The Preliminary Exam is taken by all incoming mathematics graduate
students at the University of Pennsylvania, just prior to the start of the
fall semester (generally in late August). It plays three roles:
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). Those students take the Proseminar
(Math 504, 505) during
their first year, to strengthen their problem solving ability, their
ability to present proofs, and their familiarity with material on the
- It serves as a placement exam, to help determine whether students
should begin with 500-level courses or with 600-level courses (or with a
- It is a requirement for each of the graduate degrees in mathematics,
in order to ensure that those who receive graduate degrees have a solid
- It provides an incentive for incoming grad students to review basic
which will then help them in their beginning graduate classes.
The prelim exam focuses on the key material from an undergraduate
mathematics program that is most important to those entering a
mathematics graduate program. The first half of the exam is given in the
morning, and the second half in the afternoon. Each of these two parts
consists of six problems, and students are given two and a half hours for
The exam consists of problems in algebra (including linear algebra) and
analysis (including basic topology).
Some problems are computational,
some ask for proofs, and some ask for examples or counterexamples. Each
part of the exam (morning and afternoon) constains a mixture of types of
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:
Sample Prelim Exam.
- I. Analysis (five 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
- Sequences and series of numbers and functions, uniform convergence,
equicontinuity, interchange of limit operations, continuity of limiting
- Ordinary differential equations (separable, exact, first order linear,
second order linear with constant coefficients), applications such as
- Multivariable calculus: partial derivatives, multiple integrals,
integrals in various coordinate systems, inverse and implicit function theorems.
- II. Algebra (five problems)
- Linear Algebra:
- Vector spaces over R, C, and other fields: subspaces,
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
- Eigenvalues and eigenvectors: computation, diagonalization,
characteristic and minimal polynomials, invariance of trace and
- 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.
- III. 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.
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