# Mathematical Gemstones: A Tour

is a blog created by Dr. Maria Gillespie (Colorado State University) whose research interest lies in combinatorics, with applications to Algebraic Geometry and Representation Theory. One of the aspects I like most about the blog is the fact that the posts are organized by level of difficulty using gemstones (i.e. Amber, Pearl, Opal, Saphire, and Diamond) as indicators.  It was really interesting to see how the writing and explanations varied according to the levels and their intended audience.

I was curious to know more about the inspiration behind the ‘Mathematical Gemstones’ blog so, in this tour, I hope to give you a glimpse of two of my favorite posts and insights from Dr. Maria Gillepsie herself!

(Sapphire)

In this post, Gillespie discusses a solution to the following voting problem:

Suppose two candidates, A and B, are running for local office. There are 100 voters in the town, 50 of whom plan to vote for candidate A and 50 of whom plan to vote for candidate B. The 100 voters line up in a random order at the voting booth and cast their ballots one at a time, and the votes are counted real-time as they come in with the tally displayed for all to see. What is the probability that B is never ahead of A in the tally?

She discusses first how to enumerate all ballots as sequences and then finds the solution using the crystal functor (F1)  to count the ballot words by counting the chains of the F1 graph. I found the diagrams in the post very insightful, especially for those new to the topic but familiar with combinatorics.

Another fun post was the idea of constructing Pythagorean triples (i.e. a triple of positive integers (a,b,c) with a²+b²=c²) on a sphere! By parametrizing all triples via geometric means, one can show that (r²−s²,2rs,r²+s²) is a Pythagorean triple on the unit circle for any integers r and s.   However, questions about finding Pythagorean triple on the unit sphere remain open, and she shares,

“There is some hope, however. In this paper by Hartshorne and Van Luijk, the authors show that there are infinitely many Pythagorean triples in the hyperbolic plane, using the Poincare Disk model and some nice hyperbolic trig formulas combined with some Eulerian number theory tricks. So Pythagorean triples are not the sole property of flat Euclidean space.”

Maria Gillespie:  I’m an assistant professor at Colorado State University, currently working towards tenure. I started my blog when I was a graduate student at UC Berkeley.  At the time, there was just so much new mathematics that I was trying to learn, and I know I absorb things best when I explain things to others.  So it started out as a way for me to learn new concepts in graduate school, by typing them out to an online audience. At the same time, I’m also passionate about bringing the joy of mathematics to everyone around me – sometimes I learn or remember a fascinating nugget of truth in mathematics, and I just want to yell it to the universe and share that joy with as many people as I can.  So I decided to put together a blog that could accomplish both of these goals at once.The way I accomplished this was to make each post a self-contained “mathematical gemstone” – a shining example of mathematical beauty and truth.

Some gemstones could be “harder” than others, perhaps assuming a higher level of mathematical background for the intended audience.  In order to help the reader determine which posts would be appropriate for them to read, I sort posts into five levels of “gemstone hardness” according to actual measures of how hard real gemstones are, starting from Amber (one of the softest gemstones) and going all the way up to Diamond (a very hard stone).  Here is a description of the levels, from my About This Blog page:

• Amber: This category contains posts that anyone with very basic elementary or middle school mathematics background can appreciate.
• Pearl: For Pearl posts, some high school courses or early college courses may be helpful in understanding the content.
• Opal: These posts are aimed at undergraduates with some basic first-course background knowledge in algebra, analysis, discrete math, or topology.
• Sapphire: These gemstones would be appreciated by mathematics graduate students or professors, or those with at least an undergraduate degree in mathematics.
• Diamond: The hardest type of gemstone. These posts are highly specialized, containing content that only mathematicians who have studied the topic in depth will have the background to understand.

I later added a “Miscellaneous” category that allows me to share thoughts or discoveries that are not strictly mathematical, but which are related to mathematics as a discipline.  This category includes things like LaTeX tips, book recommendations, thoughts on social issues in mathematics, and most recently, doing math in a pandemic.  To summarize, my blog serves as an organized platform for me to share ideas and thoughts that could be enjoyable or helpful to other mathematicians and scientists, to students, and to the general population.

VRQ:  What is the most interesting thing you’ve learned through blogging?

Maria Gillespie: People can get REALLY upset when you try to suggest new ways of doing or explaining elementary mathematics.  I have a post on the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), which is often used as a tool in middle and high schools to teach the order of operations.  In the post, I explain why I feel the acronym can be confusing to students, and the incorrect answers you can get by interpreting it too literally. That post really must have hit a nerve among some readers for some reason, because I have never been at the receiving end of so much hatred and vitriol in the comments!  It just goes to show that even mathematics can become controversial if it’s out there on the internet.

VRQ:  What are some of your favorite blog posts you’ve written?

Maria Gillespie: I’d have to say the most fun one I’ve ever written was Can you prove it … combinatorially?in which I prove Binet’s formula for the Fibonacci numbers directly using a combinatorial proof, without any induction or generating functions (the usual methods).  It was just so satisfying and fun to find a combinatorial proof of a formula that involves the square root of 5, as intricate as such a proof may be.My favorite among the “soft topics” is the recent series of four posts I wrote on doing math in the pandemic, starting with (it links to the others there).  Writing all of that out really made it hit home how very much we’ve all learned from being forced into a virtual setting this past year.

While the pandemic has been a horrible natural disaster, the silver lining I see is that there’s a lot of opportunity for some of these tools to still be used after the pandemic is under control, and I’m excited to see where things go in the future.Finally, the posts that have turned out to be the most useful are the ones I wrote early in graduate school on the basics of my research area, in symmetric function theory and Schubert calculus.  I’ve looked back at them countless times to remember formulas, and other grad students in my area have appreciated them as a resource as well.

VRQ:  What are some of your favorite math blogs, if any?

Maria Gillespie: I have found so many of Qiaochu Yuan’s posts on his blog Annoying Precision so useful.  They’re well-written, precise, and often have exactly the proof of some fact that I was trying to look up at the time.  I also like since I think he has really good ideas about open access and the future of mathematics publishing.

VRQ: Do you have advice for other mathematicians interested in creating their own blog?

Maria Gillespie: My main piece of advice is to just go for it!  Write about what you’re passionate about, or something you’re learning at the moment, and post it without too much worry about who will see it and whether it’s perfect.  Blogs are an excellent outlet for mostly-correct mathematics (or mathematics-related topics) that doesn’t have to be peer-reviewed but can add a lot of value to the world.  And more likely than not, your thoughts will be valuable to someone. As for the technical side, WordPress is your friend and it will take care of you.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@VRiveraQPhD)