Hi,
I am in the first week of my high school senior year. I am planning to major in Math/CS (Math is my primary major with CS as second major or minor). I completed all the math courses available in the high school by end of junior year: AP Calc BC and AP stats are highest classes in the school. My original plan was to take Multi Variable Calc (MVC) thru a local college for the fall semester and registered in it - this will satisfy my math course requirement - this is also a popular senior year choice for the previous math students from our school in the same situation. However, I got an opportunity to work with a math professor - I have been working with this prof over summer on a research project and she just offered to continue to mentor thru school year with weekly twice meetings. She kind of indicated that, in order to continue this there should be a formal acceptance by my high school GC/principal. I talked to GC - my high school is fine with it as long as I drop the MVC class. Our school runs on a block schedule and I need to have an empty block to take either of them officially and I can have only one empty block. So I can only do either MVC or math research - what do you think is most effective from purely college admissions point of view.
More details: research topic is a bit of complex one involving linear algebra; its very interesting and am loving it but my progress is slow (learning lots of new concepts and doing math modeling which is new to me as well). So, I am not expecting any paper by admission time - but there will be some outcome later (prof suggested that I should target to present results at an undergrad research conf in their university in April). On the other hand, MVC is a straight college course - if I continue this, will be done by end of Dec. So I will have time to update the universities with a grade from a college course. Dilemma is this: research with a well known univ but no concrete output to show until very late in admission cycle; or just do an advanced math course and submit its grade (still late but at least its before final decisions are made). The typical advice I am getting is to do the one I love - problem is I love them both 
Research is slightly more appealing to me but I am worried the universities may see it as an empty claim - doing research without any paper and they may even think that I didn’t do any math course or work in the senior year. Any thoughts / advice ? (last day to dap MVC class without financial penalty is next week). TIA.
PS: the professor works at one of my targeted universities (a well known ivy) - its a reach for me and my chances of getting in are very low. Not sure if it should factor into my decision.