**The Bridge to PhD master’s program will no longer accept new students. This program has recently transitioned into a fellowship opportunity for applicants to our PhD program. Please read more about Bridge Fellowships on our graduate admissions page https://www.math.upenn.edu/graduate.**

#### ASSESSMENT

Before a prospective student is invited to Penn for an in-person interview, he or she will be asked to complete a take home written assessment which consists of 12 problems in calculus, linear algebra, real analysis, and abstract algebra. The rules of the assessment are as follows.

- Students are expected to attempt any 8 of the 12 problems that they choose.
- Students can use any textbook references for assistance.
- Students are not allowed to use the internet or to receive help from anyone while completing the assessment.
- Once they receive the assessment by email, students will have 4.5 hours to complete the assessment and to submit their responses.
- Students will submit their responses by email in a single file. For example, this could be a scanned pdf or a zip file of images of their solutions

A sample assessment can be found here. The specific subject matter the assessment is based on is detailed below.

#### Calculus topics

Limits, derivatives and integrals, fundamental theorem of calculus, intermediate value theorem, mean value theorem, extremum problems, Taylor’s theorem, series, sequences, ordinary differential equations, computing areas and volumes, gradients, double and triple integrals

**Good references:** *Calculus: Early Transcendentals* by James Stewart; *Thomas’ Calculus* by Joel Hass, Christopher Heil and Maurice Weir

#### Linear algebra topics

Solving linear systems, determinants, trace, Gaussian elimination, matrix multiplication, linear transformations, linear independence, vector spaces, eigenvalues and eigenvectors, diagonalization, Cayley-Hamilton theorem, symmetric matrices, norms, inner products, Gram-Schmidt process, orthogonal matrices

**Good references:** *Linear Algebra Done Right* by Sheldon Axler; *Linear Algebra* by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence; *Introduction to Linear Algebra* by Gilbert Strang

#### Real analysis topics

Topology of the real line, infimum and supremum, continuity, uniform continuity, epsilon-delta proofs, sequences and series of numbers, differentiability, Riemann integrability, sequences and series of functions, uniform convergence, partial derivatives, multiple integrals, Lagrange multipliers

**Good references:** *Principles of Mathematical Analysis* by Walter Rudin;* Introduction to Real Analysis* by Robert G. Bartle and Donald R. Sherbert; *Understanding Analysis* by Stephen Abbott

#### Abstract algebra topics

Finite groups, matrix groups, symmetry groups, normal subgroups and quotient groups, homomorphism theorems, abelian groups, polynomial rings, roots and irreducibility, unique factorization of integers and polynomials, greatest common divisors, ideals, integral domains, Euclidean domains, finite fields.

**Good references:** *Algebra* by Artin; *Topics in Algebra* I.N. Herstein; *A first course in abstract algebra* by John B. Fraleigh

**Department of Mathematics**

David Rittenhouse Laboratory

209 South 33rd Street

Philadelphia, PA 19104-6395

Robin Toney

rtoney@math.upenn.edu

(215) 898-8178 & 898-8627

Fax: (215) 573-4063