what are some majors at the crossroads of liberal arts and science?

economics, philosophy…?

Liberal arts includes science.

Do you mean majors that are at the intersection of the humanities/social sciences and the physical and natural sciences? I wouldn’t really say economics is there; it’s pretty squarely a social science. Higher-level economics can use a lot of math, but it’s not usually a theoretical exploration of math - math is instead used as more of a tool to understand economic (aka, social scientific) elements.

That said, you can make any field come to the intersection in subfields - psychophysics, quantitative psychology (which is about the theoretical exploration of new mathematical/statistical techniques); sociobiology; philosophy of science; computational linguistics.

Geoarchaeology? That would be a science and a social science (two liberal arts fields). Really, you’d have to tell us more about your interest in the topic for our replies to offer much meaning to you.

Energy Economics

Pure math, physics. Maybe applied math.

Geography given that it can encompass everything from hydrology to public policy to agricultural economics. Environmental science/ sustainability is another possibility. Some schools offer history of science majors as well as some variant of science, policy, and technology majors. You can also design a major if you want to approach a topic from a variety of disciplines.