First and foremost, if you are a graduate student just beginning at UTK, make sure you meet with faculty early (within the first year) and often regarding research with their group. One way of learning about a research group is to attend one of the group meetings (even if you have not yet passed the qualifying exam). Almost all faculty (including me) in the department open up their group meetings for interested graduate students. Also, consider working with the associated research group during either the summer before you begin your first year or during the summer between the first and second years.
If you want to join a particular research group, I recommend contacting the faculty member who leads that group one a semester (and once during the summer) expressing your interest. Make sure this expression of interest is not passive: read one of the research group papers and have questions, attend a group meeting, and/or meet in person with the faculty member who leads the group.
If you are reading this and you are a postdoc or graduate student and you want to join my research group, that is wonderful! For people interesting in joining my group as a postdoc, I announce job openings on twitter so you can always check that (even if you don't have a twitter account). If you are interesting in joining as a graduate student, here are a couple things that might help you:
First, understand that all of this advice comes from a particular perspective which may not be the same as other faculty.
Everyone says this, but it's worth saying again: understand your audience.
I'm less picky about looking at the audience versus your slices than I used to be. As you progress in your career, the most important thing here is to communicate that you're excited about the material that you're presenting.
Timeliness is important for graduate students because many of your first talks will be at larger meetings where session chairs are likely to be rather strict. I am of the belief that, if you haven't convinced me of your point in 20 minutes, an extra 2 minutes isn't really going to help your cause. It takes a certain skill to watch the clock and extemporaenously reframe your talk as necessary. In any case, you should always make sure there is some slide in your talk at which you plan to look at the clock and re-evaluate your talk. Then, you can present the following slides faster or slower accordingly. Also, I recommend always having a few extra slides at the end in case you're faster than you expect. Practicing your talk also helps. When I'm practicing, sometimes I like to actually type out what I intend to say. Some other method may work better for you.
Be careful about apologizing too much. Act as if you're busy and important and you carefully chose to spend exactly as much time and energy on the presentation as the situation warranted.
Be intentional about how you use mathematics, and don't apologize for it. Also, don't skip over mathematics claiming that you did so because it was too difficult. Many people (even those outside academica) do a lot of mathematics, and it's always disappointing to find people giving more weight to the common myth that math is particularly hard, or just too hard for some kinds people, or just too hard for them personally. Math requires practice, just like reading. That's it. Also, when you say you're skipping something because the mathematics is complicated, I typically interpret that as meaning that you didn't feel like taking the time to present the mathematics in a comprehensible manner.
In my experience, the nuclear physicists and astronomers tend to be completely different about mathematics. Astronomers feel no shame in writing an entire article without a single equation, yet the nuclear physicists don't feel like you've done any real work unless there's a very complicated equation up there some where. [I have received a lot of conflicting advice about this in the past.] Maybe my assessment is completely wrong, but either way you should make sure to understand what is important to your audience and act accordingly.