djysyed wrote:I'm also a senior and spent countless hours researching schools that fit my interests, which are the same as yours.
University of Illinois at Chicago (My current institution): Two commutative algebraists, four additional algebraic geometers, one Algebraic K-theorist (Cross of AT and AG).
The professors at UIC suggested I look into Utah, Notre Dame, Indiana-University Bloomington and Purdue as these three schools are also power houses in Algebraic Geometry. The faculty at these schools are all previous students of the giants of Algebraic Geometry.
My advisor even told me that schools below rank 75 are still pretty good if there is a strong mathematician working there. Some of these schools include Florida State and University of Missouri-Columbia.
One other school I would recommend is University of Illinois at Urbana-Champaign since they have 4-5 people in Homotopy Theory and another couple people in Algebraic Geometry/Commutative Algebra.
djysyed wrote: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.
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