# Course Descriptions & Syllabus

All course descriptions are listed below. Please consult the online course catalog for additional course information. Syllabus (or syllabi) are linked where available.

- 110.105 (Q) Introduction to Calculus This course is a pre-calculus course and provides students with all the background necessary for the study of calculus. Includes a review of algebra, trigonometry, exponential and logarithmic functions, coordinates and graphs. Each of these tools is introduced in its cultural and historical context. The concept of the rate of change of a function will be introduced. Not open to students who have studied Calculus in high school. 4 credits. Syllabus: 110.105
- 110.106 & 110.107 (Q) Calculus I, II (Biological and Social Sciences) Differential and integral Calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, applications for systems of linear differential equations, probability distributions. Applications to the biological and social sciences will be discussed, and the courses are designed to meet the needs of students in these disciplines. 4 credits. Syllabi: 110.106; 110.107
- 110.108 & 110.109 (Q) Calculus I, II (Physical Sciences and Engineering) Differential and integral Calculus. Includes analytic geometry, functions, limits, integrals and derivatives, polar coordinates, parametric equations, Taylor’s theorem and applications, infinite sequences and series. Applications to the physical sciences and engineering will be discussed, and the courses are designed to meet the needs of students in these disciplines. 4 credits. Syllabi: 110.108; 110.109
- 110.113 (Q) Honors Single Variable Calculus The honors alternative to the Calculus sequences 110.106-107 or 110.108-109. Meets the general requirement for both Calculus I and Calculus II in one semester (counts as only one course). A highly theoretical treatment of one variable differential and integral calculus based on our modern understanding of the real number system as explained by Cantor, Dedekind, and Weierstrass. Previous background in Calculus is not assumed. Content includes differential calculus (derivatives, differentiation, chain rule, optimization, related rates, etc), the theory of integration, the Fundamental Theorems of Calculus, with applications of integration, and Taylor series. Prerequisites: A strong ability to learn mathematics quickly and on a higher level than that of the regular Calculus sequences. 4 credits. Syllabus: 110.113
- 110.201 (Q) Linear Algebra Vector spaces, matrices, and linear transformations. Solutions of systems of linear equations. Eigenvalues, eigenvectors, and diagonalization of matrices. Applications to differential equations. Prerequisites: Calculus II or 110.113, or a 5 on the Advanced Placement BC exam. 4 credits. Syllabus: 110.201
- 110.202 (Q) Calculus III: Calculus of Several Variables Calculus of functions of more than one variable: partial derivatives, and applications; multiple integrals, line and surface integrals; Green’s Theorem, Stokes’ Theorem, and Gauss’ Divergence Theorem. Prerequisites: Calculus II or 110.113, or a 5 on the Advanced Placement BC exam. 4 credits. Syllabus: 110.202
- 110.211 (Q) Honors Multivariable Calculus This course includes the material in 110.202 Calculus III but with a strong emphasis on theory and proofs. Recommended only for mathematics majors or mathematically able students majoring in physical science or engineering. Prerequisites: B+ or better in Linear Algebra, or Honors Linear Algebra. Either of these courses may be taken as a co-requisite. 4 credits. Syllabus: 110.211
- 110.212 (Q) Honors Linear Algebra This course includes the material in 110.201 Linear Algebra with a strong emphasis on theory and proofs. Recommended only for mathematics majors or mathematically able students majoring in physical science, engineering. Prerequisites: B+ or better in Calculus II, or 5 on the Advanced Placement BC exam, or 110.113. 4 credits. Syllabus: 110.212
- 110.225 (Q) Putnam Problem Solving Problem solving course offered primarily to prepare students for the Putnam exam. 2 credits.
