# Alternative Upper-Level Courses

Below you will find brief summaries of the differences between alternative courses satisfying requirements for the math major. You should talk to an adviser to help decide on the best courses for you.

## Analysis: 500 or 765?

Both Math 500 and Math 765 are introductions to the theory of functions and calculus. Here you will learn to prove some of the basic theorems of single-variable calculus. Each course has Math 127 or 223 and Math 290 as its prerequisites. The difference is in the level of sophistication or "mathematical maturity" required. Before taking Math 765 you should have some experience in writing proofs, unless you have done exceptionally well in the calculus sequence and have strong skills in logic and writing. Math courses at the 700 level generally require a greater time commitment than those at the 500 level.

## Algebra: 558 or 791?

Both Math 558 and Math 791 are introductions to modern algebra. Here you will learn about generalizations of familiar number systems, and prove basic facts about algebra. A main difference is in the level of sophistication or "mathematical maturity" required. Before taking Math 791 you should have some experience in writing proofs, unless you have done exceptionally well in the calculus sequence and have strong skills in logic and writing. Math 558 often includes more on polynomials, complex numbers, modular arithmetic and concrete permutations groups. Math 791 often goes further into the structure of abstract groups and rings. Math courses at the 700 level generally require a greater time commitment than those at the 500 level.

## Statistics: 526, 627 - 628 or 727 - 728?

Math 526 is a brief introduction to probability and some statistical methods useful in applications. Its prerequisite is Math 116, 122 or 127. Math 627-628 (with prerequisites Math 127 or 223 and Math 290) introduces the probability theory and theoretical statistics students need to begin graduate study in statistics. The sequence includes some challenging multivariable calculus and some theorem-proof material.

A more advanced treatment of the 627-628 topics is given in courses 727-728, which prepare graduate students for the PhD qualifying exam in probability and statistics. Math 727 is also a required course in the most commonly used KU program of study leading to a MA degree in mathematics.