# Course Groupings by Area

To view course descriptions and to find out which courses are offered currently or scheduled to be offered in upcoming terms, please consult the following course catalogs:

- Algebra, Coding Theory, and Number Theory
This set of courses cannot be accessed without at least one course in Linear Algebra.

**MATH 228 - Algebra: Introduction to Ring Theory****•**The course introduces the concept of a ring. Beginning from the ring of integers and its main properties (prime factorization), the modular rings are constructed, and finally more abstract rings are considered. Along the way, the concepts of mathematical induction are introduced. **MATH 324 - Elementary Number Theory****•**Number theory refers to the study of the natural numbers. Its main goal is to discover and describe (prove) interesting and unexpected relationships between numbers. **MATH 326 - Rings and Modules****•**This class studies the theory of rings and modules in general, and fields, integral domains, polynomial rings, and Noetherian rings in particular. Emphasis is on the development of abstract concepts. **MATH 328 - Algebra: Introduction to Group Theory****•**This course introduces the most fundamental algebraic concept in mathematics, namely groups. Their basic structures are studied, as well as group actions on sets, group homomorphisms, and the construction of quotient groups. **MATH 422 - Coding Theory****•**Coding theory focuses on the problem of how to encode information that is to be transmitted over an unreliable channel such that the original information can be recovered as long as not too many errors occur during transmission, using so-called error-detecting or error-correcting codes. For this purpose, finite fields and polynomials over finite fields are introduced and their properties developed as pertaining to coding theory. **•**Web page with resources for MATH 422, including online lecture notes by Dr. John C. Bowman. **MATH 424 - Algebra: Groups and Fields****•**The main focus of this course is the theory of automorphism groups of fields and field extensions, also known as Galois theory. Prominent topics include the insolvability of the quintic equation, and constructions with rules and compass. **MATH 428 - Algebra: Advanced Ring Theory****•**Topics in this course will be chosen to illustrate the use of ring theory in another area of mathematics such as the theory of numbers, algebraic geometry, representations of groups, or computational algebra. **MATH 429 - Algebra: Advanced Group Theory****•**This course covers more advanced topics in group theory, such as the Sylow theorems. - Analysis: Real and Complex
- Calculus
Calculus is the study of change and motion, fundamental in all areas of science and engineering. Many university programs therefore require at least one course in calculus; some require a few additional ones.

**MATH 100, MATH 101, and MATH 209 - Calculus I, II, and III****•**This is the calculus sequence for Engineering students. **MATH 117, MATH 118, MATH 217, and MATH 317 - Honors Calculus I and II and Advanced Calculus I and II****•**This is the Honors calculus sequence, open to all students, including Engineering students, with a keen interest in mathematics and its theoretical foundations; please refer to the First-Year Courses page and the Honors Courses page for further information. **MATH 114, MATH 115, MATH 214, and MATH 215 - Elementary Calculus I and II and Intermediate Calculus I and II****•**This is the calculus sequence taken by most most non-Engineering students; please refer to the First-Year Courses page for further information about MATH 114 and 115. **MATH 144, MATH 146 - Calculus for the Physical Sciences I and II****•**These two courses are alternatives to MATH 114 and 115, developed for students with an interest in the physical sciences. These alternatives must be taken by student who concurrently take PHYS 144 and 146, respectively (provided they do not already have appropriate previous calculus credit or choose to take MATH 117 and 118). Successful completion of MATH 146 allows entry into MATH 214. Please refer to the First-Year Courses page for further information. - Capstone and Reading Courses
**INT D 400 - Science Internship Capstone****MATH 497 - Reading in Mathematics****MATH 499 and STAT 499 - Research Project** - Differential Equations
- Differential Geometry, Tensor Analysis, and Topology
This set of courses crowns the geometrically flavored suite of undergraduate courses. The core of differential geometry consists of the study of differential manifolds, equipped with a suitable tool to measure distances, and their curvature properties. The courses in this suite provide an introduction to this area. General topology evolved as an abstract of the study of convergence and continuity.

**MATH 348 - Differential Geometry of Curves and Surfaces****•**In MATH 348 curves in the plane and 3-space are studied. It provides excellent background for students interested in computer graphics. In addition it provides a foundation for its successor, Math 448. **MATH 447 - Elementary Topology****•**This course plays in many ways a service role: it is essential for a deeper study of subjects such as advanced analysis, nonlinear analysis, differential topology, algebraic topology, algebraic geometry, ....

