I put the list above. I chose mathematics/statistics - it’s one category.
OP will need to research most every school.
I wonder / worry if they are one sided academically. But hoping OP answers.
Welcome to College Confidential, @droskill!
I am not a math expert, but in looking at College Navigator (the feds’ website), I focused on schools with a good number of students in general math programs (as applied to financial math, applied math, statistics, etc.) as a very high-level gauge as to what type of options might be available for a student interested in pure math. In terms of assessing the math content of a school, I would rely heavily on @Twoin18’s knowledge as well as @ucbalumnus and tagging in @hebegebe.
UMass-Amherst is a great option and will be tough to beat since you have in-state pricing. But it’s always nice to have choices, so here are some additional schools your family may want to investigate:
Will Meet Budget
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Central Michigan: About 10k undergrads, offers through a PhD in math (2 conferred in SY23-24) and has no differential for out-of-state students. @2plustrio has a kid here.
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Iowa State: About 25k undergrads
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Northern Illinois: About 11k undergrads, offers through a PhD in math (3 conferred in SY23-24), and no differential cost for out-of-state students.
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Southern Illinois – Carbondale: About 8200 undergrads and this school really tends to punch above its weight in terms of the number of students going on to earn doctorates in a variety of academic fields.
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U. at Albany (NY): About 12k undergrads and the tuition would be the same as at UMass due to their flagship match program (which is now applicable for students from all states, not just a select few!).
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U. of Utah: Seconding this school of about 27k undergrads
Highly Likely to Meet Budget
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Clarkson (NY): About 2500 undergrads. This is a STEM-focused school and offers through a PhD in math (4 math doctorates were conferred in SY23-24). Tends to be very generous in doling out merit aid.
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Ohio U.: Seconding this school of about 19k undergrads
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U. of Iowa: About 22k undergrads
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U. of Missouri: About 24k undergrads
Potential to Meet Budget, but Less Likely
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Binghamton (NY): About 14k undergrads and the sticker price is about $50k. But they are trying to attract more out-of-state students, so maybe there’s the possibility of a modest scholarship to bring this within budget?
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Brandeis (MA): About 3700 undergrads here. Run the Net Price Calculator to see if this might fall within budget. If it doesn’t meet the budget via need-based aid, I doubt your son would receive sufficient merit aid to meet your price target.
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Stony Brook (NY): About 18k undergrads. Sticker price is about $51k, but perhaps your son might receive enough merit aid here.
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U. at Buffalo (NY): About 20k undergrads and a sticker price of about $49k. I suspect this school might be more likely to offer merit than Binghamton or Stony Brook, but that is just a guess on my part.
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U. of Connecticut: About 19k undergrads. If your son was interested in double majoring in something like Applied Data Analysis, Mathematics/Actuarial Science, Mathematics/Actuarial Science/Finance, or Statistics (all of which sound not like abstract math) then he could qualify for the Tuition Break program (formerly the New England Regional Program) and pay 175% of UConn’s in-state tuition (source).
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U. of Minnesota – Twin Cities: About 40k undergrads. This school may have one of the most robust math programs on this list, and its NPC does give a merit estimate to see if it will fall within budget.
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U. of Vermont: About 12k undergrads and I have heard that their NPC is pretty accurate with respect to estimating merit aid.
Rutgers, Maryland, and Wisconsin are some other schools that are reputedly very strong in math, but I am less optimistic about them coming close enough to budget. I could, however, easily be wrong. 
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My daughter is at Brandeis. She was offered very generous merit aid plus some financial aid, so we are paying nowhere near the sticker price. Don’t just rely on their net price calculator.
Such a decision may depend on how many of courses in topics such as linear algebra, real analysis, modern algebra, complex analysis, topology and functional analysis your son will have completed prior to entering college. Note as well that ambitious mathematics students at LACs may be able to enhance variety and depth through a semester in Budapest or an REU. Nonetheless, elective courses at LACs with strong mathematics departments can themselves be quite extensive.
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Interesting you mention Budapest.
Kalamazoo mentioned that’s where math students study abroad on their website.
