Needless to say I failed to meet just about all of these goals – not just a handful of times, but pretty regularly. And yet I think I have succeeded in becoming a healthier and happier version of myself, just not in the ways I expected.

The biggest change in my emotional well-being came from really establishing friendships. I started making more time for friends and the activities that make me happy. I sometimes went to karaoke on a weeknight. I went to cookouts to take a break from endless homework on the weekends. Grad students are not robots. We need more than just fuel to be productive. Having fun times to look forward to (and reflect on) makes the hard work of being a grad student so much more bearable.

Part of making time for fun activities meant learning when to put my work away and say “no more.” A younger me hated the thought of handing in an incomplete homework assignment. Accepting the fact that I might not finish my homework actually made me much less anxious about working on it. It no longer seemed like a looming task that would never be done. A friend reminded me that the point wasn’t getting a good grade; it was understanding the material. I started to see value in struggling with problems, even if I couldn’t solve them before the deadline. When working on research, too, it can be very beneficial to step away from it when it starts to become frustrating; when you come back to it with fresh eyes you’re likely to better understand the problem at hand.

Time management seems to be one of the most challenging aspects of grad school for me. As math grad students, we often have the freedom to decide when and where to do our homework, grading and class preparations, and research. That freedom can make it difficult to fall into a regular self-enforced schedule, like getting to the office by 9 am every day – something I accomplished just a few times this semester. My failure to adhere to that schedule told me that perhaps it was not the best schedule for me. In fact, often I worked at home in the morning before heading to campus for my afternoon classes, enjoying the quiet solitude. I learned that I could be most productive by finding the times and places that made me actually want to work.

Lastly, I started exercising regularly this semester (something I had done as an undergrad but let slip when I got to grad school). While it can be difficult to make time for exercise, it has many benefits to offer – including improved sleep, a better mood, a break from academic work, and a sense of accomplishment. I got into new activities by signing up for phys ed classes and by exercising with friends, both of which made exercising more fun and held me accountable for actually working out.

I am by no means a perfect grad student (after all, nobody is). I have bad days and weeks when I’m really stressed out. I still procrastinate and wish I’d managed my time better. And toward the end of the semester I didn’t get many workouts in. But I’m learning that an important part of being a successful grad student is being a happy grad student, and a happy grad student I most certainly am.

]]>In his humorous 2015 Numberphile video, Matt Parker discusses a remarkable formula by Jeff Tupper of the University of Toronto whose graph is the letters, numbers, and symbols in the formula itself. More precisely, this formula:

To find out how it’s done, check out Parker’s video, plus this background explanation and generalization by Shreevatsa R. You can use Tupper’s formula to plot your own name (or anything else you like) using this Python code provided by Kaito Einstein.

(image courtesy of Weisstein, Eric W. “Tupper’s Self-Referential Formula.” From *MathWorld*–A Wolfram Web Resource. http://mathworld.wolfram.com/TuppersSelf-ReferentialFormula.html)

]]>

The start of summer is a great time to get organized: perhaps you finally have time to focus on something other than the next problem set, or perhaps you’re too burned out from finals to do anything else productive. Back in the good old days of undergrad, mathematics was a mysterious substance that emanated from textbooks, whiteboards, the mouths of my classmates, and occasionally, Wikipedia. Now that I’m in graduate school, more and more of what I learn comes from journal articles, which can prove difficult to organize. When I wrote my undergraduate thesis, my papers were all printed copies stored in an accordion folder, but as a graduate student reading hundreds of articles, this system is hardly practical. Luckily, there are several **reference management programs **that keep track of papers you’ve read and generate bibliographies for you. After searching through many options last winter, I finally settled on Zotero, which I highly recommend (and no, I am not being paid to write this).

I chose Zotero [zoh-TAIR-oh] because of the following features:

- It’s free
- It only takes one click in Chrome, Firefox, Safari etc. to download a new paper to your collection
- You can store the actual PDFs of the papers themselves, not just the bibliographic information
- You can write notes about each item or tag them with keywords
- You can highlight or annotate the PDFs using another program (e.g. “Preview” on a Mac) and import these highlights/annotations into Zotero
- If you search for a term, Zotero will look through the titles, tags, notes, and even the highlighted passages and annotations you’ve imported for all the documents in your library
- Zotero will generate bibliographies and in-line citations in both BibTex and MS Word

In addition, Zotero lets you share your collections with others, import collections you may have already started in other software packages, and access your collections (well, the citations at least) from multiple computers. The PDFs themselves are stored on your original machine (by default, in your “Downloads” folder), unless you choose to put them in Dropbox, OneNote, Google Drive, or some other cloud storage. You should probably do this anyway to back them up.

