Math prep for econ PhD

[tutonic]

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

 

4) Advanced Mathematical Analysis

- series of real numbers

- series and sequences in n-dimensional real space Rⁿ

- limits and continuity of functions mapping between Rⁿ and R^m

- differentiation (Maxima, minima and the derivative, Rolle's Theorem and Mean Value Theorem)

- the topology of Rⁿ

- metric spaces

- uniform convergence of sequences offunctions.

[Spectrum]

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)

[tutonic]

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).

[Double Jump]

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.

[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.

[Double Jump]

Your probably right Spectrum we'll have to get more info from tutonic.

[tutonic]

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.

 

Your probably right Spectrum we'll have to get more info from tutonic.

 

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.

[Double Jump]

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).

[vulcan]

A couple of the courses seem oddly similar to this one at LSE:

MA212 Further Mathematical Methods

 

You even use the same book!

[eBopBob]

Similarly, the mathematical analysis appears very similar to:

MA203 Real Analysis

MA203 Real Analysis

[tutonic]

A couple of the courses seem oddly similar to this one at LSE:

MA212 Further Mathematical Methods

 

You even use the same book!

 

Similarly, the mathematical analysis appears very similar to:

MA203 Real Analysis

MA203 Real Analysis

 

That's because LSE is the one providing the academic direction. That's why we're using the same books and following a very similar syllabus.

[win]

The math require for economics is extensive and varied, so "sufficient" is difficult since you will always need to pick up some odds and ends on the way. That said, this looks like an excellent set of courses to prepare: calculus, linear algebra, and proofs/analysis are the core of graduate economics. A couple quick thoughts:

 

First, it looks like "further calculus" is comparable to some "calculus 2" courses in the US, given its focus on series and integrals. Have you taken multivariate calculus previously? I would imagine linear algebra might require it, but if not, this is probably the most important missing element.

 

Second, have you taken any probability/statistics or econometrics? If not, these would be the next most important missing elements.

 

Finally, while often covered in economics classes, one area that is a bit more applied, but useful to either take a course on or do some self-studying with would be optimization. This is a lower priority, but definitely would be another area to consider working on a bit once higher priority items have been covered.

[tutonic]

The math require for economics is extensive and varied, so "sufficient" is difficult since you will always need to pick up some odds and ends on the way. That said, this looks like an excellent set of courses to prepare: calculus, linear algebra, and proofs/analysis are the core of graduate economics. A couple quick thoughts:

 

First, it looks like "further calculus" is comparable to some "calculus 2" courses in the US, given its focus on series and integrals. Have you taken multivariate calculus previously? I would imagine linear algebra might require it, but if not, this is probably the most important missing element.

 

Second, have you taken any probability/statistics or econometrics? If not, these would be the next most important missing elements.

 

Finally, while often covered in economics classes, one area that is a bit more applied, but useful to either take a course on or do some self-studying with would be optimization. This is a lower priority, but definitely would be another area to consider working on a bit once higher priority items have been covered.

 

The further calc course deals with mutlivariate calculus.

 

I should be adequately prepared on the statistics & econometrics front since I have gone through the courses here and have gotten my As for them. A recent development in my masters programme allows me to switch over to an MRes and take PhD Econometrics (ref book: Hayashi 2000) so an A in that course will definitely help allay some concerns on that front. I'm currently auditing a metrics class using Wooldridge to prepare myself for next year.

 

Duly noted on the part about optimisation. I'll be auditing a Mathematical Econs class that deals with optimisation, starting next week. The course synopsis is as follows:

 

Constrained Optimisation

Topics include: Definitions of a feasible set and of a solution, sufficient conditions for the existence of a solution, maximum value function, shadow prices, Lagrangian and Kuhn Tucker necessity and sufficiency theorems with applications in economics, for example General Equilibrium theory, Arrow-Debreu securities and arbitrage.

 

Intertemporal optimisation

Bellman approach. Euler equations. Stationary infinite horizon problems. Continuous time dynamic optimisation (optimal control). Applications, such as habit formation, Ramsey-Kass-Coopmans model, Tobin’s q, capital taxation in an open economy, are considered.

Tools for optimal control: ODEs.

These are studied in detail and include linear 2nd order equations, phase portraits, solving linear systems, steady states and their stability.

[tutonic]

Also, can anyone recommend a good book to purchase that covers the calculus aspect and will also be adequate for the abstract math/analysis course I'm taking?

 

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

 

4) Advanced Mathematical Analysis

- series of real numbers

- series and sequences in n-dimensional real space Rⁿ

- limits and continuity of functions mapping between Rⁿ and R^m

- differentiation (Maxima, minima and the derivative, Rolle's Theorem and Mean Value Theorem)

- the topology of Rⁿ

- metric spaces

- uniform convergence of sequences of functions.

 

While I have numerous pdf-versions of the textbooks (like Apostol's, Spivak's, Tao, Rudin, etc), I strongly prefer a physical book for revision. Obviously, Rudin is far too advanced for me at this juncture. Since I'm financially constrained, can anyone recommend 1 book that covers most of the material in the 3 courses quoted above?

 

Thanks a lot!

[Double Jump]

If Rudin is too challenging at the moment I recommend you check out Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert. I believe a pdf version is floating around on the internet. So you could have a chance to examine it before you purchase it. It's a lot wiser to start with a simpler intro analysis book and build the foundation then looking for a book that encompasses the most.

[tutonic]

If Rudin is too challenging at the moment I recommend you check out Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert. I believe a pdf version is floating around on the internet. So you could have a chance to examine it before you purchase it. It's a lot wiser to start with a simpler intro analysis book and build the foundation then looking for a book that encompasses the most.

