Research Spotlight: Ralph Gomez and Differential Geometry

Editor’s note: This article was initially published in The Daily Gazette, Swarthmore’s online, daily newspaper founded in Fall 1996. As of Fall 2018, the DG has merged with The Phoenix. See the about page to read more about the DG.

This is the third interview in the series Research Spotlight, in which I share conversations that I have with faculty regarding their research, their journey within their field, and their field in a broader context.

Ralph Gomez is an Associate Professor of Mathematics and Statistics. His research focuses on differential geometry.

10/27/2017

AIDAN REDDY: What, in general, does math research actually look like? Is it people writing up new equations on a chalkboard? Are there computers involved?

RALPH GOMEZ: Math research can mean different things to different people, but by and large, when we talk about research in mathematics, we really mean the creation of new mathematical ideas. This could mean, for example, proving a mathematical statement — that is to say, coming up with a logical argument for a mathematical statement — or it could mean coming up with a useful mathematical definition, or it could simply mean performing a whole bunch of calculations so a new kind of pattern can manifest itself to make general conjectures. For example, suppose I give you the even number 8. Do you agree that the even number 8 can be written as the sum of two prime numbers?

REDDY: Yes, 3 and 5.

GOMEZ: What about the even number 10?

REDDY: 5 and 5.

GOMEZ: What about the number 14?

REDDY: 7 and 7.

GOMEZ: Here’s a claim: every even number can be written as the sum of two prime numbers. Nobody knows the proof. There’s a proof for certain cases, but it’s not known in complete generality if such a statement holds true. This is called the Goldbach Conjecture.

In our little discussion a moment ago, we looked at some examples but now you can ask, “What is the general proof of this?” Or is there a counterexample? It’s not enough to check the conjecture for the first hundred numbers: that’s not a mathematical proof. If I do it for the first hundred even numbers, what about the next hundred even numbers? That’s not enough. Can you come up with a logical argument to prove the conjecture is true?

That’s a small example of conducting some mathematical research

Now, the use of computers is definitely a tool that can be used to illuminate certain ideas. For example, I have used a computer program to do certain calculations which were too long to do by hand. However a computer certainly isn’t a necessary tool to do mathematical research in the areas in which I am currently interested. Most of the day-to-day operations can be just playing around with an idea, doing some calculations, seeing if those calculations reveal anything, passing the idea off to a colleague to see if the idea sounds outrageous or not.

REDDY: I’ve read that your main area of research is differential geometry. What is differential geometry?

GOMEZ: Differential geometry, roughly speaking, is the mathematical apparatus which uses vast generalizations of calculus to better understand the shape, size and curvature of a space in any dimension of interest. One of the most fruitful things that comes out of differential geometry is the notion of the curvature of a space. What do I mean by the curvature of a space? You agree the surface of a basketball will be a different kind of curvature than a sheet of paper. How do you mathematically talk about the curvature of a space? It was really Gauss first who figured out the correct mathematical approach to talk about the curvature of a two-dimensional surface.

It turns out that you can talk about the curvature of a space from a purely intrinsic point of view, ie without reference to an ambient space. This was one of Gauss’s many great breakthroughs a few centuries ago. Subsequently, Bernhard Riemann generalized Gauss’ curvature of a surface to spaces of higher dimensions. So, you can actually talk about the curvature of a higher dimensional space, which is incredibly useful.

As you may suspect, these mathematical things that I’m trying to discuss are absolutely crucial from a physical point of view. If you think about the great work that Einstein did, one of the things that he needed was a mathematical framework which described the curvature of four-dimensional spacetime, because gravitation manifests itself as curvature of space and time. What is the mathematics that you need to talk about the curvature of a four-dimensional space? This is the language of differential geometry.

REDDY: What research projects are you working on right now, and what questions in differential geometry are you trying to answer?

GOMEZ: One of the main reasons why I like differential geometry is because of its deep relationship to theoretical physics. There’s a large portion of theoretical physics that can be viewed as very differential geometric. I’ve always been very attracted not just to the beauty of the physical universe, but the relationship between mathematical structures and the physical universe.. By extension, I am attracted to mathematical problems that may have a relationship to the world of theoretical physics.

