How do I take my mathematics to the next level?

[jdinatale]

How do I take my mathematics to the "next level?"

Let me explain what I mean:

I am finishing my junior year in undergraduate. I consider myself "good" at math, in the sense that I make A's in my classes. But that's not enough.

I want to be a world class math student because my dream is to go to a top 10 graduate school one day. I've looked at the preliminary exams, say Berkeley's for example, and I am not on that level yet. Despite taking around 18 math courses, I'm not skilled enough to tackle those problems.

Also, I've taken the Putnam exam. I scored a 0. I'm not on that "next level" of mathematics yet.

I know a great deal of natural talent is required to get into a school like Berkeley, but I believe that hard work goes a long way and can make up for not being a prodigy or genius.

How do I take my math game to the next level? Obviously doing problems. But what does that entail? Drilling analysis and abstract algebra problems for a few hours everyday? What should my game plan be?

 

[Timo]

The obvious way to prepare for the math required to enter grad school seems to be learning the math taught to undergrads at university.

 

[jdinatale]

The obvious way to prepare for the math required to enter grad school seems to be learning the math taught to undergrads at university.

Yes, I'm aware of that. But does that mean I should whip open my analysis and algebra textbooks and just drill problems everyday for a few hours? What's the most efficient way?

 

[Timo]

I would recommend attending courses and lectures. That is the best answer I can sensibly offer without drifting into the realm of "anything goes"-fantasy. But I am aware that this is not the answer you are looking for. So I better refrain from this thread and hope for others having ideas more to your taste.

 

[carlgrace]

Yes, I'm aware of that. But does that mean I should whip open my analysis and algebra textbooks and just drill problems everyday for a few hours? What's the most efficient way?

I did just that getting ready for my prelims in grad school. I knew my undergraduate stuff COLD. That way you have a good base when stuff starts getting weird in your graduate courses.

 

[jimmyly]

Let me explain what I mean:

I am finishing my junior year in undergraduate. I consider myself "good" at math, in the sense that I make A's in my classes. But that's not enough.

I want to be a world class math student because my dream is to go to a top 10 graduate school one day. I've looked at the preliminary exams, say Berkeley's for example, and I am not on that level yet. Despite taking around 18 math courses, I'm not skilled enough to tackle those problems.

Also, I've taken the Putnam exam. I scored a 0. I'm not on that "next level" of mathematics yet.

I know a great deal of natural talent is required to get into a school like Berkeley, but I believe that hard work goes a long way and can make up for not being a prodigy or genius.

How do I take my math game to the next level? Obviously doing problems. But what does that entail? Drilling analysis and abstract algebra problems for a few hours everyday? What should my game plan be?

I am in no position to give advice since I know nothing about prelim exams But I do have a suggestion. Maybe you can pick up some rigorous textbooks if you haven't already done so. I see spivak mentioned a lot and has quite the rep.

 

[jdinatale]

I am in no position to give advice since I know nothing about prelim exams But I do have a suggestion. Maybe you can pick up some rigorous textbooks if you haven't already done so. I see spivak mentioned a lot and has quite the rep.

The only Spivak book that might be useful would be his differential geometry / calculus on manifolds books, but those topics aren't covered at all on the preliminary exams, so I don't really need to worry about them.

 

[jbunniii]

This book contains almost 1000 problems which have appeared on past Berkeley prelim exams, along with solutions. If I was faced with the daunting task of trying to pass that exam, it's hard to imagine that any other resource would be more efficient than this one:

https://www.amazon.com/dp/0387008926/?tag=pfamazon01-20