Reproduced below is an article I wrote to for the Science Café.

Examples of Tiling in the Alhambra from Patronato de la Alhambra y Generalife
Examples of Tiling in the Alhambra from Patronato de la Alhambra y Generalife

Religion, beauty, mathematics, and entertainment. Their intersection is not as empty as one might think. Those who have had the opportunity to play Azul, the 2018 winner of the Speil Des Jahres have, inadvertently, experienced such an intersection. This beautifully crafted game credits its inspiration to the Portuguese ceramic tiles, called azulejos,which ultimately drew their inspiration from the Alhambra palace nestled in the Andalusian city of Granada. But what was it that initially inspired the architects of the Alhambra to create patterns that would resonate with artists and game designers for centuries. And what does such an inspiration have to do with mathematics?

Andalusia is a region in southern Spain that was occupied by the Moors, or the Muslim inhabitants of the western Mediterranean, from the 8th century through the 13th century. After its capital, Córdoba, was captured by Christians, the leader of the last Muslim dynasty on the Iberian Peninsula agreed to concede tribute and even territory to the Christians on the condition that they leave his hometown of Granada alone. Construction of the Alhambra began in 1238 and was completed later in the 1300s as the Muslim dynasty exhaled its last breath on the Iberian Peninsula. Its décor is distinctly different from the Christian basilicas of the time. While the basilicas were adorned with golden statues of saints, crucifixion displays, and murals of Biblical tales, the Moors were strictly prohibited from depictions of Mohammed or Allah or any living creatures because it was seen as a form of idolatry. Without the expression of anthropomorphic representation, the Moors turned to represent the beauty of God in an abstract way.

The walls were adorned with intricate repeating patterns and tiling that held various visual symmetries. The fascinating result is that their numerous patterns and tessellations that are scaled across the walls of the palace contain rich mathematical properties, unique to the Alhambra, that have, in turn, been an inspiration to art. The ubiquitous tessellations of Dutch artist M.C. Escher were famously inspired by these wallpaper patterns.

But isn’t mathematics about numbers with operations like addition and subtraction, and not artistic patterns? Rest assured, anywhere you see patterns or structure, there is mathematics lurking. In mathematics, there is a very precise description of what a symmetry means. Loosely, given an object, say an infinite wall with repeating designs, if you can rearrange the object in some way (think translate, rotate, and reflect in the case of our infinite wall) that makes it indistinguishable from its original state, then that rearrangement constitutes a symmetry. The branch of mathematics that studies symmetry is known as group theory and the set of all operations that leave the object unchanged constitutes a group. While operations in arithmetic are things like addition and subtraction, operations in group theory are rearrangements that don’t alter the object. In our case, the three operations of translation, rotation, and reflection generate all rigid motions, that is, motions that preserve the distances between any two given points. In the plane, our infinite wall, if you have repeating patterns such as those depicted in the Alhambra or on the wallpaper in your living room, you can perform these rigid motions in certain ways such that, after performing these operations, the original state is left unaltered. With a few natural assumptions, it turns out that the mathematical structures that arise from these “wallpaper groups” can be classified into exactly 17 categories that encapsulates all possible rearrangements of the wall in terms of our rigid motions. There remains dispute as to exactly how many of these groups are present in the Alhambra, but the number ranges between 13 and 17, still a rich and unique accomplishment in architecture, especially given that it was produced well before group theory was invented.

            One can play this game in higher dimensions as well. Due to the fact that they have numerous applications to crystal structures in three dimensions, these groups are often known as crystallographic groups and have significant applications in chemistry. Although we inhabit a three-dimensional world, however, our visual interests are peaked by the two-dimensional wallpaper groups. This is a result of the fact that we mentally construct our world based on a two-dimensional projection of our three-dimensional world. In three-dimensions, the number of categories jumps to 230. The number continues to balloon as the dimensions increase. Since our cognitive faculties only allow for visual representations of objects up to three dimensions, the beautiful symmetries involved in higher dimensions can only be “seen” through mathematical analysis. Given the rate of growth of the number of distinct crystallographic groups, the precise size for dimension six and up is still unknown. With recent advances in computational power and continuing mathematical research, maybe we can come to know more about these crystallographic symmetries in higher dimensions and thus, slightly enhance our view of the beauty of God.


Reproduced below is a letter I wrote to the editor of the AMS Notices, published in the August 2020 edition.

The arrival of COVID-19 has prompted many members of our community to reset and reflect. During this time, I would like to make the urgent request that we learn from this seismic shift and commit to implementing positive changes to our current mode of operation.

As I fall into the Early Career/Graduate Student group, I am particularly mindful of the challenges those of us in this group face and the importance of each decision we make at this fragile point of our career. A budding career, more often than not, benefits from having a mathematical community, access to the current exchange of ideas, and the resources to bring these two together. To be candid, attending conferences is expensive. For those in more demanding circumstances, such as caretaking for others or having limited financial resources, even participating in nearby workshops may prove burdensome. One often sees advice about the importance of attending conferences, networking with others in the field, and participating in the market of perspectives. These indispensable interactions should not be left to the caprice of one’s financial standing, geographical location, or as is often expected, the mercy of one’s institution or advisor to demonstrate willingness to assist or provide the necessary resources.

Secondly, the negative environmental impact of frequent academic gatherings has gained increasing attention in recent years. The term social trap refers to a situation in which, based on short-term gains, potentially lethal long-term harm is inflicted. Even if one is aware of the calamity ahead, one must participate in these events or face massive concessions to the trajectory of one’s career. There have been efforts by some, seeing which way the wind is blowing, to try their hand at implementing incremental changes. Such efforts have included eco-friendly double conferences or, with the current state of affairs, virtual meetings. But these efforts have not gained the necessary traction to address the problem in the long term.

