Can you get into PhD program without publication?

No, it is not. The majority of students accepted into PhD programs do not yet have publications. Having one’s name on a published article may also not mean much, especially if there are several co-authors.

What does it take to get into a top math PhD program?

Roughly: good grades (3.8+ GPA) in difficult courses, good test scores (80+ percentile on math GRE subject test [not the regular GRE math, which you should get a ~perfect score on without studying]), strong research background and good letters corresponding to it.

Is it normal to start a PhD right after your undergrad?

In fact NSERC funding for your PhD will last only 3 years if you have a Masters, and 4 years if you do not: so yes, there is a way for you to start your PhD right after your undergrad. People have mentioned that it’s normal in Denmark and Australia too.

Is it possible to do a PhD in Australia without masters degree?

In Australia, only Bachelor is officially required for PhD. There is no such thing as a master degree requirement in the country. I have done exactly that, I had a Bachelor from Switzerland and did a PhD in the UK.

How much should you earn as a PhD student?

Earn a PhD salary (e.g. ~$35,000/year in Canada) for 3-4 years, then a post-doc salary (e.g. $40,000-$70,000/year in Canada) for the remaining 2-3 years? These days, it is getting harder and harder to get a stable job while our bodies are still in their prime condition for raising a family (or doing whatever else we enjoy).

How do I become a smart math major?

That means you need them to say you are smart as in PhD-smart. Advanced coursework: Take ALL the higher level courses if possible. Take some graduate level courses if possible. Now about grad school: don’t think about the top 1 or 2 or even 3 for now. If your goal is math, then your goal is math, not the school.