My Mathematics Curriculum - Guidance Needed

[ohmymath!]

My university is about to start the enrollment phase for Fall Semester. I have grown weary of searching 'good' Mathematics curriculum on Google for the past few days, so I finally decided to bump into 'Physics Forum' to get academic guidance. I am a Mathematics major and will be entering my sophomore year soon. I will, 'technically' speaking, become junior by the end of Fall Semester. Though Mathematics major is a four year degree over here in my university but I would be finishing it within 3 years. This, along with the fact that our university is poor at providing good academic guidance, I have decided to plan well in advance which courses I would/should take.

Kindly look at the following plan and tell me if it will provide good enough foundation to let me get into a good grad school (Mathematics). It would be great if anyone over here could also discuss from the viewpoint of Cambridge Mathematical Tripos Part III.

1st Year (Freshman): (Taken)
Calculus - I
Calculus - II (vector)
Linear Algebra - I
Differential Equations
Logic
Programming: C++ / MATLAB / VB
+Compulsory University Courses and few electives

2nd Year (Sophomore/Junior) Year:
Fall: (About to take)
Physics - I (Mechanics)
Calculus - III (Advanced level)
Real Analysis - I
Probability
Number Theory
Discrete Mathematics

Spring: (Planned)
Physics - II (Electricity and Magnetism)
Statistics
Real Analysis - II
Set Theory
PDEs
Topology / Algebraic Topology

Summer: (Planned)
Linear Algebra - II
Independent Study Course

3rd Year (Junior/Senior) Year:
Fall: (Planned)
Physics - III (Heat and Thermodynamics)
Algebra-I
Numerical Analysis / Advanced Statistical Analysis
Graph Theory /or/ Waves and Optics /or/Astrophysics
Functional Analysis
Quantum Mechanics - I

Spring: (Planned)
Physics - IV (Relativity and Modern Physics)
Research Project - I
Algebra-II
Complex Variables
Combinatorics /or/ Quantum Mechanics - II
Fuzzy Logic

Summer: (Planned)
Research Project - II

 

[SW VandeCarr]

I'm just curious as to the way your algebra courses are spaced out. You've taken Linear Algebra I last year and will not take a second course until next summer. Then you're taking two algebra (basic?) courses (I and II) in your junior/senior year. I don't know about the content of these courses, but you will need to have a fair amount of group theory for your advanced physics courses.

 

[Coto]

It does seem a little light on the algebra. I recall taking 2 linear algebra courses, 2 group theory courses, and 1 ring theory course.

Complex variables also seems to come pretty late. Also, with a background in real analysis you should see if you can take a complex analysis course instead of complex variables (in my school anyway, the difference between the two was the approach taken towards the subject). You also have a functional analysis course, which is quite a broad term nowadays. Depending on its topics it might not help you much.

I would suggest a higher level classical mechanics course that goes over the Lagrangian and Hamiltonian formulations.

Lastly, going by the general flow of what you're taking, discrete math and number theory don't seem to fit in very well.

 

[ohmymath!]

I'm just curious as to the way your algebra courses are spaced out. You've taken Linear Algebra I last year and will not take a second course until next summer. Then you're taking two algebra (basic?) courses (I and II) in your junior/senior year. I don't know about the content of these courses, but you will need to have a fair amount of group theory for your advanced physics courses.

I have planned my courses keeping in mind as to when my university offer them. Linear Algebra II is offered once after every two years (strange, I know) and this time it is likely to be offered in Summer. Algebra I and Algebra II are not basic courses. I agree that they must be properly titled Abstract Algebra I and II. These two course cover:

Basic number theory, Sets, Relations, Binary Operations, Groups, Cyclic Groups, Subgroups, Direct Products, Functions, Symmetric Groups, Equilance Relations,Cosets, Theorem of Lagrange, Normal Subgroup, Homomorphisms and Normal Subgroups, Direct Products and Finite Abelian Groups, Sylow Theorems, Ring, Fields and Integral Domain, Subrings, Ring Homomorphism ... Fermat’s and Euler’s Theorems, The Field of Quotients of an Integral Domain, Rings of Polynomials, Factorization of Polynomials over a Field, Noncommutative Rings, Factor Rings and Ideals, Homomorphisms and Factor Rings, Prime and Maximal Ideals, Unique Factorization Domains, Euclidean Domains, Gaussian Integers and Norms, Extension Fields, Vector Spaces, Algebraic Extensions, Geometric Constructions, Finite Fields, Modules, Automorphisms of Fields, The Isomorphism Extension Theorem, Splitting Fields, Separable Extensions, Totally Inseparable Extensions, Galois Theory, Illustrations of Galois Theory, Cyclotomic Extensions, Insolvability of the Quintic.

All this doesn't sound basic to me but I hope it would be fun to study.

 

[ohmymath!]

It does seem a little light on the algebra. I recall taking 2 linear algebra courses, 2 group theory courses, and 1 ring theory course.

Complex variables also seems to come pretty late. Also, with a background in real analysis you should see if you can take a complex analysis course instead of complex variables (in my school anyway, the difference between the two was the approach taken towards the subject). You also have a functional analysis course, which is quite a broad term nowadays. Depending on its topics it might not help you much.

I would suggest a higher level classical mechanics course that goes over the Lagrangian and Hamiltonian formulations.

Lastly, going by the general flow of what you're taking, discrete math and number theory don't seem to fit in very well.

Hmm...interesting. Complex Analysis is not offered in our university. Our 'Complex Variable' course is a compulsory course that every math major has to take. Its syllabus is similar to that of MIT ('Complex variables with Applications': http://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-fall-2003/calendar/) and even includes a few more advanced topics.

I have taken into consideration your suggestion to take Classical Mechanics course. I will arrange a meeting with my advisor soon.

I can't drop 'Discrete Mathematics' and 'Number Theory' out of my plan. I really love both these courses, especially 'Number Theory'. In fact, my 'Independent Study' course will perhaps be based on Number Theory.