In one fell swoop, virtual meetings offer an opportunity to level the playing field and address the social trap of environmental catastrophe. Local seminars, collegial meetings, and the like, that have continued in this crisis, have done so through the technological miracles of platforms such as Zoom and WebEx. Circumstances, bleak as they are, have forced our hand. We have been required to adapt to these technologies. Upon exiting this chapter of history, why not perpetuate this adaptation and, as far as is conceivably possible, convert all future conferences and gatherings to such virtual formats?

Whether we implement these changes now or in the future, the arc of time will mandate that we update the way we operate.


In the end, we will remember not the words of our enemies, but the silence of our friends” Martin Luther King Jr.

Quite a few years ago I had visited New Orleans, LA where I happened upon a rather large and imposing statue of the American Confederate general Robert E. Lee. I specifically recall my middle school textbooks telling me that, although he was a general for the confederates, General Lee was an honorable and respectable man, even though he fought for the South. I was taught that the American Civil War was fought for an array of nuanced reasons, one of many being slavery. General Lee, said to be a good man, just happened to be on the losing side. In fact, many people are named after him. There is a mathematical teaching method of Socratic inquiry known as the Moore Method, named after Robert Lee Moore who, in turn, was named after General Robert E. Lee.

As I stood in New Orleans, next to this statue of Robert E. Lee, I noticed the majority of people around me were African American. I wondered what they thought about that statue and this strange juxtaposition they had to live with each day. After all, the true history of Robert E. Lee is not so kind to his beliefs. The American Civil War was about slavery and Lee was firmly on the front lines of the wrong side of history. Yet his legacy seems to live on, in part through the naming of children in a way that generals of the North, such as Ulysses S. Grant, do not. I’m sure it, in part, has to do with the aesthetics of the name Robert Lee relative to Ulysses. Even conceding this point, it doesn’t detract from the discomfort of being named after someone with a legacy such as Lee.

These issues still resound today. The recent protests in reaction to the murder of George Floyd has prompted the downfall of many confederate statues. This momentum has even led to the renaming of an American football team (formerly known as the Washington Redskins). In the mathematical community, we give honor by naming theorems, conjectures, and so forth after mathematicians of history. We saw that there is a popular pedagogical strategy known as the Moore method. It is an shameful reality that Robert Moore refused to teach African American students. Even in my narrow field of study, the study of complex maps of the unit disk into itself, the problem is pervasive. One important topic of study is known as the Nevanlinna-Pick interpolation problem which asks the following question: for a set of n  initial points in the disk given by \{z_i\}^n_1  and n  target points in the disk given by \{w_i\}_1^n  , when is there a holomorphic function \phi  from the disk into itself such that \phi(z_i)=w_i  for all 1 \leq i \leq n  ?

This problem is named after Rolf Nevanlinna and Georg Pick. Nevanlinna was a documented Nazi sympathizer while Pick, a Jew, died in a concentration camp. At the time Pick died in 1942, Nevanlinna, a Finnish citizen, was chair of a committee to improve relations with Nazi commanders and Finnish volunteers fighting for Germany (see, e.g., The Scholar and the State by A. Soifer). Turning to another example, there is an important and interesting class of these self maps of the disk, called inner functions, which can be thought of as maps that send the boundary of the disk onto the boundary. The reason for the name “inner” is unclear but everyone has gotten used to calling them by this name. There is an important subset of inner functions, called Blaschke products, that are named after Wilhelm Blaschke, an Austrian mathematician who described himself as “a Nazi at heart” and signed a vow of allegiance to Hitler. It’s not clear to me why we can’t just call them something like “boundary products” instead. After all, it makes more sense than inner functions. There is also the famous Bieberbach conjecture (After being proven by de Branges, it is now called the de Branges theorem), named after Ludwig Bieberbach, an enthusiastic Nazi who held firm the view that the German race was superior. All of these reside in my narrow field of study.

I have heard the argument that the political opinions of these mathematicians are irrelevant to the mathematics they did. And this is true enough. Our books should continue to acknowledge the scientific contributions of historical figures, independent of their beliefs. We should not rewrite history. Nevanlinna should be credited with his work on the problem. Blaschke should be credited with introducing the class of functions currently known as Blaschke products. But this does not mean we should do the honor of naming these things after them, so that we as mathematicians must repeat their names every time we refer to the problem.

One rather common response is that fault can be found with any historical figure and renaming one theorem, taking down one statue, will bring the whole lot down. Anyone who has taken any sort of critical thinking class will immediately identify this line of reasoning as a prime example of the logical fallacy known as the slippery slope argument. After all, this could be applied to anything. Freedom of speech is important but you cannot run into a theater and shout “fire!”. Like everything else, we should start with the cases that are clear, those who, in great excess to their peers, were racist, sexist, and so forth. We then proceed to debate the intermediate cases just like we do in every other outlet of human discourse.

We ought to take down racist statues and rename the things that are named after people who went out their way to support ideologies that were actively oppressive to different demographics, even by the standards of their peers. The sooner this step is taken, the sooner we put an end to an erroneous practice. Some steps have been taken. The statue of Robert E. Lee was taken down in 2017. The prestigious Nevanlinna prize was renamed as the IMU Abacus Medal in 2018, although the IMU failed to specify why the change was made. We are moving in the right direction, but we can move faster. As part of the mathematical community, I do not want us to be seen as a community that dragged our feet on progress or took our time to remove barriers that are insulting to so many, including members of our own community.