One of the things I’ve recently gotten interested in is so-called “generalized geometry”. Together with my colleague Janet Talvacchia, who is also a differential geometer here at Swarthmore, we’ve gotten really interested in generalized geometry for a variety of reasons. One of the reasons is the prominent role generalized geometry has taken in formulating various mathematical issues in string theory. Generalized geometry is a mathematical framework discovered by Nigel Hitchin, in I think 2002 or 2003 and continued by some of his students. What can generalized geometry do?

Imagine we lived in a world in which the only interesting thing was the line y=5x. Suppose someone was doing mathematical research on the line y=5x. Maybe one breakthrough was that they found out the slope of this line was 5, maybe another breakthrough was that it makes a certain angle with the x-axis, and then maybe a couple of decades later someone is able to study the line y=-x. They find out that the slope of that line is -1, and that it makes a certain angle with the x-axis, and so on. So here you have someone studying the line y=5x, and someone else studying the line y=-x. It would be interesting if someone came along and said, “Hey, let’s consider the line y=mx+b.” That takes into account not only the line that you had been studying, y=5x, but also the other line y=-x. Those are all manifestations of the single enterprise y=mx+b. That’s kind of what generalized geometry has done. It says, “There are many geometric structures out there, but they’re all manifestations of the same underlying geometric framework, namely generalized geometric things.” That’s one of the exciting things about generalized geometry; it can incorporate many existing geometric structures as manifestations of a single unifying geometric framework.

Theoretical physicists working on string-theoretic problems have actually discovered some very fascinating properties occurring in generalized geometry that the mathematicians have not picked up on. What Janet and I are trying to do is look at those string-theoretic formulations and see how they make sense from a purely mathematical point of view and moreover see how they reveal new geometric structures.

REDDY: How did you develop an interest in math, and how did you end up doing the particular type of work you do?

GOMEZ: It has definitely been a windy road. The standard thing that people think is since I’m a mathematician, I must have been good at mathematics from the very beginning, but that’s definitely not true with me. In fact, I didn’t even plan on going to college. No one in my family went to college. My oldest brother was in and out of jail and my other brother made a collection of bad decisions which prevented him from leading a responsible life as well. What is more, I lived in a neighborhood where gangs were prominent. The idea of going to college wasn’t something I really thought about, and it was not talked about at home either. It was a concept you only saw on TV. Although I did take college prep classes in high school, I didn’t imagine myself going to college. In fact, I think in my sophomore year of high school, I actually thought about joining the gang that was living in my neighborhood. The main gang that lived in my neighborhood was called the XIVers (Fourteeners). At the time I actually wanted to be in their gang, but there were a couple of reasons why I decided not to take that route.

The first reason was skateboarding. Skateboarding actually deterred me from wanting to become a gangster. The second is this: I remember when I was 8 or 9 going with my mom and sister every other Saturday to the county jail to visit my oldest brother who was incarcerated there. One of the things that made a profound impression on me was seeing the severe disappointment in my mom’s face from having to see her son in jail. I remember thinking to myself, “Because the poor choices made by my brothers cause immense heartache for my mom, I should therefore do the exact opposite of what my brothers do.” I at least knew not to be a “bad” kid; that is, I shouldn’t break the law, I should stay out of jail.

Being immersed in the sport of skateboarding and having close friends who also enjoyed skateboarding kept me out of trouble. I continued to do well in high school, but nothing special. I went up to precalculus in my high school and didn’t understand what was going on there. I could do some of the calculations but I had no interest in mathematics. Finally, during my senior year, I was getting worried that maybe I should think about doing something after I graduate, so I enrolled in the local community college. I couldn’t go to the more fancy community college, which was maybe 30 minutes away from my house, because my family couldn’t afford the gas. So, I had to go to the small community college a couple of blocks from my house, which allowed me to walk to and fro. I enrolled in classes there, and guess what the first math course I took there was? It wasn’t precalculus; it was even lower than that — trigonometry. My first math course at the college level was trigonometry! It was there that I started to take mathematics a little bit more seriously. With the problems in trigonometry, I started to be more methodical, to be more clear in how I wrote up solutions, and I started to enjoy it a little bit.