Another option might be to consider Canadian universities, where students do a higher percentage of their coursework in their chosen major, compared to US schools.
I’m not an expert on the academic strengths, admissions requirements, and costs of the various Canadian U’s. (I believe Toronto, McGill, and UBC may be too expensive and/or have too high a GPA threshold; not sure about Waterloo.) But as one data point, I saw Queen’s University in Ontario (a 6.5 hr drive from Boston) cited for strength in pure math. The current cost for US students is under 45K/year, and the GPA required for Arts & Sciences (including math) is comfortably below a 3.5. Just one example, and not necessarily the best one, but seems like proof-of-concept.
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I don’t agree with you. Your son is a fine test taker as can be evidenced by his very strong math score. If he was a poor test taker, that would not have been as high.
His issue is that his verbal was much lower. Have you considered a tutor for that type of testing? It seems to me he might benefit from knowing why he made the mistakes he made, and how to improve that part of his SAT.
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I expect he’ll take some loans in addition to our contribution, but yes we’re looking at the UMass Amherst (I went there).
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This is an excellent idea - thank you.
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Yup - he’s got some verbal issues that we’re testing for - and we’ve got a tutor for the verbal portion. We’ll see how he’s doing as he just took his PSAT. We’ve also applied to College Board for extra time on all tests - he’s been rejected twice but we’re going back for a third round.
Brandeis is one of the schools we’re looking at - but I haven’t looked at their math program yet. How is your daughter enjoying it?
Thank you so much for all of this information. I’ll admit to feeling a bit hopeless about where our son can go, and you’ve given me a bunch of great ideas. I agree with you about UMass Amherst being hard to beat - University of Minnesota keeps coming up (as does Stony Brook), but University of Connecticut, Clarkson are new ideas. Can’t thank you - and, well, everyone here, for all of your thoughts.
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Please check on how likely admission is to UMass-Amherst, using scattergrams from SCOIR or Naviance if your school system has that. That will largely depend upon your son’s rank within the class.
Will do - thanks for the pointer.
Totally agree with your comment - he doesn’t really know - he’s done some of the proof stuff but hell, I thought I wanted to be a math major until I got to my second year of college and realized that my brain couldn’t do it.
Thanks for the comments - always a good reminder that he’s still a kid and doesn’t know what he doesn’t know.
Yeah sorry I didn’t include this - he’ll have all of that.
So you have medical documentation or a diagnosis to support the more time request ?
Does the student meet the UMASS requirements I posted above ? You haven’t answered but rigor outside of math will be a concern, if not there
I think if the $36k in state will require loans, then OOS publics like UMN need to be out as they will cost much more. Some SUNYs that match could work or you can go further south or west. For example, C Michigan was mentioned. Its full price is under $30k as is W Carolina. If the SAT comes up, schools like Alabama, UAH and Miss State could be mid 20s and U Kansas high 20s.
Loans = bad. If you need more than $6 or $7k a year, UMASS may not be the right choice.
I’m guessing schools like Brandeis and Clarkson, you can try, but they are having financial issues so your need profile may spook them.
UMass Lowell is cheaper than UMass and may be a desirable sub.
Not trying to drive a wedge into your situation but these are issues you need to be aware of and address.
Good luck.
My interpretation of @droskill’s comment was that the family could contribute $45k and that if their son wanted a school that was around $50k or so that loans would be necessary. Of course, I could be wrong, so hopefully @droskill will clarify what the situation is.
They have very strong math department with many strong students. I know couple.
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Agreed but I thought, maybe incorrectly, they are Mass residents. So the loan comment through me as UMASS will be less than $45k. Even so, a UMN (likely of met the core requirements) or a UMD (reach) will be too pricey. As we all know, there are flagship subs that may cost less, SAT / ACT dependent.
I wouldn’t push aside the high merit LACs. They come with strong money and are large on the per capita PhD lists. Maybe combo with study abroad in Hungary which may give more heft, they work. The classes will likely be smaller. Umass is huge - is that a concern ?
Also there are schools that are more focused on LD kids if that might be a nice feature - whether they measure up in math curriculum I don’t know - the Marists, Hofstras of the world. Arizona likely too expensive.