The Preliminary Exam is taken by incoming mathematics masters graduate students at the University of Pennsylvania, just prior to the start of the fall semester. It is also required of submatriculant students. It plays a double role:
- It serves as a placement exam, to help determine which courses are appropriate for the student.
- It provides an incentive for incoming grad students to review basic material, which will then help them in their beginning graduate classes.
Students who do not pass the exam the first time will have a second chance to pass it at the end of the spring semester (generally in late April or early May). This exam is no longer required of PhD students starting in 2023 and beyond.
The prelim exam focuses on the key material from undergraduate mathematics that is most important to those entering a mathematics graduate program. It is given in two parts (I & II), in the morning and afternoon of the same day. Each of these two parts consists of six problems, and students are given two and a half hours for each part (which is intended to be more time than is necessary for those who know the material).
The exam consists of problems in algebra (including linear algebra), analysis (both real and complex), and basic geometry-topology. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples. Each part (I & II) of the exam contains a mixture of types of problems, and a mixture of analysis and algebra problems.
The following list of topics gives a general idea of the material that is covered on the exam:
I. Calculus and Real Analysis (four problems)
Continuity, uniform continuity, properties of real numbers, intermediate value theorem, epsilon-delta proofs.
Differentiable functions of one variable: differentiation, Riemann integration, fundamental theorem of calculus, mean-value theorem, Taylor's theorem
Sequences and series of numbers and functions, uniform convergence, equicontinuity, interchange of limit operations, continuity of limiting functions.
Ordinary differential equations (separable, exact, first order linear, second order linear with constant coefficients), applications such as orthogonal trajectories.
Multivariable calculus: partial derivatives, chain rule, multiple integrals, integrals in various coordinate systems, inverse and implicit function theorems.
II. Complex Analysis (Two problems)
- Basic properties of complex numbers including roots of unity. Basic properties of the more common complex functions including the geometric series, complex logarithm, exponential, square root, and trigonometric functions. Analytic continuation, the argument principle, the maximum modulus principle, computation of integrals using the Cauchy residue theorem.
III. Algebra (four problems)
- Linear Algebra:
- Vector spaces over R, C, and other fields: subspaces, linear independence, basis and dimension.
- Linear transformations and matrices: constructing matrices of abstract linear transformations, similarity, change of basis, trace, determinants, kernel, image, dimension theorems, rank; application to systems of linear equations.
- Eigenvalues and eigenvectors: computation, diagonalization, characteristic and minimal polynomials, invariance of trace and determinant. Jordan canonical forms.
- Inner product spaces: real and Hermitian inner products, orthonormal bases, Gram-Schmidt orthogonalization, orthogonal and unitary transformations, symmetric and Hermitian matrices, quadratic forms.
- Abstract algebra
- Groups: finite groups, matrix groups, symmetry groups, examples of groups (symmetric, alternating, dihedral), normal subgroups and quotient groups, homomorphisms, structure of finite abelian groups, Sylow theorems.
- Rings: ring of integers, induction and well ordering, polynomial rings, roots and irreducibility, unique factorization of integers and polynomials, homomorphisms, ideals, principal ideals, Euclidean domains, prime and maximal ideals, quotients, fraction fields, finite fields.
IV. Geometry-topology (two problems)
- Vector calculus: vector fields in Euclidean space (divergence, curl, conservative fields), line and surface integrals, vector calculus (Green's theorem, divergence theorem and Stokes' theorem).
- Point-set topology: metric spaces, compactness, connectedness, topological spaces, convergence in metric spaces and topological spaces.
Sample & Old Preliminary Exams: (Warning: starting with the Spring 2022 Prelim the syllabus has been changed to include complex analysis, and a few other topics like Fourier series, which do not appear in previous prelims; some problems of this type may be found in copies of recent prelim exams for our AMCS program here: https://www.amcs.upenn.edu/exams/written-preliminary-exam )