Marissa Loving: A very brief explanation

Aloha!  My name is Marissa Loving and I’m incoming mathematics PhD student here at UIUC.  I hail from Hawaii where I was born, raised, and attended college.  I am a bit out of my comfort zone here in the Midwest, but so far it has been smooth sailing.  Perhaps I should use a different expression now that I'm surrounded by fields of crops rather than the Pacific Ocean.  I’m quite excited to be part of SPI since this is the first research program I've participated in that wasn’t composed purely of STEM, if not solely math, students.  I love my math, but it’s wonderful to get fresh perspectives from people in English, psychology, or library sciences, just to name a few.

I first became interested in math research during my freshmen year of college while taking an introductory topology course.  Topology was my first proof-based math course and I enjoyed the flexibility that proofs gave me in constructing solutions to problems compared to the fairly singular method of problem solving employed in algebra, trigonometry, or calculus.  The other thing I really enjoyed about topology was the way we formalized an everyday concept, the distance between two points, to create an abstract structure, a metric space.  We then took another step into abstraction by considering how we could maintain the concept of distance or “close together-ness” that was available to us in a metric space, but without the use of a metric.  Ultimately, this led us to the idea of a topological space.  I apologize if it seems as though I’m turning this post into a math lecture, but the process I'm attempting to illustrate, that of taking a very mundane concept and abstracting it into an extremely general structure that can then be studied and understood, is what really drew my attention to the possibility of researching mathematics and led me to my first real research experience during an undergraduate summer program at the California Institute of Technology.  My project was in the area of spectral geometry, which stems from the intersection of mathematics and theoretical physics, and it really fascinated me.  
It was a combination of this experience and my enjoyment of my courses in analysis and topology that have fueled a growing interest in geometric analysis and topology and I am hoping to do my doctoral research in one of these areas.  However, as an undergraduate I only had limited or, in some instances, no exposure to many of the diverse research areas present in mathematics.  As a result, it is quite possible that there are other fields of mathematics that I would enjoy even more than geometry or topology.  I decided to use this summer as an opportunity to broaden my horizons by participating in a research project in the area of Mathematical Logic and its application to other fields of mathematics such as fractal geometry.  My research advisor is currently out of town so I can’t speak to the specifics of my project just yet, although I’m looking forward to sharing them with you later in the summer.