If you’re looking for some sure things, these are some schools in the western half of the U.S. that would meet your budget and that offer a PhD in math.
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New Mexico State: NMF covers full tuition + $5k/year stipend and an additional $4k for housing the first year. So this would be about $3k for the first year and then about $7k/year for subsequent years. Here’s its math website: https://math.nmsu.edu/
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Oklahoma State: NMF covers tuition, $6500/year toward housing, about another $4k/year in stipends, and an additional $3k for the first year. Essentially, this comes very close to a full ride. Here’s its math department: Department of Mathematics | Oklahoma State University
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U. of Houston: NMF would cover tuition & fees, leaving about $11k for room & board. Here’s the math department: Department of Mathematics - University of Houston
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U. of Idaho: NMF would bring a full ride here. Here’s the math department: Department of Mathematics and Statistical Science - College of Science - University of Idaho
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U. of Nebraska - Lincoln: If he becomes a NMF then he would get free tuition here, leaving room & board costs of around $14k/year. Here’s its math department: https://math.unl.edu/
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U. of Oklahoma: It has a rather complex NMF award, but essentially it looks like you’d pay about $8k/year for tuition (it’s $2k off of in-state tuition) and all years would get a reduction of $5500/year for room & board & books, which would bring the costs down to about $16k/year. But the first year there’s an additional $5k for housing and there’s an additional cash stipend (average of about $1250/year) and first year NMFs get an additional $1k. So it won’t be the least expensive NMF offer, but it is one that will bring the school within budget. Here’s its math department: https://math.ou.edu/
UT - Dallas and Texas Tech have already been mentioned and should be within budget (and I’m too tired to keep pulling the numbers/sites). I would take a careful look at Washington State’s course offerings. In College Navigator (the fed’s website based on IPEDS data) the math offerings were listed as applied math rather than just regular math, but it might have been a coding error on their end.
From my review of the schools’ websites, these institutions would not provide sufficient automatic merit aid to bring these schools within budget. Whether they have competitive scholarships of sufficient size, I do not know:
- Colorado State
- Kansas State
- Montana State
- North Dakota State
- U. of Hawaii at Manoa
- U. of Kansas
- U. of Montana
- U. of Nevada – Reno
- U. of New Mexico (its website no longer listed a scholarship for NMF award)
- U. of Utah
- U. of Wyoming