Once I started learning about the work of Einstein, I got really attracted to physics. I started thinking, “I want to learn these ideas from physics.” I quickly realized, “If I want to learn these things in physics, I better be good at mathematics.” So, it’s actually the pathway of physics that lead me to wanting to learn more mathematics.

When I transferred to UC-Santa Cruz, I actually transferred not as a math major, but as a biochem major because I wanted to eventually do a PhD-MD program. As I took the courses in biology, chemistry and physical chemistry, I was really missing the purely theoretical aspects of mathematics. I realized I could switch my major to mathematics and it would make me much happier, so I switched to the math major. Then I realized, “If I’m a math major, then I can actually be a mathematician and be a professor, because I like thinking about these things and I also like to explain things, so why don’t I give that a try?”

I earned my degree at UC-Santa Cruz and then I got my Masters in Applied Math at UC-Santa Cruz after taking a year off. After that, I wasn’t sure if I was smart enough or capable enough to get a PhD in mathematics. Thus, I took a year off and decided to teach at the community college where I started out. I wanted to thank that institution because, if that community college didn’t exist, I would not have gone to college at all. I then went back to that institution and taught college algebra courses, really elementary math courses, and it was extremely enlightening. In that year, I thought about whether or not I was capable enough to get a PhD in mathematics. By discussing things with friends and really thinking about what I was interested in, I decided to give it a try.

The first semester I arrived at the University of New Mexico to start the PhD program, my dad passed away. Before he passed away, he made me give him a deathbed promise that I would not let his death interrupt my schooling. I was about to drop out of the PhD program and head back home to help out with my family because of various things that happened as a result of his death. But, I made the promise and stayed in the PhD program and picked up the PhD after five and a half years. One of the crucial things that happened while studying for my PhD is that my advisor, Charles P. Boyer, showed me that he believed I was capable of being a mathematician by giving me a research grant. The fact that a professional and successful mathematician believed in me and wanted to see me succeed really meant the world to me. That really was a turning point. I really immersed myself in mathematics like I had never done before after that.

REDDY: What would you say to someone else who, like you used to, believes that math isn’t for them?

GOMEZ: I don’t believe that. I don’t believe that mathematics isn’t for anyone. Just like I think that everyone can appreciate music that sounds beautiful, I think it’s the same thing with mathematics. Absolutely everyone is capable of learning, appreciating, and understanding the beauty of mathematics, but you have to be willing to invest time into it.

Some people have these brick walls that they have developed as a result of mathematical anxiety. In my opinion, that is one of the main hindrances to being able to mathematically understand something. If one can only somehow destroy that wall that creates anxiety, only then will the many splendors of mathematics reveal themselves.

One of the main reasons why I am convinced of this is the year that I taught at the community college. I taught a really remedial course in mathematics, college algebra or something. The bulk of the audience was forty, fifty, and sixty year-old adults who were coming back to learn basic mathematics because of the following: their children or grandchildren would ask, “Could you help me with my math?”, and then the adult would have to say, “Sorry, but I don’t understand that stuff.” The heartbreak in telling their child or grandchild that they can’t help them was enough to make them want to learn the mathematics. When I started teaching these adults these mathematical things, I immediately detected these massive mathematical anxiety walls. The bulk of my work was not in teaching them the mathematics. The bulk of my effort was actually in destroying the mathematical anxiety walls so that the mathematics could freely flow. Once that wall is broken, people can more easily absorb mathematics.

I really think anyone can understand mathematics and appreciate the beauty of it, but one must ask oneself, “Do I want to invest the time, and do I want to figure out how to destroy those mathematical anxiety walls?” A lot of the time, it is true; you’re going to be frustrated with the mathematics. Are you willing to accept that frustration and try to navigate it? Or, are you going to give up? Ultimately, it depends on your choice in the matter.

Featured image courtesy of commons.wikimedia.org