Example Syllabus - MIT BSc Pure Mathematics
Pure mathematics is the study of the basic concepts and structures that underlie mathematics. Its purpose is to search for a deeper understanding and an expanded knowledge of mathematics itself.
Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students may also wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.
The subject Real Analysis is basic to the program. Since this subject is strongly proof-oriented, some students find it useful to take an intermediate subject such as Linear Algebra or Introduction to Mathematical Reasoning or Linear Algebra, before taking Real Analysis.
The subject Algebra I is more advanced and should not be elected until the student has had some experience with proofs (as in Real Analysis).
-- https://math.mit.edu/academics/undergrad/major/course18/pure.html
Calculus 1 - "Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions."
Calculus 2 - "Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications."
Differential Equations - "Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams."
Real Analysis - "We will study variable-coefficient elliptic PDE, looking at regularity estimates like the Schauder inequality and the De Georgi-Nash-Moser inequality. We will use these estimates to solve elliptic PDE, starting with linear PDE and then turning to non-linear PDE. We will study applications of Fourier analysis, including a little combinatorics/number theory as well as applications to PDE. We will use Fourier and harmonic analysis to prove L p estimates for operators such as the Calderon-Zygmund inequality for the Laplacian. In dispersive PDE, we will study the Schrodinger equation. We again look at estimates, and build up to the Strichartz inequality. We will use the Strichartz inequality to study solutions of non-linear dispersive PDE. We will also practice different strategies for proving estimates. For instance, when is it a good idea to break a sum into pieces, and which pieces should we use? When is it good to use Holder’s inequality? When do we need to exploit cancellation, and how do we prove that cancellation happens? When we look at a complex expression, how do we decide which terms are important and which are less important?"
Introduction to Topology - "Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group."
Algebra I & II &Linear Algebra - "Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses linear algebra software."