Graduate School Funding

Mathematics PhD programs in the United States are fully-funded. This means

The department pays your tuition and fees
You are paid a salary for teaching and research while there (usually around $20,000 - $25,000).

Tuition and Fees

For most PhD programs, the cost of tuition and any associated frees are paid directly by the math department to the university. At most universities, this also includes the cost of any health insurance the university offers.
For Masters programs in the US, you must pay the tuition and related fees. For this reason, it's often better to enroll in a PhD program (which will also grant you a Masters degree).

Salary / Stipend

PhD programs in the US will pay you a salary or stipend for a fixed number of years. This will be carefully spelled out in your offer letter. Depending on the University and the cost of living in an area, the amount ranges from $20,000 to $26,000 or more.

This tends to be a nine month salary (running over the academic year), with many Universities giving you additional money or opportunities to earn money during the summer.

Depending on your university and offer, you may see different kinds of descriptions and duties:

TA (Teaching Assistant) - you will be expected to lead a discussion section or teach your own class. This is usually 1 or 2 classes per semester, and the majority of funding comes in this form.
Fellowship, RA (Research Assistant), Grant - you will not be teaching during this period. Instead you will focus on your research.

Most institutions will try to give you a mixture of some semesters with teaching and then some supported by fellowships.

Outside Funding

You are welcome (and encouraged!) to find your own funding from outside of the university. This often results in decreased teaching responsibilities or a larger salary. The usual source for such grants is your country's government, though there are some private opportunities also available.

Select PhD Programs in Algebraic Topology

Here is a list of universities with active algebraic topology programs, broken down by region, together with a list of researchers in algebraic topology at those institutions. When applying to these programs, it can be helpful to list individual resaerchers you would be interested in working with by name (together with a sentence or two about why).