11

Take your time Andrew, don't worry so much about classes. Learn Python and solve your calculus problems with it. Learn R and solve your Stats problems with it. If you can do this by the end of your freshman year you've accomplished an incredible amount. Next year, if you can take a discrete class that includes some basic work with matrices this would be ...


10

I will vote for complex analysis. The main theme of complex analysis is that differentiable complex functions are incredibly rigid, which is completely different from the behaviour of differentiable real functions. The main results of complex analysis are truly astonishing, when you think about them properly. In my opinion, complex analysis is the first ...


8

An argument can be made in favor of Cryptography, not because it is more fundamental than Complex Analysis, or more beautiful, but because: (1) You have not yet taken Number Theory, and you will learn quite a bit of Number Theory in a course on Cryptography. Here's a typical course: "Introduction to Cryptography and Computer Security", Brown Univ., 2014. (...


7

Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right? When I was an undergraduate at Penn State, these two courses were the only 300 level math courses, both designed to be taken first semester junior year. The introduction to abstract algebra used "Numbers, Groups, and Codes", ...


6

Despite the names of these fields, as a student I found real analysis more abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I can justify this, but let me give two examples: Lagrange's theorem in abstract algebra: The order of a subgroup $H$ of a finite group $G$ divides the order of $...


4

This same question has arisen at my current institution even earlier (i.e., after AP Calculus AB). The traditional next course after the Calculus is Real Analysis (constructing the real numbers in one of the four ways - Dedekind cuts, infinite decimal expansions, rational sequences modulo Cauchy, or with the axiom of completeness: usually via one of the last ...


3

What you likely will not have experienced after Calculus and Statistics is: proofs. At most institutions, both Discrete Math and Linear Algebra introduce, and may even concentrate on, proofs. The same is true of Theory of Computation in computer science. Multivariate Calculus (sometimes called Calc III) often does not emphasize proofs. So that can serve as ...


2

That depends on what you're interested in. If you're thinking about biology or chemistry as a career, differential equations would be a good choice. If you're interest is in computer science then I would go with the discrete math option. For electrical engineering, you might think about a class in complex analysis. (Complex refers to complex numbers not ...


2

Seem you have two problems. The students resistance and the misunderstandings of which chapter you are referring to. To overcome the student's resistance, suggest that they (the students) are part of an experiment to improve the order of the chapters in the book. Tell them when you finish, the class will help you write to the publisher why your order is ...


2

As a number theorist I am biased, but I think a conjecture-proof study of elementary number theory could be a good place to start. Many summer programs for mathematically-inclined high school students (e.g., the Ross Program, the Hampshire College Summer Studies in Math, PROMYS) focus on number theory and these have a long track record of producing ...


1

I think you should focus on learning practical and interesting skills that will help you to get a coding job. For example, how about learning how to code in C#, Java or even Javascript? Coding is a practical and real world skill. It's not a specialty reserved for research scientists anymore. I suggest you begin by focusing on real world situations before ...


1

The difficulty of the courses depends on the content and execution of the courses in question, and as such, the question does not have an answer.


1

You use the phrase "chapter[s] in my course" (as opposed to the book), which makes no sense linguistically. I would recommend that you find some other way of expressing what you're trying to say. This itself may be the major point of confusion for your students.


1

In my experience, this hasn't been an explicit reason for students finding the subject harder. In fact, it was quite common to teach subjects (not just maths) in a different order than what is given in the textbook in my own education. I'd suggest the following for you (with the notion of encouraging students to get used to a different order from middle ...


1

Complex variables has meaningful application in many other courses, on the other hand, while PDEs come up in much of differential geometry etc. it is usually the case that the methods needed to solve that PDE are specific to the world in which it arose. In other words, take complex first. However, if only one course is available, I'd take whichever I could. ...


1

Also consider linear algebra. Given your interests one emphasizing matrix manipulation rather than theory is preferable, for now at least. P.s. In general, when accelerating in high school try to take courses that are mainstream in both topic (used in many majors) and treatment (how commonly taught) as that keeps your options most open. Also consider ...


1

My advice is to take complex analysis, but a more applied version (weak on proofs, strong on contour integration). Just based on your question and what you said about yourself, I think you will enjoy this more. And you can take more theoretical analysis versions later (if you choose). But at least you will have some exposure to a fundamental field.


Only top voted, non community-wiki answers of a minimum length are eligible