Typical four-year schedule of a math major

WillJ

For a math major who takes Calc I first semester of freshman year, what might his 4 year schedule look like (only with regard to math classes, not electives and stuff)? Obviously this varies a lot from person to person (and a bit from school to school), but in general, what's a normal sequencing of the commonly taken courses?

In high school, math is almost completely linear: algebra 1 ---> geometry ----> algebra 2 ----> precal/trig ----> calc is how it's almost always done. I'm mostly wondering how less linear it is in college (what courses can be taken at the same time, what are the biggest choices in course-taking that math majors make, etc.). I'm talking about pure math, not applied.

Bitter

I can only really give you the first two years since after that, the math department here starts to focus on your interest.

Calculus I and discrete mathematics

Calculus II and algebraic structures

Calculus III and college geometry

Diff Eq

trinitron

Calculus, Linear Algebra
Analysis, Probability
Algebra, Analysis on Manifolds, Differential Equations
Topology, Complex Analysis, more Real Analysis (lebesgue integration)
Differential Topology, Algebraic Topology, Geometry

I tried to put things you could take at once on one line. This is just my opinion/a result of my own experience, but I'm still a student so you should probably wait for some more informed people to respond.

andytoh

This is me:

First Year:
Calculus I
A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor’s theorem; sequences and series; uniform convergence and power series.

Linear Algebra I
A theoretical approach to: vector spaces over arbitrary fields including C,Zp. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.


Second Year:
Calculus II
Topology of Rn; compactness, functions and continuity, extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers. Integrals; Fubini’s theorem, partitions of unity, change of variables. Differential forms. Manifolds in Rn; integration on manifolds; Stokes’ theorem for differential forms and classical versions.

Linear Algebra II
A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra. Examples of symmetry groups and linear groups, stochastic matrices, matrix functions.

Ordinary Differential Equations
Ordinary differential equations of the first and second order, existence and uniqueness; solutions by series and integrals; linear systems of first order; non-linear equations; difference equations.

Introduction to Number Theory
Elementary topics in number theory: arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.

Groups, Rings and Fields
Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hölder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic.


Third Year:
Partial Differential Equations
Diffusion and wave equations. Separation of variables. Fourier series. Laplace’s equation; Green’s function. Schrödinger equations. Boundary problems in plane and space. General eigenvalue problems; minimum principle for eigenvalues. Distributions and Fourier transforms. Laplace transforms. Differential equations of physics (electromagnetism, fluids, acoustic waves, scattering). Introduction to nonlinear equations (shock waves, solitary waves).

Complex Analysis I
Complex numbers, the complex plane and Riemann sphere, Mobius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchy’s theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarz’s lemma, residue theorem and residue calculus.

Complex Analysis II
Harmonic functions, Harnack’s principle, Poisson’s integral formula and Dirichlet’s problem. Infinite products and the gamma function. Normal families and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem.

Real Analysis I
Function spaces; Arzela-Ascoli theorem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, Lp spaces. Applications to probability.

Real Analysis II
Measure theory and Lebesgue integration; convergence theorems. Riesz representation theorem, Fubini’s theorem, complex measures. Banach spaces; Lp spaces, density of continuous functions. Hilbert spaces; weak and strong topologies; self-adjoint, compact and projection operators. Hahn-Banach theorem, open mapping and closed graph theorems. Inequalities. Schwartz space; introduction to distributions; Fourier transforms on Rn (Schwartz space and L2). Spectral theorem for bounded normal operators.

Point-Set Topology
Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Topology of function spaces. Fundamental group and covering spaces. Cell complexes, topological and smooth manifolds, Brouwer fixed-point theorem.

Differential Geometry I
Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.


Fourth Year:

Set Theory
The basic principles of axiomatic set theory, leading to the undecidability of the continuum hypothesis. We will also explore those aspects of infinitary combinatorics most useful in applications to other branches of mathematics.

Differential Topology
Smooth manifolds, Sard’s theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

Algebraic Topology
Introduction to homology theory: singular and simplicial homology; homotopy invariance, long exact sequence, excision, Mayer-Vietoris sequence; applications. Homology of CW complexes; Euler characteristic; examples. Singular cohomology; products; cohomology ring. Topological manifolds; orientation; Poincare duality.

Differential Geometry II
Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.

Symplectic Geometry
A variety of concepts, examples, and theorems of symplectic geometry and topology. These may include, but are not restricted to, these topics: review of differential forms and cohomology; symplectomorphisms; local normal forms; Hamiltonian mechanics; group actions and moment maps; geometric quantization; a glimpse of holomorphic techniques.

complexPHILOSOPHY

That list is intense andytoh. I am wayyyyyyyyyyyyyyyy behind.

What university are you at? My CC sucks.

andytoh

The fourth year courses are cross-listed with graduate courses, but it's good to take them before going into graduate studies, in order to gain an edge to the PhD programme.

But my problem solving is not as good as some undergraduate students here, like AKG. I believe problem solving is more of a talent than knowledge.

