I think I can help. I am an undergraduate at UCI. I've taken a lot of grad courses and know a lot of professors. Let me see if I can break down the quality of the departments by subject:
Logic: Very good logic program. We have Matt Foreman whose is very highly ranked. Martin Zeman is also a great logic professor. If you want to do descriptive set theory or model theory, this might be a good option for you.
algebra/number theory: There are only 4 professors working in this area. Professor Baranovsky (algebraic geometry) has a lot of students already, so I'm not sure if he is willing to accept any more. The other three number theorists are not accepting students at this time, but that doesn't mean that they won't change that for you . We have some very talented number theorists. Professor Rubin and Professor Silverberg both graduated from Princeton and have A LOT of accomplishments. Professor Wan solved a very substantial problem of Dwork's and he is very highly ranked.
Differential Geometry: I don't know of any graduate students working in this area. We have a really good faculty (Professors Lee, Palais, Terng, Lu), however, I don't think they are actively accepting graduate students. If any were accepting students, it would be professor Lu. They have had multiple postdocs since I've been here. You should email for best results.
Analysis: I know we have some really talented analysts like Yeh, but I'm not too familiar with the department.
Dynamical Systems: I know of at least one professor (Anton Gorodetski) who works in this area. He is very talented. He also did his postdoc at Caltech. Matt Foreman and Donald Saari also work in this department.
Mathematical Physics: We have Abel Klein. He has done a lot of amazing work with spectral theory and functional analysis in Quantum Mechanics. He graduated from MIT. We also have at least one other professor in this area but I don't remember her name.
Applied Math: The applied math at Irvine is generally very good especially if you are interested in mathematical biology. If you are interested in this area I would suggest working with Komarova. She is very approachable and very talented. There are a lot of professors that do differential equations. There is also at least one professor who does image processing. Because we have such a great chemistry and biology departments, I know there is a lot of work that is interdisciplinary. Check this link for more information: http://www.math.uci.edu/research/applied.math.php
Topology: I think that Stern is the only real (as in actual) topologist. I don't think he is around very much, though.
Exams:
The exams at UCI are a little odd. When you enter you have to take two advisory exams in algebra and basic real analysis. You can find copies of past exams on UCI's website. If you don't pass algebra, you are forced to take a beginning algebra course. Same with the analysis exam. If you do pass them, or if you pass the courses you are forced to take, you must take the comprehensive exams in analysis and algebra. These are generally a little harder than the advisory exams. Once you pass the comprehensive exams, you must take the qualifying exams. The qualifying exams are usually much harder than the comprehensive exams. I've been told the exam situation at UCI is very stressful.
Sorry for the inevitable bad grammar throughout this post. I've written this up in a hurry. If there is anything else you would like to know, let me know.