 

I found the pdf version online. I'll delve deeper into it. Thanks a lot.

[win]

The further calc course deals with mutlivariate calculus.

 

I should be adequately prepared on the statistics & econometrics front since I have gone through the courses here and have gotten my As for them. A recent development in my masters programme allows me to switch over to an MRes and take PhD Econometrics (ref book: Hayashi 2000) so an A in that course will definitely help allay some concerns on that front. I'm currently auditing a metrics class using Wooldridge to prepare myself for next year.

 

Duly noted on the part about optimisation. I'll be auditing a Mathematical Econs class that deals with optimisation, starting next week. The course synopsis is as follows:

 

Constrained Optimisation

Topics include: Definitions of a feasible set and of a solution, sufficient conditions for the existence of a solution, maximum value function, shadow prices, Lagrangian and Kuhn Tucker necessity and sufficiency theorems with applications in economics, for example General Equilibrium theory, Arrow-Debreu securities and arbitrage.

 

Intertemporal optimisation

Bellman approach. Euler equations. Stationary infinite horizon problems. Continuous time dynamic optimisation (optimal control). Applications, such as habit formation, Ramsey-Kass-Coopmans model, Tobin’s q, capital taxation in an open economy, are considered.

Tools for optimal control: ODEs.

These are studied in detail and include linear 2nd order equations, phase portraits, solving linear systems, steady states and their stability.

 

In my opinion, given all the courses you've mentioned, you should be very well prepared. While there is always more you can learn, I would suggest focusing more on making sure to learn the topics in your classes deeply and thoroguhly as opposed to trying to cover more subjects. Even at the expense of other topics. For example, learning the material in these courses you initially listed should take priority over auditing an optimization course, which is a nice bonus but something that you can do with relative ease if you really understand the math.

 

As for books suggestions, Rudin's Principles is the obvious choice, but I think most people's first encounter is a little intimidating given its sparing prose, lack of examples, and so on. I think once you are in class you will find that less the case, but I think two useful bridge books are Understanding Analysis by Abbott, which is a gentler introduction which I think is more accessibly written. It wont replace rudin, but it would probably be an easier introduction to the most important topics.

 

Another one, which is not specifically analysis, but I think is very helpful is Thomas Sibley's Foundations of Mathematics. I have not come across an electronic version (if you do, let me know), but it is really a very accessible introduction to a lot of the topics that rudin either takes for granted or moves through quickly but are fundamental later on. For example, it spends a fair about of time explaining the basics of set theory, of how to write proofs, the foundations of functions and relations.

 

Anyways, it sounds like you are well covered. Good luck with your courses.

[tutonic]

In my opinion, given all the courses you've mentioned, you should be very well prepared. While there is always more you can learn, I would suggest focusing more on making sure to learn the topics in your classes deeply and thoroguhly as opposed to trying to cover more subjects. Even at the expense of other topics. For example, learning the material in these courses you initially listed should take priority over auditing an optimization course, which is a nice bonus but something that you can do with relative ease if you really understand the math.

 

As for books suggestions, Rudin's Principles is the obvious choice, but I think most people's first encounter is a little intimidating given its sparing prose, lack of examples, and so on. I think once you are in class you will find that less the case, but I think two useful bridge books are Understanding Analysis by Abbott, which is a gentler introduction which I think is more accessibly written. It wont replace rudin, but it would probably be an easier introduction to the most important topics.

 

Another one, which is not specifically analysis, but I think is very helpful is Thomas Sibley's Foundations of Mathematics. I have not come across an electronic version (if you do, let me know), but it is really a very accessible introduction to a lot of the topics that rudin either takes for granted or moves through quickly but are fundamental later on. For example, it spends a fair about of time explaining the basics of set theory, of how to write proofs, the foundations of functions and relations.

 

Anyways, it sounds like you are well covered. Good luck with your courses.

 

I concur that Rudin is too advanced for me right now. To be quite frank, I can't make any sense of what I see in there. I'll look into Abbot. It was actually one of the shortlisted ones

 

So you're suggesting I hold off on auditing the mathematical econs/optimisation course? The reason I am of the mind to do it is because the course is a relatively easy A. So another A to add to my collection will increase my overall profile, right?

 

Another reason is that doing 'applied' math in the form of mathematical econs course will make the year much more pleasurable, as compared to doing just pure math courses. So, taking it will add a factor of fun, for lack of a better term.

 

Lastly, with the mathematical econs course, I'll only be attending 9 classes a week. True, it's considerably higher than the usual 4 classes but it seems quite manageable.

[win]

I concur that Rudin is too advanced for me right now. To be quite frank, I can't make any sense of what I see in there. I'll look into Abbot. It was actually one of the shortlisted ones

 

So you're suggesting I hold off on auditing the mathematical econs/optimisation course? The reason I am of the mind to do it is because the course is a relatively easy A. So another A to add to my collection will increase my overall profile, right?

 

Another reason is that doing 'applied' math in the form of mathematical econs course will make the year much more pleasurable, as compared to doing just pure math courses. So, taking it will add a factor of fun, for lack of a better term.

 

Lastly, with the mathematical econs course, I'll only be attending 9 classes a week. True, it's considerably higher than the usual 4 classes but it seems quite manageable.

 

You definitely know more about the courses and workloads than I do, but it seems like you've got plenty of hard math to cover in your courses. If you are just planning on auditing the class, I guess you can probably just stop going if you have too much work. My point is just that doing well in something like Analysis (which I read as the subjects of Abstract Math and Advanced Math Analysis) is much more important than the bonus class of optimization. Either way, I'm sure you'll find a balanced work load and it sounds like you are taking all the right classes. Good luck!