My workflow so far looks like this:

- Search for a paper or author on Google Scholar
- Follow the link to the website of the journal article and click the Zotero button to download both the citation and, if your university can get you access to it, the PDF. The Zotero download button circled in this picture is available for all webpages once you install the Chrome extension, which I found to be quick and painless. Other browsers are also supported.
- The article is happily stored in Zotero. On the left, you have the option of organizing your library into folders. On the right, the article metadata is listed. The menu above the metadata allows you to add notes or tags, which are then searchable.
- Using the Mac “Preview” program, I highlight and annotate the PDF.
- When I’m done, back in Zotero I extract the highlights and annotations. I might do this once a month or so, since I can do it for many documents simultaneously.
- Extracted annotations and highlights now appear in the righthand panel.
- I can search my entire library for the word “heck,” and Zotero will find the annotations where I wrote “what the heck?”
- If you use LaTex, simply export everything to BibTex and do your bibliography and citations from there. If you need to write something in MS Word, install the Zotero extension in Word, which will allow you to add citations in just a couple clicks:
- Upon clicking Word’s “Zotero Insert Citation” button, you’ll be taken to Zotero where you easily search for the paper you want, and it will be added in whatever citation format you prefer.
- And with one more click, you can generate a bibliography.

I’ve found this system extremely helpful this semester, and if you don’t have a system of your own that you like, I suggest you give it a try. Zotero is quick to learn, and I have yet to run into any bugs. Granted, things don’t always go as smoothly as the example above, so here’s what to do in some other situations:

- If you just have the PDF (say, it was emailed you to) you can have Zotero extract the citation information (e.g. author, title, year) from the PDF metadata to allow you to make citations.
- If the PDF doesn’t have any metadata encoded (perhaps you have a scanned book chapter, for instance), you can download the citation information from the internet and then attach your PDF to it.
- As a last resort, if you can’t get the complete citation information online in a downloadable format (as is sometimes the case with books or websites), you can enter the missing information by hand.
- Conversely, if you find the citation online but not the PDF (or if you don’t have free access to it), you can still download the citation into Zotero. You can always add the PDF later if you get access to it.
- The most annoying situation is when I can’t highlight or annotate the PDF (this often happens if it’s a scanned copy that was laid slightly diagonally in the scanner). This isn’t a Zotero issue of course, but Zotero will still allow you to type notes about the document as a whole in Zotero’s own notes section, even if you don’t have a way to annotate specific passages.

Lastly, if you’re still not convinced and want to try out other reference management software, here’s a good place to begin: https://en.wikipedia.org/wiki/Comparison_of_reference_management_software

Do you have a reference manager that you particularly like? Let us know in the comments below which one is your favorite and why!

]]>*By Tai-Danae Bradley*

Welcome to part 2 of our series where we’re taking a candid peek into the world of mathematical research. Last time we chatted about the often laborious process of doing math which, as we heard from Andrew Wiles, is much like stumbling in a dark room while searching for a light switch. The late William Thurston also described math in a similar fashion in his Math Overflow profile. Perhaps you’ve seen it? Here’s a screenshot:

I love that first sentence of his second paragraph! In my own experience, I’ve found that perseverance really is key as I’m well acquainted with *fog, muddle, *and* confusion. *In fact, mathematical fog is one of the main reasons why I started my own blog. But if I’m to be honest, there are times when I’m not sure how much of that fog is simply “the nature of the beast” and how much of it is, well, “encouraged.”

Quite naturally, this brings me to mathematician** **Piper Harron’s epic thesis, without which no foray into mathematical candor would be complete. No doubt you know what I’m talking about. If you don’t, then you *must* go check it out. Seriously. Right now. Just browse through the table of contents. With numerous *laysplanations* (explanations for the layperson) and chapter titles like “Getting Mathy With It,” you can’t tell me you won’t want to read more! Harron’s thesis is quite the epitome of* math-made-accessible, *and I was especially drawn in by the transparent honesty in her prologue. There she states that her thesis was written for

People who, for instance, try to read a math paper and think, ‘Oh my goodness what on earth does any of this mean why can’t they just say what they mean????’ rather than, “Ah, what lovely results!” (I can’t even pretend to know how ‘normal’ mathematicians feel when they read math, but I know it’s not how I feel.)

