According to your experience as students and professors, what are (and why) the courses that should be part of a math undergraduate degree, but that are missing in most institutions?
Before I answer your question, I want to preface my answer with the statement that, what courses are available at any one particular university are contingent on the faculty. If there is nobody appropriate to teach a branch of mathematics, then it is difficult to run such a course!
In America at least, I feel as though there is such a heavy emphasis on the continuous track of mathematics. The discrete track, I feel, is underrepresented. Courses such as Graph Theory, Game Theory, and Combinatorics are classes I have taken but gained so much value from, and are more difficult to come by. I've been blessed by attending two schools (undergraduate and graduate) that have had a wide variety of courses to choose from. Even then, these courses were frequently only offered every few semesters. There are a wide number of sub-genres of topics that could be developed into a full class, if the right person were available to create the course.
I want to answer another question that seems relevant: What courses should be a part of a mathematics education degree program?
I have long yearned for a course that focuses specifically on the methodology of a particular commonly taught mathematics course. A course on teaching high school algebra, teaching high school geometry, teaching college group theory or teaching college analysis would do wonders, I think. To my knowledge, no such courses exist.
A course in number theory.
Number theory is one of the "big four" topics in mathematics (with algebra, geometry, and calculus/analysis). It has a long history, equal to that of geometry. It is foundational to algebra and calculus, inspiring and giving many examples to those topics. It is the first application of set theory to the "big four." It has strong links to non-big-four topics such as set theory, probability, statistics, and combinatorics. It is simple enough to be understood by elementary school students but deep enough to provide many of our famous unsolved problems (Goldbach conjecture, twin-prime conjecture, etc.). It forms many puzzles that can be understood by non-mathematicians. It inspired the most famous current unsolved problem, the Riemann hypothesis. Finally, it has important real-world applications in topics such as cryptography.
I was not able to take a number theory class when I was a math undergrad at Cornell University. A professor tried to get a class together but could not get the minimum number of students. My main exposure to number theory was in "math camp" during a high school summer, and it was the only required topic since it has practically no prerequisites but is studied in university.
At large research schools, or moderately large state universities there is often an undergraduate or graduate course which deals with the following topics. However, at a small liberal arts school, too often one, two or all three of the following are either missing or under-emphasized:
- elementary topology
- differential geometry
- complex analysis
Actually, topology is very common, but, I list it for my own reason. More important than all of these:
- a second course in linear algebra including modules and cannonical forms, multilinear algebra and other fun things we dare not cover in the linear course for the masses.
A course in experimental mathematics could be valuable so that students are introduced to making conjectures and testing their plausibility. This would give them a taste of an aspect of research mathematics and might help counter the notion that all the mathematics there is has already been discovered. Appropriate subject matter, depending on level and interests, could include combinatorics, dynamical systems, discrete geometry and/or number theory.