My question is mainly for mathematics Ph.D. application in the United States. However, the answers from any other field (or other countries) where undergraduate/master research doesn't play a critical role in admission are also welcomed (by "critical" I mean if one has no research experience, then the chance of acceptance is small).
As we know, in mathematics or theoretical physics, a solid background in some areas (like algebraic geometry) need several years to lay down and some applicants may not have research experience in the area they are interested in.
Suppose the main reason that applicants want to apply to school A is that the math department of school A has a strong algebraic geometry group, but they have no research experience in it. How would they convince the graduate committee that they are truly interested in algebraic geometry in the statement of purpose?
Please note under some circumstances it is important to convince the graduate committee that one is really interested in a particular field. For instance, if they had changed interest from another field, say analysis to algebraic geometry or they are transferring from one school to another to pursue a Ph.D. degree in algebraic geometry, or the department to which they are applying has only a narrow range of research focuses (some elite private schools, for example).
To put in another way, for an applicant without research experience in a certain field, what kind of experiences/characteristics could make the graduate committee believe that they are really interested in this field?