This is SPOT ON! As I mentioned in our previous post, I’m often in the stumbling-in-the-dark-while-groping-for-the-light-switch phase. But there are times when it feels like the switch was installed in the most remote, most inaccessible location possible, like, I don’t know, behind the drywall of the ceiling in the attic. I can’t tell you how many times I’ve thought *Geez! Why can’t mathematicians just say what they mean?!* And since this happens so frequently, I can’t help but wonder if there’s a secret math-committee out there whose goal is to obfuscate math as best as they can. But of course that’s silly. (Right?) There is no such committee. (*Right?!)* But it really is nice to hear others voice the same thoughts that have crossed my mind.

And finally, I’d be remiss to close our discussion without mentioning Terry Tao’s excellent blog post wherein he answers the question, “Does one need to be a genius to do math?” with an emphatic “NO.” In it he writes,

In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the ‘big picture.’ And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does

notneed some sort of magic “genius gene” that spontaneously generatesex nihilodeep insights, unexpected solutions to problems, or other supernatural abilities (emphases original).

Since Tao is one whom most of us would without hesitation call a genius, I found his words to be even more significant.

And this brings me right back to that Slate article I referred to at the beginning of part 1 of this series. After the author reminds us that “no one is born knowing the axiom of completeness,” he goes on to say, “even the most accomplished mathematicians had to learn how to learn this stuff.” How true! And I especially love how math teacher and self-proclaimed bad-drawer Ben Orlin illustrates this in his ‘How to Edit your Math Pessimism’ series:

I think it’s comforting to know that we students can replace “negatives” with just about *any* mathematical concept we’ve struggled with today and be in such good company! Pretty neat, huh?

I’ll leave you now with a few more links that relate to the topics we’ve chatted about today. And to everyone about to enter final/qualifying exam season, I wish you all the best of luck!

- Not too long ago, Evelyn Lamb wrote a fantastic article in which she contrasts Piper Harron’s thesis with Shinichi Mochizuki’s proof of the abc conjecture.
*I can’t recommend it highly enough.* - In this article, Marcus du Sautoy points out that
*everyone*– not just those whose brains are ‘wired’ mathematically – is capable of doing mathematics. - In a similar vein, Benjamin Braun wrote this piece for the AMS Blog in which he addresses “the secret question (Are we actually good at math?).” At the same blog, Evelyn Lamb wrote about math and the genius myth. Both posts contain several great links.
- Over at Baking and Math, Yen Duong notes that students too often think they are too “stupid/slow/dense” to do math, while the problem is often in the
*communicator*

*This article was adapted from Bradley’s **Snippets of Mathematical Candor**, originally published at **Math3ma** on April 4, 2016.*

*By Tai-Danae Bradley*

A while ago at my blog Math3ma, I wrote a post in response to a great Slate article reminding us that math – like writing – isn’t something that anyone is good at without (at least a little!) effort. As the article’s author put it, “no one is born knowing the axiom of completeness*.”* Since then, I’ve come across a few other snippets of mathematical candor that I found both helpful and encouraging. And since final/qualifying exam season is right around the corner, I thought it’d be great to share them here for a little *morale-boosting.
*

The first comes from a fantastic post written by University of Illinois at Chicago’s recent PhD Jeremy Kun (also blogger at the excellent Math ∩ Programming) in which he answers the question *What is it ‘really’ like for a mathematician to learn math? *In short, his answer is contained in the post’s title: “Mathematicians are chronically lost and confused (and that’s how it’s supposed to be).”

Of course, he means this in a *good* way and elaborates by sharing this colorful metaphor of mathematical research by Andrew Wiles:

You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are a culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them.

I definitely feel like I’m stumbling in the dark whenever I tackle a new subject. (I believe the technical term for this is *learning.*) And I’m not even at the research stage yet!* *In response to this occasional feeling of lostness that we students tend to feel, Kun says:

Sometimes the answer is to pinpoint one very basic question I don’t understand and try to tackle that first. Other times it’s to learn what I can and move on.

This is great advice for graduate students. Personally, I find that there’s just not enough time to learn it all, so often I have to be okay with not being able to understand a topic as well as I’d like. But occasionally I end up “wasting” an entire day just grappling with a *single* problem/idea. Although those days are frustratingly slow, they actually end up being the most productive and satisfying and not a waste at all! So I was encouraged when Kun mentioned that one of his colleagues has similar sentiments:

If I spend an entire day and all I do is understand this one feature of this one object that I didn’t understand before, then

that’sa great day.