**MATH 448 - Introduction to Differential Geometry and Tensor Analysis****•**Building upon profound insights of Gauss and Riemann, differential manifolds in n-space are introduced and their local and global curvature properties are studied. The necessary multilinear algebra, called tensor analysis, is developed along the way; an excellent preparation for students who want to understand Einstein's theory of general relativity, and a good foundation for the study of differential topology and certain aspects of the theory of partial differential equations. - Discrete Mathematics
- Elementary Education
These courses are offered to support undergraduate programs in the Faculty of Education. These courses are restricted to students in Elementary Education.

**MATH 160 - Higher Arithmetic****MATH 260 - Mathematical Reasoning for Teachers** - Engineering Service Courses
These courses are offered to support undergraduate programs in the Faculty of Engineering. These courses are restricted to students in Engineering, except MATH 201 and MATH 300, which also are open to students in Specialization Computing Science, Specialization Geophysics, and Specialization Physics.

- Geometry
- Linear Algebra
Linear algebra is the study of vector spaces and their transformation properties. Linear algebra can be applied to least squares fitting, to study rotations in space (such as might be useful in computer graphics and engineering for example), to develop powerful search algorithms such as employed by Google, and so forth. The linear algebra courses therefore serve as prerequisites for many advanced courses (not just for mathematics and statistics, but also in physics and engineering), including the courses in algebra, coding theory, and number theory listed above.

**MATH 125 - Linear Algebra I****•**See First-Year Courses for more information. **MATH 127 - Honors Linear Algebra I****•**See First-Year Courses for more information. **MATH 225 - Linear Algebra II****•**This course follows MATH 125. The main theme of this course is linear transformations of vector spaces and all that it entails (e.g., abstract vector spaces, diagonalization of matrices, inner products, etc.). **MATH 227 - Honors Linear Algebra II****•**This course follows MATH 127. The course is similar to MATH 225, but there is more emphasis on the theoretical foundations. Also may include new algebraic structures such as fields other than the real or complex numbers. **MATH 325 - Linear Algebra III****•**This course emphasizes algorithmic aspects of the theory of a linear operator. For instance, modules over a polynomial ring are studied and used to develop methods to find the Jordan and rational canonical forms of a matrix. - Mathematical Finance, Life Contingencies, and Risk Theory
These courses are offered in support of the Honors and Specialization programs in Mathematics and Finance, but also will be of interest to any students interested in learning more about how mathematical and statistical tools are used to study problems in finance.

**MATH 253 - Theory of Interest****•**The theory of interest deals with calculating present and accumulated values for various streams of cash flows. This is relevant in applications such as the calculation of amortization schemes or pensions and serves as basis for courses in mathematical finance. **MATH 356 - Introduction to Mathematical Finance I****•**This first part of the introduction to mathematical finance starts by analyzing option pricing in a basic one-period model of a financial market, subsequently studies how risky assets can be modelled over several time periods, and then derives implications for option pricing in such multi-period models. **MATH 357 - Introduction to Mathematical Finance II****•**This second part of the introduction to mathematical finance considers in more detail the pricing and hedging forwards, futures and options. In particular, the famous Black-Scholes formula is presented as a limit case of the option pricing in multi-period models. The analysis of models for variable and stochastic interest rates rounds off this introduction to mathematical finance. **MATH 408 - Computational Finance****•**Combining computational and numerical methods with financial applications, the course gives an introduction to computational finance, which has become very important in both academia and practice. A main focus of the course is to learn about Monte Carlo methods in financial engineering. **MATH 415 - Mathematical Finance I****•**Using tools from probability theory, the course analyzes the fundamental concepts of mathematical finance in discrete-time models, and their implications on optimal consumption and investment problems as well as the pricing and hedging of financial contracts. **STAT 353 - Life Contingencies****STAT 453 - Risk Theory** - Mathematical Modelling, Numerical Methods, and Optimization
- Mathematical Physics
These courses are offered in support of the Honors in Mathematical Physics program, but also will be of interest to any student interested in learning more about the deep connection between mathematics and physics. MA PH 343 and MA PH 451 are taught on a rotating basis by the Department of Mathematical Sciences and the Department of Physics.

**MA PH 343 - Classical Mechanics I****•**This course studies modern formulations of classical mechanics, which are essential for understanding quantum mechanics. Topics that will be discussed include the Lagrangian and Hamiltonian formulations of classical mechanics, including canonical transformations, Poisson brackets and the Hamilton-Jacobi equation. The motion of rigid bodies and systems with small oscillations around equilibrium also will be studied. **MA PH 451 - Mathematical Methods of Physics I****•**The main focus of this course is to learn about how Lie theory solves problems in quantum physics, for example spin and angular momentum in quantum mechanics, and understand the connection between Lie theory and gauge theories. Note that this description deviates from the calendar description, and only applies to offerings of the course when the course is offered by the Department of Mathematical Sciences. - Probability and Stochastics
- Statistics