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The Interview:

Karen: Tell me about your research. 

Daley: I’m currently working under the direction of Profs. James Waterman and Yongquan Zhang in an area known as complex analysis, investigating the properties of certain classes of complex functions. Our problem concerned figuring out what can be said of arbitrarily high-order derivatives for some given infinitely-differentiable complex function satisfying certain conditions, or in other words, determining the behavior of the infinite family of derivatives for holomorphic or meromorphic functions. This research was carried out over the summer as part of the Summer Math REU, and forms the basis of my senior honors thesis that I am working on.

I also did a URECA project last summer with Prof. Evita Nestordi where I focused on probabilistic questions related to random generating sets and the diameters of their respective Cayley graphs for matrix groups over finite fields.

When doing mathematics research, are you working most of the time by yourself? Or are you doing collaborative work?

It’s actually a bit of both. For the Math Department’s REU program, I worked collaboratively with a small group of undergraduates on our project, but each of us had a slightly different focus on a different aspect of the project; for example, in my case, I mainly focused on value distribution theory. However, we explored a lot of topics together as a group and worked through problems together.

How have your mentors helped you to progress in your understanding in these complex topics?

One of the important roles research mentors or advisors do for you is to help you find a problem and then guide you on the right path. The most difficult part of doing math research actually can be trying to find an adequate problem that is both interesting and suitable to your abilities, and not too obscure. Honestly, it’s almost as difficult as the research itself to identify a good question.

And how has being involved in research enhanced your education overall?

Just the experience of doing research is very valuable. There are a number of math majors who are considering graduate school, so doing research as an undergraduate is a good way to figure out if that path is right for you. It's not the same as being in graduate school, certainly, but it gives you a taste of what research is like. 

Before I did research, I was convinced for a while that I might be better off just spending my summers reading through textbooks, learning more on my own. After connecting more with faculty and other students in my department involved with undergraduate research, this gave me the motivation to pursue research opportunities. And I now have a much better understanding once I got a taste of what actual research was like. It also gave me more confidence to pursue the path I’m on.

What are your current plans?

I’ve just finished applying to graduate school programs for a Ph.D. in math. Generally beyond that, I’d like to keep doing math.

What advice would you give to a freshman math major with regards to undergraduate research?

I've been running the Math Club for almost two years now, and due to my position, I often get asked for advice about undergraduate research. One of the things that I have put a lot of energy into is getting students more aware and involved in applying for opportunities. For example, every week I send out a list of research positions or internships that just opened up, and I have seen this effort benefit others. At one point, a student came to thank us for sending out an announcement for an REU program in the city that they ended up applying and being accepted to.So when I get asked for advice, I try to point to some resources and say: “Just apply to stuff! Find opportunities and explore what’s out there.” It’s important to keep an open mind.

Tell me about your involvement on campus.

I’m currently president of the Math Club, but it’s been a big part of my academic experience since I was a freshman. In general, we like to act as a sort of community for motivated students interested in mathematics, and also as a resource for academic opportunities. This could include announcements for research or fellowship opportunities, or going on conference field trips, and so on.

I also started a student chapter of the Association for Women in Mathematics at Stony Brook, called the Women in Mathematics Club. Very few undergraduate women elect to take the more advanced math courses even when they are qualified, which in turn can make them less  competitive for research programs, scholarships, and so forth. So the idea is to build some support from the inside and promote higher attainment of these things.

Did you come to Stony Brook knowing you wanted to be a math major?

Yes, but I would say that I didn’t really know what it entailed when I started out. The most advanced course I had taken before university was a calculus class, and so I applied to colleges as a math major. It wasn't until much later that I actually found I enjoyed math as much as I do, when I began self-studying real analysis in my freshman year. Once you’re doing advanced mathematics, it is more like doing puzzles or brain teasers as opposed to performing actual calculations.

What do you most like about doing math research?

It's all about trying to solve a particular problem and coming up with approaches to it. Sometimes, if you have a tough nut to crack, it can be a little exasperating, and you can get a little obsessive about it. But the drive this engenders is what I enjoy about mathematics.