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Perhaps you came to Davis with a specific adviser in mind. It's entirely reasonable to reach out to that person now, even if you don't have time for a research project yet. They might have a working group you could attend, or they might be interested in meeting and talking anyway. Getting to know them, their research interests, and their research style is still useful.
Perhaps you came to Davis without a specific adviser in mind. That's common and fine. Here are some ideas on finding one.
Students in the applied math program (GGAM) are usually advised by GGAM faculty. Students in the math program (GMAT) are usually advised by faculty members of the math department. There's overlap between these two groups. Here are some relevant directories:
Many faculty maintain departmental websites. Many have little research bios on their directory page. Here are some other sources of information:
Sometimes it's good to talk to other people. Here are some ways of finding a faculty members other students, current and past:
Here is what the graduate council has to say about the advisee/advisor relationship: Mentoring Guidelines
To see how to prepare for the Qual, go to The Qualifying Exam tab.
The timeline described here can be too late for some people. To find another version of the timeline, go to Wish I'd Known... tab.