Which books or subjects would you recommend for undergrads for grad school?

I am an undergraduate mathematics student in my third year. Most undergrad programs don't completely prepare you for grad school. I know Ph.D. students are telling me that there is a lot of crucial material that got left out. Here are the subjects I have or plan to study:

Calc sequence and Analysis 1 & 2

Abstract Algebra

ODEs

Topology

Linear alg.

Number theory

Combinatorics

Mathematical logic

Calculus on Manifolds

Set theory (I will study separate from M. Logic)

I want to be ready for grad school simply because I struggle with depression and it will take a lot of load off if I can prepare somewhat ahead of time. Which other courses or texts would you recommend to have a solid foundation for grad school? Is learning Mathematica or similar languages a considerable advantage also in grad school?

I would ask faculty but they are always busy and there are two advisors really one of which doesn't have a background in math and the second is a bit of a curmudgeon in the department. Thanks if you can help.

edited Jul 8 '16 at 12:30

asked Jul 6 '16 at 23:10

Red

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$\begingroup$ The first major hurdle in a graduate program (after admission) is the qualifying exam, so anything you can do now to prepare would be advantageous. If your department offers graduate courses and allows you to take them as an undergraduate, you could consider enrolling in one or more of them. Specifically, whichever of analysis, algebra, and/or topology interests you the most, as you will very likely have to take qualifying exams in these subjects. $\endgroup$ – Bungo Jul 6 '16 at 23:27
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$\begingroup$ Another recommendation: get and read this book: A Mathematician's Survival Guide: Graduate School and Early Career Development $\endgroup$ – Bungo Jul 6 '16 at 23:32
$\begingroup$ @Red: math.stackexchange.com/questions/94827/…, math.stackexchange.com/questions/148165/…, math.stackexchange.com/questions/394639/…, math.stackexchange.com/questions/22861/… $\endgroup$ – Moo Jul 7 '16 at 0:55
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$\begingroup$ I would start with All the Mathematics You Missed But Need to Know for Graduate School by Thomas A. Garrity (2001). It's designed to give you an overview of many topics and it has a decent bibliography for each of these topics. $\endgroup$ – Dave L. Renfro Jul 7 '16 at 13:56

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