A great day, indeed. In fact, I find that math often comes with an initial “shock factor” that (usually!) goes away once I make a conscious decision to focus and, well, *do the work*. It’s sort of like going for an early-summer swim at the ocean or pool. The water looks inviting, but as soon as you jump in, its unexpectedly frigid temperature takes your breath away, and your only thought is* Agh! What am I doing here?!* But after wading around a bit, your body adjusts and the cold doesn’t feel so bad anymore. And that’s when the real fun begins.

Case in point: I was recently working through some of my school’s old topology qualifying exams and came across a question which I *initially* had no idea how to solve. In fact, here’s the problem (You don’t actually have to read it!):

*But here’s what my brain saw:*

Yep. You see, my first thought was, *Is this even English?!* But after a few minutes of pondering, I realized just how simple both the question *and *its solution were! Of course, in other instances, minutes turn into hours, days, or even semesters, and the end result doesn’t *always* turn out to be simple. But in general, I find that once I put in the work, the math isn’t as terrifying as it seems at first. And as Wiles observed, it’s okay – *necessary,* even! – to stumble around in the dark before finding the light switch.

Pretty encouraging, right? Well it doesn’t end there! I’ve got more quotes from great mathematicians that I’d love to share with you. So be sure to stay tuned for part 2! In the meantime, I’ll leave you with a few links that touch on some of the things we’ve talked about today:

- This Quora answer to “What is it like to understand advanced mathematics?” and this article from the
*Princeton Companion to Mathematics*emphasize the amount of hard work one must do in order to produce good results. - Here’s a pretty funny but all-too-real answer to the question “What do grad students in math do all day?”
- At the 2013 Joint Mathematics Meetings, Francis Su gave an excellent talk on why teachers should not value students based on their accomplishments and academic performance. I also love this article he recently wrote for the MAA Focus news magazine.
- Speaking of performance, both Cathy O’Neil and Dan Lee have written about the ugly side of math competitions.
- And though not math-specific, this tweet needs to be tattooed to the backs of eyelids of students everywhere. (Or at least framed and hung on a wall!)

*This article was adapted from Bradley’s **Snippets of Mathematical Candor**, originally published at **Math3ma** on April 4, 2016. *

Born in India in 1887, Ramanujan’s story is one of the most intriguing and famous stories in all of mathematics, not least because of his unusual mathematical style. Although he excelled in mathematics from a young age, Ramanujan did poorly in other subjects while in college (presumably because he spent all his time focusing on math) and eventually dropped out. Without a degree, Ramanujan had to endure several years of extreme hardship and poverty before finding a stable job. Despite the fact that he had little to no formal training, during these years he continued doing independent research in mathematics, and eventually managed to send samples of his results to the leading mathematicians of England at the time. Although initially viewed as a crank, Ramanujan’s work caught the eye of G. H. Hardy, one of the pre-eminent mathematicians of the era. After reading some of Ramanujan’s theorems, Hardy is famously quoted as saying: “They must be true, because, if they were not true, no-one would have the imagination to invent them.”

After overcoming the objections of his parents, Ramanujan sailed to Cambridge, where the movie really picks up. Dev Patel and Jeremy Irons do a stellar job of capturing the friction and mathematical differences between Ramanujan and his mentor Hardy. Ramanujan, who many biographers describe as devoutly orthodox, often simply wrote down fantastic identities and formulas without formal proof, claiming that he had been inspired by religious figures in his dreams. This often infuriated Hardy, who insisted on a more rigorous approach. Nevertheless, Hardy recognized Ramanujan’s genius, and indeed the significance of many of their discoveries is only being understood and explored today. It is unclear how Ramanujan, especially with no formal training, could have possibly anticipated several decades of work carried out by number theorists over half a century after his time.

One of the great selling points of the movie is that its discussion of both mathematics and the mathematical process feels deeply authentic, with the contrived presentation of mathematics often found in popular science movies kept to a comparative minimum. (Indeed, the movie was written with input from actual number theorists – Professors Ken Ono and Manjul Bhargava!) In addition to superb acting (I cannot stress how much I enjoyed Jeremy Irons’ performance), “The Man Who Knew Infinity” manages to capture the spirit of Ramanujan’s work and genuinely intertwine it with the human element of the story in a way that is accessible but still not watered-down.

If my brief foray into movie-reviewing has convinced you, check out the new Ramanujan movie in more detail at http://www.ifcfilms.com/films/the-man-who-knew-infinity! **This is an independent film playing select theatres, so be sure to see it while it lasts.** Trailer and movie times available here.

