## Honors Sequence in Mathematics

The Honors Sequence in Mathematics is for highly motivated and mathematically talented freshmen and sophomores who are interested in mathematics, engineering, and sciences. The sequence rapidly introduces students to the rigorous and abstract thought processes of higher mathematics.

Students in the Honors Sequence enjoy small classes and close contact with research-active faculty from the very first day. And most students develop close working relationships with classmates because they study together for up to four semesters.

### Courses in the Honors Sequence

#### Math 424. Honors Real Analysis (3 hours)

Continuity, compactness, completeness, connectedness and uniform convergence are developed in the metric space setting, and are then applied to rigorously establish the fundamentals of calculus, including the theory of differentiation, Riemann-Darboux integration, sequences and series of functions, and interchange of limiting operations.

#### Math 425. Honors Advanced Analysis (3 hours)

This course provides a theoretical treatment of differential and integral calculus in higher dimensions. Topics include inverse and implicit function theorems, submanifolds, the theorems of Green, Gauss and Stokes, differential forms, and applications.

#### Math 427. Honors Abstract Algebra (3 hours)

Group theory elucidates structures ranging from elementary clock arithmetic to symmetries in theoretical physics. This abstract algebra course displays the power and also the beauty of groups, covering counting formulae, factorization, and modules, with applications to the structure of Abelian groups and linear operators.

#### Math 428. Honors Topics in Mathematics (3 hours)

A capstone course whose topic can vary. Possibilities include geometric analysis, algebraic geometry, and computational algebra.

## Entry requirements for the Honors Sequence

Recommended prerequisites for entry into the Honors Sequence are: Multivariable calculus (Math 241), and at least one of Fundamental Mathematics (Math 347) or Abstract Linear Algebra (Math 416). Choose honors sections whenever available.

All students entering the Honors Sequence must be approved by the Director of Undergraduate Studies after an interview.

Entry requirements are flexible, and students are welcome to enter at any point and take only part of the Honors Sequence, depending on their background, interests and degree requirements. Taking an honors section of Math 347 or of Math 416 is a good way to decide whether the Honors Sequence will appeal to you. Math 347 and 416 count towards the Major and the Minor in Mathematics.

Math 424 and 427 are offered in the Fall semester; Math 425 and 428 are offered in the Spring semester.

## Registering for the Honors Sequence

Registration for any course in the Honors Sequence must be approved by the Director of Undergraduate Studies. Continuation in the Sequence will depend on performance in previous Honors Sequence courses.

Recommended prerequisites for specific courses:

- Math 424: an honors section of Math 347 or of Math 416.
- Math 425: Math 424 and linear algebra (Math 415 or Math 416).
- Math 427: an honors section of Math 416.
- Math 428: This topics course is primarily intended for students who have taken courses in the Honors Sequence.

These recommended prerequisites are flexible. When thinking about taking an Honors Sequence course, it is a good idea to meet with the Director of Undergraduate Studies to discuss your own situation.

### Requirements satisfied by the Honors Sequence

Courses in the Honors Sequence count toward the Mathematics Major and Minor.

Students in other majors can consult the following list to determine how Honors Sequence courses might count towards their degree requirements:

- Math 424 meets any requirement for Math 447.
- Math 425 does not relate to any specific requirement.
- Math 427 meets any requirement for Math 417.
- Math 428 does not relate to any specific requirement.

### Further Information

To learn more
about the Honors Sequence, or to ask advice on how to fit it into
your course of study, please email or telephone:

Director of Undergraduate Studies in Mathematics

313 Altgeld Hall

1409 W. Green Street

Urbana, IL 61801

telephone (217) 333-1108

fax (217) 333-9576

e-mail: mathadvising@illinois.edu