- 110.302 (Q,E) Differential Equations with Applications This is an applied course in ordinary differential equations, which is primarily for students in the biological, physical and social sciences, and engineering. Techniques for solving ordinary differential equations are studied. Topics covered include first order differential equations, second order linear differential equations, applications to electric circuits, oscillation of solutions, power series solutions, systems of linear differential equations, autonomous systems, Laplace transforms and linear differential equations, mathematical models (e.g., in the sciences or economics).Prerequisites: Calculus II or 110.113. 4 credits. Syllabus: 110.302
- 110.304 (Q) Elementary Number Theory This course provides some historical background and examples of topics of current research interest in number theory. Includes concrete examples of some of the abstract concepts studied in 110.401-402. Topics include primes and prime factorization, congruences, Euler’s function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley’s theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, and Dirichlet’s theorem on primes. Prerequisites: Calculus II and Linear Algebra. 4 credits. Syllabus: 110.304
- 110.311 (Q) Methods of Complex Analysis This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions.Prerequisites: Calculus III. 4 credits.
- 110.328 (Q) Non-Euclidean Geometry For 2,000 years, Euclidean geometry was the geometry. In the 19th century, new, equally consistent but very different geometries were discovered. This course will delve into these geometries on an elementary but mathematically rigorous level. Prerequisites: High school geometry. 4 credits.
- 110.401 (Q) Advanced Algebra I An introduction to the basic notions of modern algebra. Elements of group theory: groups, subgroups, normal subgroups, quotients, homomorphisms. Generators and relations, free groups, products, commutative (Abelian) groups, finite groups. Groups acting on sets, the Sylow theorems. Definition and examples of rings and ideals. Introduction to field theory. Linear algebra over a field. Field extensions, constructible polygons, non-trisectability. Prerequisites: Linear Algebra. 4 credits. Syllabus: 110.401
- 110.402 (Q) Advanced Algebra II This is a continuation of 110.401. Theory of fields (continued). Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals. Modules over a ring. Principal ideal domains, structure of finitely generated modules over them. Prerequisites: 110.401. 4 credits. Syllabus: 110.402
- 110.405 (Q) Real Analysis I This course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.Prerequisites: Calculus III, Linear Algebra. 4 credits. Syllabus: 110.405
- 110.406 (Q) Real Analysis II This course continues 110.405, with an emphasis on the fundamental notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem. Functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral. Prerequisites: 110.405, or 110.415. 4 credits. Syllabus: 110.406
- 110.407 & 110.408 (Q,N) Geometry and Relativity Special relativity: Lorentz transformation, Minkowski spacetime, mass, energy-momentum, stress-energy tensor, electrodynamics. Introduction to differential geometry: theory of surfaces, first and second fundamental forms, curvature. Gauss’s theorema egregium, differentiable manifolds, connections and covariant differentiation, geodesics, differential forms, Stokes theorem. Gravitation as a geometric theory: Lorentz metrics, Riemann curvature tensor, tidal forces and geodesic deviation, gravitational redshift, Einstein field equation, the Schwarzschild solution, perihelion precession, the deflection of light, black holes, cosmology. Prerequisites: Calculus II, Linear Algebra, General Physics II. 4 credits.
- 110.413 (Q) Introduction to Topology The basic concepts of point-set topology: topological spaces, connectedness, compactness, quotient spaces, metric spaces, function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits. Prerequisites: Calculus III. 4 credits. Syllabus: 110.413
- 110.415 (Q) Honors Analysis I This highly theoretical sequence in analysis is reserved for mathematics majors and/or the most mathematically able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics. Prerequisites: Calculus III and Linear Algebra. 4 credits.
- 110.416 (Q) Honors Analysis II This course continues 110.415 Honors Analysis I, with an emphasis on the fundamental notions of modern analysis. Topic here include functions of bounded variation, Riemann-Stieltjes integration, Riesz representation theorem, along with measures, measurable functions, and the lebesgue integral, properties of Lp- spaces, and Fourier series. Prerequisites: 110.415. 4 credits.
- 110.417 (Q,E) Partial Differential Equations for Applications Characteristics. classification of second order equations, well-posed problems. separation of variables and expansions of solutions. The wave equation: Cauchy problem, Poisson’s solution, energy inequalities, domains of influence and dependence. Laplace’s equation: Poisson’s formula, maximum principles, Green’s functions, potential theory Dirichlet and Neumann problems, eigenvalue problems. The heat equation: fundamental solutions, maximum principles. Prerequisites:Calculus III. Recommended: 110.405 or 110.415. 4 credits.
