Obligatory program in maths

[Giallo]

A question to all of you my fellow students: in what consists the obligatory program of the faculty of maths in the first years in you universities?

In mine (ETH Zürich):

1^st year:
- analysis 1 and 2 (real analysis: limits, series, sequences, derivates, integrals in Rⁿ, Stokes/Gauss/Green's theorems, vector analysis)
- linear algebra 1 and 2 (matrices, linear forms, something on modules, all in finite dimensions)
- informatics (a basic course in C++)
- numerical mathematics 1 (polynomial interpolations, approximation of roots, eigenvalues and all that stuff + programming in MATLAB)
- physics 1 (classical mechanics) and 2 (special relativity and electromagnetism)

3^rd semester:
- complex analysis (from complex numbers to residue theorem and Riemann mapping theorem)
- algebra 1 (a little of group theory, ring theory and field theory)
- algorithms and complexity (study of algorithms, their runtime and classes P/NP at a basic level)
- mmp 1 (mathematical methods in physics, I still don't know what really should be the course on...)
- physics 3 (geometrical optics, statistical physics and quantum physics, all at introductive level)

4^th semester:
- algebra 2 (Galois theory in finite extensions and a bit of representation theory)
- topology (point-set topology and a bit of algebraic topology)
- measure theory (well... measure theory)
- numerical mathematics 2 (ODE's, more MATLAB)
- probability and statistics (an introduction to probability theory and a bit of statistics)

From the 5^th semester on we can choose the classes we want to attend, but we still have to do some credits in pure maths and some in applied maths.

[gorcee]

If I'm not mistaken, Swiss/European students typically begin university around age 20, right?

So there might be some age offset here. Also, programs in the United States span a pretty wide range of emphasis.

A "traditional" program at, say, an average university in the US would look something like this (this is not by any means standard):

1st Semester
Calculus 1 (Single variable differential calculus, series, limits, Taylor series)
Humanities Elective
Science Elective
Computer Science 1
Some First-Year student course

2nd Semester
Calculus 2 (Single variable integration, applications, conic sections, introduction to multivariable calculus)
Discrete Mathematics
Humanities Elective
Science Elective
Science Laboratory


3nd Semester
Calculus 3 (Multivariable Calculus)
Introductory Linear Algebra
Introductory Differential Equations
Humanities Elective
General Elective

4th Semester
Introduction to Analysis/Advanced Mathematics (usually an introduction to proofs)
Introduction to Algebra
Math Elective
Math Elective
Humanities Elective

There's a lot of electives, typically, since programs of study don't specialize until the 3rd and 4th years. Many students come in already at the Calculus 1 or 2 level, so they can skip ahead a bit. With the concentrations, students are generally required only to have introductory courses in: Differential Equations, Multivariable Calculus, Linear Algebra, Analysis, and Algebra. Some schools require students to take at least one computer science course and one physics course. Others have more general requirements. Most universities require students to take at least a 5 credit elective humanities sequence.

Courses such as Complex Analysis, Numerical Methods, Dynamical Systems, Probability, Abstract Algebra, Vector Calculus, PDEs, etc. are not required, but some of these will likely be required in the concentration.

More specialized programs/universities have a slightly accelerated program. But otherwise, the options for the focus/concentration remain the same.

[OverBored]

Here in the UK we had:

1st Year:
Vectors and Matrices (mainly a simple look at linear algebra)
Numbers and Sets (a little set theory, but mainly about proof methods and some modular arithmetic)
Groups
Differential Equations
Analysis I
Probability
Dynamics and Relativity
Vector Calculus

We break our year into three terms, the third of which is mainly exam term, and so there are no first year lectures. There are however a few second year lectures that people either attend then or a year later, and most attend in first year.


These are
Metric and Topological Spaces
Variational Principles

From then onward, there is an increasing amount of choice, although it is quite common to do all or almost all second year courses as well.

[Giallo]

If I'm not mistaken, Swiss/European students typically begin university around age 20, right?

Right, 19/20. In the US?

By the way, all our classes are proof based.

[Tirian]

If I'm not mistaken, Swiss/European students typically begin university around age 20, right?

Right, 19/20. In the US?

By the way, all our classes are proof based.

College typically starts at 18 in the United States. The ordinary incoming freshman oriented towards math, sciences, or engineering will come in knowing analysis up through the Fundamental Theorem of Calculus.

Just for lulz, here is the recommended schedule for (pure) math majors at Carnegie Mellon (my alma mater, although I took the honors track):

1st semester: differential and integral calculus, physics 1 (mechanics), computer programming, biology, rhetoric
2nd semester: differential equations and approximation, formalistic foundations, "matrix theory", physics 2 (electromagnetism), humanities elective
3rd semester: discrete math (or combinatorics or graph theory), multidimensional calculus, chemistry, two electives
4th semester: ODE, abstract algebra, three electives
Junior year: two semesters of real analysis, probability, and linear algebra

I should make clear that those electives are largely but not fully expected to come from the humanities in the first three semesters, although one could focus on mathematically "practical" branches of the liberal arts like philosophy, economics, or statistics for the most part.