And now, an exclusive behind the scenes interview with Diana Davis about how she put this incredible project together without any prior experience:

Diana Davis: “One challenge was that we recorded all the pentagon dancing once, using the Math Department video camera, but the resolution was not good enough to make a great video, so we had to do the whole thing again. So then the question is, what video camera can we get that has good resolution? It turns out that an iPad was the best option. But then, how do you use an iPad to record video from directly above? So we filmed in a building at Brown that used to be a church, and put the camera in the organ loft pointing straight down. To do this, I bought an 8-foot plank from Home Depot, which I tied to a chair using an Ethernet cord, and taped the iPad to the end of the plank, and then leaned out over the railing to press Record and Stop. We taped the pentagons to the floor directly below the iPad.

The second challenge was making the magic happen. Sometimes people ask who did the video editing. It was ME! And I had never done any video editing before. I learned, by putting in hundreds of hours and failing and trying again, how to use these two powerful video editing programs, Adobe After Effects and FinalCut Pro. Adobe After Effects is what makes the magic happen, where Libby dances across one edge and reappears on the other side — basically you can make part of the screen show one video and another part of the screen show another video, so I made the two parts of the screen show two parts of the same video. Making this work perfectly took forever. If you pause the video at any point, you will find that it looks perfect, with about 105% of a Libby on the screen at any one time. I had to blend in the shadows so that they didn’t disappear all of a sudden. I think that Libby dancing on the double pentagon surface is a work of art! And then I used FinalCut Pro to put the pentagon videos together with the Math Hatters and the text.

The best place to start is this FAQ video: https://vimeo.com/47273811 because as you know, it’s better to *show* you how we made it than to *tell* you how we made it.”

Suppose you need to walk through a wet parking lot. The lot is covered with puddles and you would like to keep your shoes as dry as possible. If you know the depth of the puddles at every point, how do you choose the path that minimizes the maximum depth of the puddles you cross? A hiker might want to solve a similar problem if they want to avoid fatigue by seeking low elevations. How do you traverse a mountainous area while remaining as low as possible?

To make the problem more formal, say the area of interest is and the elevation of the ground (or puddle depth) at each point is , with continuous. Choose areas that are acceptable start and finish points and call them and . If you want to walk from the north end of the square to the south, then you could choose and . The problem can be stated as the minimax

over continuous paths such that and .

Initially, you might suppose that this is a fundamentally continuous problem since there is a continuum of possible routes . However, we can make the problem discrete by considering sublevel sets of .

The -sublevel set of is the set of points where is below elevation , that is

.

The minimax problem is solved by finding a that lies in the lowest sublevel set connecting and . That lowest elevation is

.

Watch the animated figure below and notice the first time that the white sublevel set connects the top and bottom. A path between top and bottom must first appear when the boundaries pass through a saddle point of because saddle points are where connected components of sublevel sets join each other. This is a consequence of the “mountain pass theorem”: if is optimal and is maximized by , then is a stationary point of . That stationary point is a normally a saddle point (though it could be part of a flat-topped area).

Now we can write a discrete process for finding an optimal path:

- Locate all saddle points of .
- Order the saddle points by elevation so that .
- For each saddle point , determine whether its sublevel set has a connected component intersecting and .
- If the lowest such saddle point is , then is the minimum value for the minimax problem, and the optimal path is any contained in that connects and .

For something else to think about, let’s consider a connection to graph theory. If you watch the above animation a few times, you’ll notice that connected components of the white sublevel sets are created at local minima. They spread out and meet other components at saddle points. You might imagine a graph (or network) being created between the local minima: when two components touch, an edge is created between two minima (one in each component).

Is there a natural way to choose a representative from each component to connect? If a hole forms so that the component is no longer simply connected, should an edge be added to make a cycle? Also, a graph on the maxima could be created in the same way. Should the graph on the maxima be the dual of the graph on the minima?

Post any thoughts or questions in the comments below!

]]>“Goodbye, mission control. Thanks for trying.” ~aiken_~

to the lighter

“I leave. Dog panics. Furniture shopping.” ~Reed~

That said, there seem to be very few–I found two or three–flash fiction stories pertaining to math. With this in mind I want to propose a challenge:

**Write your own six-word story/stories capturing the life, experiences, or work of mathematicians and graduate students, and post them in the comments below. The best ones will be highlighted in my post next month!**

To give some sense of what I mean, and maybe to help get your creative thoughts rolling, here are two six-word stories I wrote about common experiences in the lives of math graduate students.

