# Thread: Math prep for econ PhD

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## Math prep for econ PhD

Hi all,

I'm planning on taking these 4 courses this coming academic year and am wondering if it's sufficient, in terms of math prep for an econ PhD programme. Below are the topics covered in each course.

1) Further Linear Algebra

- Diagonalisation and Jordan normal form, applied to systems of differential equations.
- Inner products, orthogonality, quadratic forms, and orthogonal diagonalisation.
- Direct sums and projections, with applications to least squares.
- Generalized inverses.

- Complex numbers. Complex matrices and vector spaces. Hermitian and unitary matrices, unitary diagonalisation and spectral decomposition.

2) Further Calculus

- Limit of a function of one variable, continuity.
- Riemann integral, Fundamental Theorem of Calculus.
- Improper Integrals, Test for convergence.
- Double Integrals.
- Dominated convergence.
- Laplace Transforms.

3) Abstract Mathematics

- Mathematical statements, proof, logic and sets
- Natural numbers and proof by induction
- Functions and counting
- Equivalence relations and integers
- Divisibility and prime numbers
- Congruence and modular arithmetic
- Rational, real and complex numbers
- Supremum and infimum
- Sequences and limits
- Limits of functions and continuity
- Groups
- Subgroups
- Homomorphisms and Lagrange's Theorem

- series of real numbers
- series and sequences in n-dimensional real space Rn
- limits and continuity of functions mapping between Rn and Rm
- differentiation (Maxima, minima and the derivative, Rolle's Theorem and Mean Value Theorem)
- the topology of Rn
- metric spaces
- uniform convergence of sequences offunctions.

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## Re: Math prep for econ PhD

As usual, I have no clue about what you should do for PhD admissions. However, that Further Calculus class looks odd to me. I'm guessing it's not proof based given the courses that come after it, but it seems like in that case it's a regular calculus course. If you've already taken the standard calculus series then I'd look into this course a little more. Laplace Transforms are covered in differential equations courses but not the other topics usually, in the US.

Your real analysis class looks like it covers the standard topics, although it could be said to be missing reimman integration (although this is not to say the class looks less rigorous at all - we can't tell the level of rigor from what you've posted, so don't take what I just said as the class being inferior to that which most people on this forum take)

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## Re: Math prep for econ PhD

Side note: I'm not from the US. Hence, the course titles don't necessarily correlate with those that you have there.

In any case, reimman integrals are covered in further calculus instead of my analysis course (since courses 1-3 are pre-requisites for the 4th one).

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## Re: Math prep for econ PhD

Your "Further Linear Algebra" class seems like an applied class? I'm guessing the proofs won't be too bad relative to your other classes. Your "Further Calculus" calculus class seems like a mix mash of Riemann integration and multivariable stuff. Your "Abstract Math" class is definitely an intro to proof class, it'll be a mix of number theory, advanced calculus, and group theory. Your "Advanced Mathematical Analysis" class seems like a regular advanced calc class.

How are you planning on sequencing those classes out? I have no idea if your list is sufficient for a Phd program.

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## Re: Math prep for econ PhD

I think you're wrong Double Jump, but tutonic will have to confirm this.

As he's European, I'm guessing what we assumed was a basic intro to proofs class is actually a relatively rigorous first semester of real analysis with 2 chapters of group theory (or something like this). The Further Calculus class is a second semester of analysis, and the fourth class is a more rigorous edition of a first semester in real analysis (most likely with baby Rudin, I'm guessing?) intending to prepare students to excel in graduate math courses.

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## Re: Math prep for econ PhD

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## Re: Math prep for econ PhD

Originally Posted by Spectrum
I think you're wrong Double Jump, but tutonic will have to confirm this.

As he's European, I'm guessing what we assumed was a basic intro to proofs class is actually a relatively rigorous first semester of real analysis with 2 chapters of group theory (or something like this). The Further Calculus class is a second semester of analysis, and the fourth class is a more rigorous edition of a first semester in real analysis (most likely with baby Rudin, I'm guessing?) intending to prepare students to excel in graduate math courses.
Originally Posted by Double Jump
Hi guys. Thanks a lot for the replies. Abstract Math serves as a first look at proof while Advanced Math Analysis focuses on real analysis. Unfortunately, my institution doesn't have any analysis course that uses baby Rudin. Below, you'll find the brief course descriptions (if it's of any help), along with the accompanying textbook(s).

I'm planning on taking all 4 of them this year.

Further Calc: This half course provides students with useful techniques and methods of calculus and enables students to understand why these techniques work. Throughout, the emphasis is on the theory as well as the methods.

Reference books:
Adam Ostaszewski Advanced Mathematical Methods & Ken Binmore and Joan Davies Calculus:Concepts and Methods

Further Linear Algebra: In Algebra, students have met many of the key concepts of linear algebra. In thiscourse, we study further theoretical material and look at additional applications of linearalgebra.

Reference books: Anthony, M. and M. Harvey, Linear Algebra:Concepts and Methods

Abstract Math: This course is an introduction to formal mathematical reasoning, in which proof is central. It introduces fundamental concepts and constructions of mathematics and looks at how to formulate mathematical statements in precise terms. It then shows how such statements they can be proved or disproved. It provides students with the skills required for more advanced courses in mathematics.

Reference books:
Biggs, Norman L. Discrete Mathematics, Eccles, P.J. An Introduction to Mathematical Reasoning; numbers, sets and functions & Bryant, Victor. Yet Another Introduction to Analysis.

Advanced Math Analysis: This is a course in real analysis, designed for those who already know some real analysis (such as that encountered in Abstract mathematics). The emphasis is on functions, sequences and series in n-dimensional real space. The general concept of a metric space will also be studied.

Reference Books
: Bartle, R.G. and D.R. Sherbert Introduction to Real Analysis, Binmore, K.G. Mathematical Analysis: A Straightforward Approach & Bryant, Victor Yet Another Introduction to Analysis.

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## Re: Math prep for econ PhD

Your "Further Calc" class seems really interesting. Skimming through the books, the Advanced Mathematical Methods book teaches a lot of concepts that can be applied to differential equations (i.e. learning about the Jacobian helps figuring out the inverse function theorem that can be used to solve stuff). And the Calculus: Concepts and Methods book is super useful for learning econometrics, I bet you could actually start reading through Greene and at least understand what's going on with the matrices after you learn Binmore's book.

The "Further Calc" class will complement your "Further Linear Algebra" class. Your "Abstract Math" course is an intro proofs course. Don't let "intro" deceive you, these types of classes can be very difficult because they sacrifice depth for breadth so you may lack awareness of nuances on problems.

Yup Advanced Math Analysis is just Advanced Calc or real analysis or whatever (schools call it by different names).

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## Re: Math prep for econ PhD

A couple of the courses seem oddly similar to this one at LSE:
MA212 Further Mathematical Methods

You even use the same book!

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## Re: Math prep for econ PhD

Similarly, the mathematical analysis appears very similar to:

MA203 Real Analysis

MA203 Real Analysis