The issue these days with Algebraic Topology is that there are very few areas of mathematics which fall purely under the scope of Algebraic Topology. Most Algebraic Topology heavy areas fall under Differential Geometry or Algebraic Geometry. That being said, Chromatic Homotopy Theory is pretty large but people who study it are at top 20 institutions (Harvard, MIT, Northwestern). You could also look into Topological K-theory but most Topological K-theorists also take part in CHT. Again, not many Topological K-theorists exist at non-top 20 programs. A last option would be pure category theory which is studied by John Baez or Emily Riehl.
If you want to combine Algebraic Topology and Algebraic Geometry, your best bet is Algebraic K-Theory. Simply put, K-theory is a cohomology theory and, as you might know, the cohomology groups of a space have a ring structure via cup product. As such, Algebraic Geometric methods are quite useful. There are many schools with Algebraic K-theorists and most of them aren't ranked top 20.This
reddit post gives more details about which areas of Algebraic Topology are open for research.