction to differential equations for students in the mathematical or other sciences. Topics include first-order equations, second-order linear equations, and qualitative study of dynamical systems

*Applies to requirement(s): Math Sciences**The department**Prereq: MATH-211. *

**MATH-339 Topics in Applied Mathematics**

**MATH-339FM Topics in Applied Mathematics: 'Rigidity Theory'**

*Not Scheduled for This Year. **Credits: 4*

A framework constructed from fixed-length bars attached at flexible joints is either rigid or flexible. Such structures arise in many applications in architecture, engineering, robotics, and biology and provide a model for understanding related problems in areas including computer-aided design, sensor networks, and statistics. We will use linear algebra and graphs to develop the theory needed to analyze frameworks and make connections to applications.

*Applies to requirement(s): Math Sciences**The department**Prereq: MATH-101, MATH-211, and either MATH-206 or MATH-232. *

**MATH-339PD Topics in Applied Mathematics: 'Partial Differential Equations'**

*Fall. **Credits: 4*

Partial differential equations (PDEs) are often used to describe natural phenomena arising in a wide variety of contexts including physics, biology, and economics. Our focus will be on basic yet representative linear partial differential equations such as the heat and wave equations. We will explore the motivation behind each model we study and emphasize methods of finding solutions and analyzing their behavior. Techniques will include transform methods, separation of variables, energy methods, and numerical computations.

*Applies to requirement(s): Math Sciences**L. Mrad**Prereq: MATH-203 and MATH-211, or PHYS-205. *

**MATH-339PT Topics in Applied Mathematics: 'Optimization'**

*Not Scheduled for This Year. **Credits: 4*

Mathematical optimization involves finding the best solution to a problem from a set of feasible solutions defined by mathematical constraints. It has an elegant theory and applications in fields like management, economics, engineering, and computer science that require decision making under constraints on time or other resources. We will begin by studying linear optimization, including duality, the simplex algorithm, and the geometry of linear programming. Other topics will include discrete optimization, network optimization, and nonlinear optimization.

*Applies to requirement(s): Math Sciences**The department**Prereq: MATH-211. *

**MATH-339SP Topics in Applied Mathematics: 'Stochastic Processes'**

*Spring. **Credits: 4*

Stochastic processes are mathematical models that evolve with time and include an element of randomness. They involve a collection of states-for example, the weather in a geographical location, the size of a population, or the length of a queue-and a description of how the system evolves from one state to the next. This course is devoted to the study of a class of stochastic processes called Markov chains, and we attempt to study their behavior using tools from probability theory and linear algebra in beautiful, interconnected ways. Topics will include Markov chains in discrete and continuous time, branching processes, queuing theory, and Markov chain Monte Carlo.

*Applies to requirement(s): Math Sciences**T. Chumley**Prereq: MATH-211 and MATH-342. *

**MATH-342 Probability**

*Fall and Spring. **Credits: 4*

This course develops the ideas of probability simultaneously from experimental and theoretical perspectives. The laboratory provides a range of experiences that enhance and sharpen the theoretical approach and, moreover, allows us to observe regularities in complex phenomena and to conjecture theorems. Topics include: introductory experiments; axiomatic probability; random variables, expectation, and variance; discrete distributions; continuous distributions; stochastic processes; functions of random variables; estimation and hypothesis testing.

*Applies to requirement(s): Math Sciences**T. Day, A. Hoyer-Leitzel**Prereq: MATH-203. *

**MATH-395 Independent Study**

*Fall and Spring. **Credits: 1 - 8*

*The department**Instructor permission required.**Notes: The permission of The department is required for independent work to count towards the major or minor.*