AKG

1.
*Analysis I
*Algebra I

2.
*Analysis II
*Algebra II
*Intro to ODE
Probability and Stats

3.
*Real Analysis I
*Complex Analysis I
*Intro to Topology
*Groups, Rings, and Fields
Intro to Number Theory
^Combinatorial Methods
Intro to Mathematical Logic
Intro to Differential Geometry

4.
^*Real Analysis II
^*Complex Analysis II
^Set Theory
Classical Geometries
Seminar in Math
General Relativity (applied math)

* - denotes "core" course
^ - denotes courses cross-listed as graduate courses

You can find descriptions for all these courses here except for Probability and Stats.

I guess andytoh is at the University of Toronto, because the descriptions he's put above are identical to the one's you'll find in my link above.

Anyways, in that link above, you can see what courses are prerequisites for others; that way you can perhaps tell what courses can be taken simultaneously. In fact, if you go here and find the heading "Mathematics (Science program)" then look at the subheadings "Specialist program" and "Major program" you can see what kinds of choices pure math specialists and majors have to make. The courses that end in Y1 are worth 1 credit, and are full year (2 semester) course, and the ones that end in H1 are 0.5 credits (half-year, i.e. 1 semester).

If you look at the requirements for the specialist program, most of the choices you have to make are pretty "narrow" in the sense that you have to pick at least 3 out of 6 options, or 1 out of 2, etc. The only "big" decisions come in your third and fourth year. You have to take 2.5 300- or 400-level credits including 1.5 400-level credits. If you go to the first link I gave, you can see all the 3rd and 4th year courses. At that point, the decision becomes based on:

i) the scheduling of the courses
ii) what you like
iii) what you're good at
iv) what's useful to know
v) etc.

in no particular order.

hrc969

andytoh: so are you taking those courses listed as 4th year cousres right now?

andytoh

I've already graduated, and am studying for pleasure, concentrating mainly in problem solving. I haven't gotten stuck in a textbook problem for about a month so far, but I'm sure I will when I start symplectic geometry.

As an aside, this is something I've noticed whenever you get stuck in a proof question. The question is almost always asking you to prove a theorem that isn't in the book. You can find the proof of that theorem (and hence the solution) in a more advanced textbook of the same subject. I'm building quite a library at home as a result of this.

arunma

I recently got my math degree; I'll list my math courses, which should give you a decent idea of what's in store for you. But bear in mind, during my junior year I dropped my math major (got bored with it and filled in the gaps with more physics), and picked it up again senior year, so that's why I took very few math courses for awhile, followed by a lot of math courses. Also the math courses I took were more spread out, since I also had the physics major to worry about. If you're just doing math, then your schedule may be more densely packed.

Freshman Year
Fall:
1. Calculus 2
Spring:
1. Multivariable Calculus

Sophomore Year
Fall:
1. Sequences, Series, and Foundations (essentially a class on formal math proofs)
2. Linear Algebra and Differential Equations
Spring:
1. Differential Geometry

Junior Year
Fall:
1. Complex Analysis
Spring:
(This is the empty spot I warned you about.)

Senior Year
Fall:
(Also nothing.)
Spring:
1. Applied Linear Algebra
2. Advanced Calculus
3. Mathematical Analysis of Biological Networks

Fifth Year
Fall:
1. Computational Algebraic Geometry
2. Numerical Analysis
3. Theory of Probability and Statistics

Well, aside from the fact that you probably won't go a full year with no math courses, this is a reasonable approximation of what you'll do.

leon1127

1st Year
Calculus 2
Vector Calculus
ODE
Linear Algebra
Intro to Calculus (proof-based)
Method of Applied Maths

2rd Year
Algebra
Analysis 1 & 2
Topology
Complex Analysis
PDE
Stat
Data Analysis
Stochastic Processes
Research in Clifford Algebra
Numerical Analysis

As you might realise, it doesnt contain anything about modern geometry and number theory! that is correct! my school doesnt even offer them at graduate level (but a class call geometry and topology which is just intro to manifold theory....)

Nolen Ryba

I'm a double in math/physics, so this may vary from what a strict math major would take, but oh well, these are my last two years. The only junior/senior math courses I took before this time were Cryptography, PDEs, Integral Transforms, and Complex Variables. My first years were occupied with getting in the lower level math and physics and the general education requirements. I also started as a EE and consequently wasted 8 credits my first year.

