I am wondering if anyone here could say if it is worth going into law,
particularly when coming from a good law school (and assuming I have a
real interest in the subject and enjoy research, etc.). Is it true
that the profession is contracting, and that it could be hard to find
a decent job? Is there decent upward mobility in the profession, or
should one expect a sub-60-70k salary for many many years after
school? Is there anyone here that thinks not going into law (ie
pursing a Ph.D. instead) is a better choice?
I went into a top law school (the University of Michigan, ranked #8 when I matriculated, graduated in the top 25%, cum laude, with an editorship of a law journal under my belt) with almost the same academic background (undergraduate math major) and a similar LSAT score to you.
It was a somewhat easier choice for me. I was a solid A- math student, but didn't have the chops and talent to pursue a PhD in math and make an academic career out of it, even though I was something of a math prodigy. I also didn't have the passion for it. I saw that I was spending my free time focused on the humanities, social sciences and campus politics and journalism, rather than on math (although tutoring and grading paid my way for all of my personal and living expenses).
A legal job definitely provides a secure lifetime of decent employment, can be intellectually challenging in some subfields (other kinds of practices not so much), and provides a certain amount of interpersonal interaction and immediate, easily understood relevance that you can't secure as an academic mathematician.
It isn't that hard to find a decent job for a graduate of a top law school, and the profession is not meaningfully contracting. Indeed, almost no occupation has been less impacted economically by the pandemic. Post-law school compensation is bimodal. A minority (maybe 30-40%) start at large law firms (sometimes after a judicial clerkship) and make very good money (low 100s) right off the bat. The rest get decent middle class jobs at first. Most, from both routes, end up eventually self-employed in small and medium sized law firms, although a lucky few (maybe 5%-10%) end up as partners in big law firms and a similar share end up as senior civil servants.
The problem is that the instincts you learn getting as far into math as you have are not very advantageous to a Big Law career, which places a huge premium on social skills, upper middle class to upper class social capital, and hard work as what amounts to being a super-bureaucrat at relatively menial details for long uncreative hours that are only dimly connected to results. A lawyer needs to be smart, but being a "genius" intellectually doesn't provide much marginal benefit. Most economically successful lawyers have quite narrow and specialized practices that present fewer intellectual challenges as you mass produce the same kind of work over and over, and lawyers derive a lot of their income from their capacity to market their services effectively to the affluent and the powerful. Also, a lot of your compensation in law is basically for your marketing, for taking on highly stressful responsibility, and for dealing with very unpleasant situations. It often isn't the most enjoyable life style unless you have a very particular type A, competitive, extraverted personality who understands people extremely well but isn't academically oriented.
Corporate law, in particular, values your interpersonal skills very highly and doesn't place much of a premium on your intellectual legal knowledge and research ability. Those things are factors of production in corporate law but they aren't what leads to success there and are often pawned off on junior associates who never have a shot at making partner.
I could have done better economically (I basically took what amounted to a mommy track for various reasons), but didn't understand the profession, or what the work involved, or what was critical to get ahead at the time and in my early career and had other priorities and a set of values and world views ill suited to the work.
If I was doing it all over again, I would have chosen a quantitative heavy but non-math PhD path (maybe Economics or physics or operations research or statistics) or would have become an actuary, rather than becoming a lawyer. I love knowing the things that I know because I went to law school (which I loved) and because I've have an incredibly diverse (although not terribly well paying) legal practice for 25 years. I was a professor (in a gradate estate planning program for financial planners) for a while, and it was the best job I've ever had and I still enjoy teaching a lot. I also spend lots of time in math related hobbies to exercise and enjoy math related talent that I have but can't use very often at work.
If I were in your shoes, with publications already and an acceptance in a top graduate math PhD program, I would definitely take that path. It is a field within academia with a healthy trend line of stability or growth within academia, and being a professor (which you have a viable shot at doing) is a wonderful way to live. There are fewer job seekers per open position for PhDs in math than in most academic disciplines. I've never met a math prof whose regretted his choice (and I know many, having grown up all my life as a child of a professor and a college administrator in a small college down and having been a math major).