- 110.421 (Q) Dynamical Systems A basic introduction to the general theory of dynamical systems from a mathematical standpoint, this course studies the properties of continuous and discrete dynamical systems, in the form of ordinary differential and difference equations and iterated maps. Topics include contracting and expanding maps, interval and circle maps, toral flows, billiards, limit sets and recurrence, topological transitivity, bifurcation theory and chaos. Applications include classical mechanics and optics, inverse and implicit functions theorems, the existence and uniqueness of general ODEs, stable and center manifolds, and structural stability. Prerequisites: Calculus III, Linear Algebra, and Differential Equations. 4 credits.
- 110.423 (Q) Lie Groups for Undergraduates This course is an introduction to Lie Groups and their representations at the upper undergraduate level. It will cover basic Lie Groups such as SU (2), U(n) , the Euclidean Motion Group and Lorentz Group. This course is useful for students who want a working knowledge of group representations. Some aspects of the role of symmetry groups in particle physics such as some of the formal aspects of the electroweak and the strong interactions will also be discussed. Prerequisites: Calculus III. Prior knowledge of group theory (e.g., 110.401) would be helpful. 4 credits.
- 110.427 (Q) Introduction to the Calculus of Variations The calculus of variations is concerned with finding optimal solutions (shapes, functions, etc.) where optimality is measured by minimizing a functional (usually an integral involving the unknown functions) possibly with constraints. Applications include mostly one-dimensional (often geometric) problems: brachistochrone, geodesics, minimum surface area of revolution, isoperimetric problem, curvature flows, and some differential geometry of curves and surfaces. Prerequisites: Calculus III. 4 credits.
- 110.429 (Q) Mathematics of Quantum Mechanics The basis of quantum mechanics is the Schrodinger equation. The focus of this course will be on one dimensional Schrodinger equations. Topics include eigenvalue problems, bound states, scattering states, tunneling, uncertainty principle, dynmaics, semi-classical limit. The ideas will be illustrated through many examples. Prerequisites: Differential Equations or the permission of the instructor. 4 credits.
- 110.431 (Q) Introduction to Knot Theory The theory of knots and links is a facet of modern topology. The course will be mostly self-contained, but a good working knowledge of groups will be helpful. Topics include braids, knots and links, the fundamental group of a knot or link complement, spanning surfaces, and low dimensional homology groups. Prerequisites: Calculus III. 4 credits.
- 110.439 (Q) Introduction to Differential Geometry Theory of curves and surfaces in Euclidean space: Frenet equations, fundamental forms, curvatures of a surface, theorems of Gauss and Mainardi-Codazzi, curves on a surface; introduction to tensor analysis and Riemannian geometry; theorema egregium; elementary global theorems. Prerequisites: Calculus III, Linear Algebra. 4 credits.
- 110.443 (Q,E) Fourier Analysis and Generalized Functions An introduction to the Fourier transform and the construction of fundamental solutions of linear partial differential equations. Homogeneous distributions on the real line: the Dirac delta function, the Heaviside step function. Operations with distributions: convolution, differentiation, Fourier transform. Construction of fundamental solutions of the wave, heat, Laplace and Schrödinger equations. Singularities of fundamental solutions and their physical interpretations (e.g., wave fronts). Fourier analysis of singularities, oscillatory integrals, method of stationary phase. Prerequisites: Recommended: Calculus III, Linear Algebra. 110.405 or 110.415. 4 credits.
- 110.462 (Q) Prime Numbers and Riemann’s Zeta Function This course is devoted to such questions as: How many prime numbers are there less than N? How are they spaced apart? Although prime numbers at first sight have nothing to do with complex numbers, the answers to these questions due to Gauss, Riemann, Hadamard) involve complex analysis and in particular the Riemann zeta function, which controls the the distribution of primes. This course builds on 110.311 and is an introduction to Analytic Number Theory for undergraduates. Prerequisites: 110.311. 4 credits.
- 110.599 Independent Study, Undergraduate Topics vary based on an agreement between the student and the instructor. Credits vary.