Fall 05
Linear Algebra (senior)
Combinatorics (junior)
Intro Abstract Algebra (junior)
Quantum Physics I (junior)
Theoretical Mechanics I (junior)
Optics (junior)

Spring 06
Number Theory (senior)
Abstract Algebra (senior)
Intro Real Analysis (junior)
Quantum Physics II (junior)
Statistical Mechanics (junior)
*Differential Geometry (senior)

Fall 06
Nonlinear Dynamics and Chaos (senior)
Real Analysis (Senior)
Finite Groups and Fields (Graduate)
Quantum Mechanics (Graduate)
Electricity and Magnetism I (Senior)
*Topology (senior)

Spring 07
Rings and Modules (Graduate)
Functional Analysis (Graduate)
Statistics (low level requirement needed for graduation)
Electricity and Magnetism II (senior)
Theoretical Mechanics II (junior)


The starred courses in topology and differential geometry indicate independent study; for the differential geometry, the course here wasn't offered last spring so I found the outline and problem sets from it and followed them. For the topology, we don't offer a course here so I picked up an introductory book (Mendelson) and worked through almost every problem in it. This spring I will be continuing with the independent study of these subjects using more sophisticated books, but right now I have a huge project that has been eating up 20-30 hours per week so I have been putting that off. If you go to a smaller school like me you may have to do similar independent studies to get the necessary background.

hrc969

Well I don't know about getting a "typical schedule" because typical depends on whether you are a student that is looking of going to a top grad school, any grad school or just finish a BS in math.

But about being linear or not:
Taking math classes at the upper division level is highly non-linear.

Anyways here is my (math) schedule so far and what I expect it to be in the future:
* means graduate class

Fall 04:
Multivariable Calculus A

Winter 05:
Multivariable Calculus B

Spring 05:
Honors Linear Algebra (Lower Division)
Differential Equations (Lower Division)

Fall 05:
Honors Analysis A
Complex Analysis

Winter 06:
Honors Analysis B
Differential Geometry A
Honors Linear Algebra (Upper division)

Spring 06:
*Complex Analysis A
Differential Geometry B
Topology

Fall 06:
*Differentiable Manifolds
*Complex Manifolds
*Complex Analysis B
Honors Algebra A

Winter 07:
*Algebraic Topology
*Riemannian Geometry A
Several Complex Variable(reading course)
Advanced Linear Algebra
Honors Algebra B

Spring 07:
*Topics in Manifold Theory
*Riemannian Geometry B
Fourier Analysis
Number Theory
Honors Algebra C

Fall 07:
*Algebra A
*Real Analysis A
*Partial Differential Equations
*Functional Analysis
*Fourier Analysis A

Winter 08:
*Algebra B
*Real Analysis B
*Algebraic Geometry A
*Fourier Analysis B
*(Reading course or research in something related to complex geometry)

Spring 08:
*Algebra C
*Real Analysis C
*Algebraic Geometry B
*Commutative Algebra or Number Theory
*(More geometry)

Yeah, that not typical but maybe it shows the difference between lower division and upper division.

You can still catch up, if that's what you want do. Of course you would have to take alot of classes per quarter.

But really you should go at a pace that you are comfortable with.

mattmns

Wow, you guys have some insane schedules compared to mine (granted I go to a terrible school, so this is no shock). I have never taken more than 3 math classes any semester. To get a BS in math here you basically need 9 upper level math classes (note: we are on a semester system).

Here would be a minimal (pure math) schedule at my school:

Semester 1: Calc 1
Semester 2: Calc 2
Semester 3: Calc 3
Semester 4: Linear Algebra, Number Theory
Semester 5: Advanced Calc 1, Discrete Structures
Semester 6: Advanced Calc 2, Complex Variables
Semester 7: Abstract Algebra 1, Topology
Semester 8: Math Elective, Math Elective (Approved electives are: Abstract Algebra 2, Fourier Analysis, Probability, ODE Theory, Diff Geometry, though others can be apporved)
-------------

Here is what my schedule looks like:

Semester 1: Calc 2
Semester 2: Calc 3
Semester 3: Abstract Algebra 1, Discrete Structures
Semester 4: Linear Algebra, Vector Analysis, Intro prob & stats
Semester 5: Advanced Calc 1, Combinatorics, Complex Variables
Semester 6: Advanced Calc 2, Graph Theory, Number Theory
Semester 7: Honors Thesis I, Topology, and 1 of: *Measure Theory or *Probability or maybe *Fourier Analysis
Semester 8: Honors Thesis II, Abstract Algebra 2, maybe Differential Equations.
-----------

Currently I am in semester 6, so 7 and 8 are expected, and I changed my major to math at the end of semester 4.

mathwonk

are you guys kidding? when do you have time to take so many courses?


for laughs, at harvard in the sixties a better than average math major would take something like:

1) freshman year: a spivak type calculus course.
2) sophomore year: a loomis sternberg type course, and a birkhoff maclane course in algebra.
3) a real analysis course like Royden or Rudin, and a complex analysis course like ahlfors or cartan.
4) an algebraic topology course like spanier or dold, and an algebra course like lang.

hrc969

No kidding.

Well posted, my schedule http://physicsforums.com/showthread....38#post1221338
So, yeah, not as much sleep and minimizing time wasted throughout the day.

gravenewworld

calc I
calc II
calc III
diff eq. & lin algebra
linear algebra
foundations of mathematics
advanced calculus
modern algebra
senior seminar
elective course on analysis
4 other electives above the 3000 level