Working hard, checked arXiv. Start again.

From: Grad Admissions. “We are sorry…

]]>*Originally published by Scientific American *

From the profound revelations of the shape of space to the furthest explorations reachable by imagination and logic, the history of mathematics has always been seen as a masculine endeavor. Names like Gauss, Euler, Riemann, Poincare, Erdös, and the more modern Wiles, Tao, Perelman, and Zhang, all of them associated with the most beautiful mathematics discovered since the dawn of humanity, are all men. The book Men of Mathematics, written by E.T. Bell in 1937, is just one example of how this “fact” has been reinforced in in the public consciousness.

Even today, it is no secret that male mathematicians still dominate the field. But this should not distract us from the revolutionary contributions women have made. We have notable women to thank for modern computation, revelations on the geometry of space, cornerstones of abstract algebra, and major advances in decision theory, number theory, and celestial mechanics that continue to provide crucial breakthroughs in applied areas like cryptography, computer science, and physics.

The works of geniuses like Julia Robinson on Hilbert’s Tenth Problem in number theory, Emmy Noether in abstract algebra and physics, and Ada Lovelace in computer science, are just three examples of women whose contributions have been absolutely essential.

**Julia Robinson (1919-1985)**

At the turn of the twentieth century the famed German mathematician David Hilbert published a set of twenty-three tantalizing problems that had evaded the most brilliant of mathematical minds. Among them was his tenth problem, which asked if a general algorithm could be constructed to determine the solvability of any Diophantine equation (those polynomial equations with only integer coefficients and integer solutions). Imagine, for any Diophantine equation of the infinite set of such equations a machine that can tell whether it can be solved. Mathematicians often deal with infinite questions of this nature that exist far beyond resolution by simple extensive observations. This particular problem drew the attention of a Berkeley mathematician named Julia Robinson. Over several decades, Robinson collaborated with colleagues including Martin Davis and Hillary Putnam that resulted in formulating a condition that would answer Hilbert’s question in the negative.

In 1970 a young Russian mathematician named Yuri Matiyasevich solved the problem using the insight provided by Robinson, Davis, and Putnam. With her brilliant contributions in number theory, Robinson was a remarkable mathematician who paved the way to answering one of the greatest pure math questions ever proposed. In a Mathematical Association of America article, “The Autobiography of Julia Robinson”, her sister and biographer Constance Read wrote, “She herself, in the normal course of events, would never have considered recounting the story of her own life. As far as she was concerned, what she had done mathematically was all that was significant.”

**Emmy Noether (1882-1935)**

Sitting in an abstract math course for any length of time, one is bound to hear the name Emmy Noether. Her notable work spans subjects from physics to modern algebra, making Noether one of the most important figures in mathematical history. Her 1913 result on the calculus of variations, leading to Noether’s Theorem is considered one of the most important theorems in mathematics—and one that shaped modern physics. Noether’s theory of ideals and commutative rings forms a foundation for any researcher in the field of higher algebra.

The influence of her work continues to shine as a beacon of intuition for those who grapple with understanding physical reality more abstractly. Mathematicians and physicists alike admire her epoch contributions that provide deep insights within their respective disciplines. In 1935, Albert Einstein wrote in a letter to the New York Times, “In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

**Ada Lovelace (1815-1852)**

In 1842, Cambridge mathematics professor Charles Babbage gave a lecture at the University of Turin on the design of his Analytical Engine (the first computer). Mathematician Luigi Menabrea later transcribed the notes of that lecture to French. The young Countess Ada Lovelace was commissioned by Charles Wheatstone (a friend of Babbage) to translate the notes of Menabrea into English. She is known as the “world’s first programmer” due to her insightful augmentation of that transcript. Published in 1843, Lovelace added her own notes including Section G, which outlined an algorithm to calculate Bernoulli numbers. In essence, she took Babbage’s theoretical engine and made it a computational reality. Lovelace provided a path for others to shed light on the mysteries of computation that continues to impact technology.

Despite their profound contributions, the discoveries made by these three women are often overshadowed by the contributions of their male counterparts. According to a 2015 United Nations estimate, the number of men and women in the world is almost equal (101.8 men for every 100 women). One could heuristically argue, therefore that we should see roughly the same number of women as men working in the field of mathematics.

One large reason that we don’t is due to our failure to recognize the historical accomplishments of female mathematicians. Given the crucial role of science and technology in the modern world, however, it is imperative as a civilization to promote and encourage more women to pursue careers in mathematics.

*Originally published by Scientific American *