I am a senior and would apply for PhD in US universities for the next Fall. I major in Algebra. My academic interest includes Commutative Algebra, Homological Algebra and Algebraic Topology. But I'm open to other aspects of algebra as well.

I want to ask: do we have a list of univer that are good at studying algebra? Moreover, my profile is not exceptionally good, so I hope you could recommend me under top 30 univer. I really appreciate it.

## Universities with good algebra department

### Re: Universities with good algebra department

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.

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.

### Re: Universities with good algebra department

Thank you for your comprehensive answer. Anyway, I just want to ask, it seems like all univer here are good at Algebraic Geometry, do u know some in the case of Algebraic Topology?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.

### Re: Universities with good algebra department

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.

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.

### Re: Universities with good algebra department

again, super detailed answer ). I just want to ask because I am currently reading algebraic topology. I'm open to other aspects of algebra as well.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.

One ques, I guess I will apply to all 8 univer u recommended. What do u think of other univer under top 30. I mean, it's really hard since looking at the websites can hardly tell if their department is good at PDE or Algebra, for example. I hope u understand what I mean. I want to ask for more recommendation.