I'll take a stab at it, but this is just a general, hypothetical, answer since every university in the US will have its own procedures and even different departments in the same university will. But, for a variety of reasons, attention is taken to fairness in analyzing candidates, partly because of law and partly to avoid scandal. Note that this isn't restricted to math alone.
Initially, the work is likely to be clerical. Some applications can be rejected without much analysis for having missing parts or (some places) falling below a GPA expectation. I'd guess that such clerical decisions are only made when values are very bad, so that missing GPA by a few hundredths probably doesn't result in a clerical rejection, but 0.5 might. Such decisions might just be initial sorting ("likely reject") and serve as advice to the committee rather than being final.
In the clerical period there might or might not be room to try to get missing parts (letters...) into the student's file, making the deadline a bit soft.
Now there is a set of "viable" candidates that must be considered by the committee. The committee membership is a matter of "service", one of the expectations of faculty and fairly often newly tenured faculty are "rewarded" with committee membership (ha). But people can volunteer for the committee as well and it likely has a mix of junior and senior faculty.
Whether advice can be sought outside the committee or not is a matter of local rules. Can the committee ask a named professor to comment? Possibly, but it depends. I doubt that the fairness rules permit too much weight to be given when such consultations are allowed. But things vary as noted above.
In any case, the committee will schedule a series of meetings. It will have a target for the number of admissions and that is also likely a bit soft.
The process is one of winnowing. First you make three piles. The obvious accepts, the obvious rejects, and the "further consideration" group, which is probably fairly large. Individuals on the committee might object to the placement of one or more candidates and will keep their own lists of people who they want to revisit later if possible. This winnowing may take one or more meetings, but the meetings are likely long and people are busy with other things so it may take more than one meeting and the meetings may be spread out.
Subsequent meetings revisit (mostly) the "further consideration" applicants and people comment and files may be moved to other piles. I would guess (weasel word) that in many, perhaps most, such committees it is a matter of consensus rather than formal votes that cause a file to be moved. But in large committees that may not be possible and voting is used.
As the deadline to announce acceptances approaches the harder cases get considered and more attention is paid to the allowed number of acceptances and, perhaps, finances for student awards (TA and such). A new pile will also start to grow, which is "alternates"; people who might be offered a slot if folks in the accept pile don't accept, some of which is certain. The committee will have some historical information about how many such there are likely to be.
At the final meeting there may be a call for further comments on individual cases (or not) and, by consensus, probably, the committee declares its work done.
Note that the work is hard. Every applicant is different in some way. There isn't an obvious way to choose between some pairs of candidates so much of the work is guesswork. Predictions need to be made about the likely success of candidates in the program. In the modern age some balance is usually sought for different groups of people so that the marginal decisions don't, for example, always favor males or young people, or a particular ethnic group. This later can become very contentious, outside academia at least.
In the above, note all the "weasel words" such as "likely", etc. Things vary. Some places will be more rigid, some more flexible, but care will be taken to avoid conflicts of interest and unfairness. And note that at any time, a member can raise issues of the distribution, which can lead to a discussion and perhaps a rebalancing of the weight given to various things.
I also know of one program that differs radically from the above, but it isn't math, it is very small, and it isn't "top-level". There, the director of graduate studies is, alone, responsible for accepting students but with advice and consultation with faculty willing to advise students. There is no committee at all. The distribution of applicants to that program is broad enough that the fairness question doesn't arise.
