Our standard graduate courses are offered on a two-year cycle, with students typically taking two classes each semester for their first two years. After that, students usually begin working with their research advisors on an individual basis and/or as part of a research seminar. Course descriptions follow, below.**Two-Year Sequence in Introductory Graduate Study**

*Beginning Fall Semester of an Even Year:*

Math 312: Topology*

Math 501: Graduate Analysis I

Math 502: Graduate Analysis II

Math 505: Graduate Topology I*Beginning Fall Semester of an Odd Year:*

Math 322: Complex Analysis*

Math 503: Graduate Algebra I

Math 504: Graduate Algebra II

Math 506: Graduate Topology II

*Two of our standard graduate courses, Topology and Complex Analysis, are offered as dual undergraduate and graduate courses, with all students attending the same lectures, but the graduate students doing additional advanced work.

Graduate students whom we admit based on their strengths and our belief that they are a good fit for our program, but who lack the needed background for succeeding in our graduate courses, may spend up to one year taking advanced undergraduate courses, with additional work, as appropriate. Example courses include:

Math 302: Real Analysis II

Math 304: Abstract Algebra II

Advanced graduate students may enroll in certain upper-level undergraduate courses, as is appropriate to their their research topics. Graduate students complete additional work in these courses. Example courses include:

Math 311: Partial Differential Equations

Math 390: Algebraic Number Theory

Students pursuing an M.A. degree before the Ph.D. work with an advisor in either of the following, each semester. Similarly, students who have completed the introductory courses, pass their preliminary exams, then focus on independent research with their dissertation advisory in either of the following.

Math 701: Supervised Work

Math 702: Research Seminar

When appropriate, students may also enroll in courses at the University of Pennsylvania.

### Course Descriptions

**MATH 302 Introduction to Real Analysis ****II**

This course is the second semester of a year-long sequence in real analysis covering the real number system, elements of set theory and topology, continuous and differentiable functions, uniform convergence, the Riemann integral, power series, Fourier series and other limit processes.

**MATH 304 Abstract Algebra II**

This course is the second semester of a year-long sequence in abstract algebra covering groups, rings, and fields, and their homomorphisms; quotient groups, quotient rings, and the isomorphism theorems. Standard examples including symmetric groups, free groups, and finitely generated abelian groups; integral domains, PID's and UFD's, and polynomial rings; finite and infinite fields; Sylow theory and field extensions. Additional topics may include: Galois theory, modules and canonical forms of matrices, algebraic closures, and localization.

**MATH 311 Partial Differential ****Equations**

Heat and wave equations on bounded and unbounded domains, Laplace's equation, Fourier series and the Fourier transform, qualitative behavior of solutions, computational methods. Applications to the physical and life sciences.

**MATH 312 Topology**

General topology (topological spaces, continuity, compactness, connectedness, quotient spaces), the fundamental group and covering spaces, introduction to geometric topology (classification of surfaces, manifolds).

**MATH 322 Functions of Complex ****Variables**

Analytic functions, Cauchy's theorem, Laurent series, calculus of residues, conformal mappings, Moebius transformations.

**MATH 390 ****Algebraic Number Theory **

Algebraic number fields and rings of integers, quadratic and cyclotomic fields, norm and trace, ideal theory, factorization and prime decomposition, lattices and the geometry of algebraic integers, class numbers and ideal class groups, computational methods, Dirichlet's unit theorem.

**MATH** **501****& 502 Graduate Real Analysis I & II**

In these courses we study the theory of measure and integration. Topics include Lebesgue measure, measurable functions, the Lebesgue integral, the Riemann-Stieltjes integral, complex measures, differentiation of measures, product measures, and L^p spaces.

**MATH 503 & 504 Graduate Algebra I & II**

This is a two course sequence providing a standard introduction to algebra at the graduate level. Topics in the first semester include categories, groups, rings, and modules. Topics in the second semester will include linear algebra, fields, Galois theory, and advanced group theory.

**MATH 505 Graduate Topology ****I**

Math 505 and 506 offer an introduction to topology at the graduate level, and can be taken in either order. Math 505 covers the basic notions of algebraic topology. The focus is on homology theory, which is introduced axiomatically, via the Eilenberg-Steenrod axioms, and then studied from a variety of points of view (simplicial, singular and cellular). The course also treats cohomology theory, duality on manifolds, and the elements of homotopy theory.

**MATH 506 Graduate Topology II**

Math 506 focuses on differential topology. Topics covered include smooth manifolds and smooth maps, transversality and intersection theory, differential forms, and integration on manifolds.