Mathematical Inclusion Principle: Emmy Noether

This is a post in an ongoing series which features people, events, and organizations that demonstrate and/or support the importance of inclusivity and diversity in the mathematical sciences. I will share these posts with the students in courses I teach, and I encourage you readers to do the same. The series title may be considered a "play on words" with the method from combinatorics known as the "inclusion-exclusion principle".

Emmy Noether born March 23, 1882; Erlangen, Germany died April 14, 1935; Bryn Mawr, PA, USA

Amalie Emmy Noether was born in 1882 in a city in the state of Bavaria in Germany. Her father, Max Noether, was a professor at the university in Erlangen, although his daughter's achievements in mathematics would far surpass his, in time. Throughout high school, she studied English and French, and took certification examinations to become a language teacher. Although her exams were deemed sehr gut ("very good"), she actually decided to continue her studies at the university instead.

It was not at all common for females to study at university at this time in Germany. In fact, the university's administration had said that this would "overthrow all academic order" and, during Emmy's time, she was one of only two women out of almost 1000 students. [1] In fact, she was only allowed to audit courses, sitting in on lectures (if the professor even allowed this) without officially enrolling. Still, she gathered knowledge and passed the graduation exam in 1903 [2].

Thereafter, Emmy Noether was devoted to mathematics. She attended lectures by some of the most well-known mathematicians at the time (all men, of course) at the University of Göttingen in 1903, and then returned to Erlangen to work on a thesis, which she published in 1907. Fortunately, the rules for women were being changed, and she was allowed to complete this work and receive her degree, which was summa cum laude, mind you.

For the next 25 years, she worked and taught in universities in Germany and Russia. At first, she stayed at Erlangen, occasionally substituting for her father when he was ill. In 1915, she returned to Göttingen to teach, at the behest of world-famous mathematician David Hilbert, who intervened on Noether's behalf to ensure the sexist university policies could not prevent her from doing so. One faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" Hilbert's retort shows that his interest was in the mathematical prowess of the candidate and nothing else: "I do not see that the sex of the candidate is an argument against her admission as privatdozent. After all, we are a university, not a bath house." [3] It was at Göttingen that she produced her most famous work about symmetry and conservation laws. (You can read more about that work here in Science News [4].) And, although it sounds wonderful that Noether was provided this opportunity to teach, was able to apply for habilitation (something like tenure), and received a citation from the Prussian Minister for Science, Art, and Public Education, it is important to note her professorship position came with no salary! 

Emmy Noether visited Moscow State University from 1929-1930 to teach and work with mathematicians there on their research in abstract algebra. Many of her colleagues, all of whom were well aware of her talent and passion for mathematics, were frustrated at the university systems and cultures that prevented her from receiving proper recognition and compensation for her work. While the wider world was rather discriminatory, at least we can say that her corner of the mathematical world recognized her as an outstanding individual, regardless of gender. Indeed, at the 1932 International Congress of Mathematicians, in Zurich, Switzerland, Noether delivered a plenary address. Furthermore, that same year, she shared the Ackermann–Teubner Memorial Award with Emil Artin, in recognition for their "advancement of mathematical knowledge". [5]

Unfortunately, the rise of Nazism and widespread anti-Semitism in Germany impacted the university system and led to the removal of Noether, a Jew, from her teaching position. The very same Prussian Ministry for Sciences, Art, and Public Education, from whom she had received a glowing citation earlier in her life, sent her a notice: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen." [6] Luckily, scientific colleagues in the rest of the world were working to help scholars like Noether, and she found a position at Bryn Mawr, an all-female liberal arts college in southeastern Pennsylvania. While in the United States, she also visited and lectured at Princeton University and their Institute for Advanced Study.

Emmy Noether passed away somewhat suddenly in April 1935, at the age of 53. Doctors had found a pelvic tumor and an ovarian cyst during surgery. While on bed rest, she suffered a high fever and died. Doctors were not entirely sure what had happened, citing a possible infection as the cause of the fever and untimely death. Support from the mathematical and scientific community was swift: Noether's colleagues and other eminent scholars wrote their condolences and tributes, including Albert Einstein and Herman Weyl. Below is a quote from Einstein's letter published in 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. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. [7]

Why do I write this now? Well, I think Nature put it best with their blog post last week [8]:

Celebrate the mathematics of Emmy Noether: An algebra pioneer who faced discrimination deserves wider recognition on the centenary of her namesake theorem.

From the article, because they summarize it all better than I could:

The results that Noether published 100 years ago were, for her, a rare foray into physics, in which she was not particularly interested. Albert Einstein had just developed his general theory of relativity, and was struggling to understand how energy fitted into his equations. [German mathematicians David] Hilbert and [Felix] Klein were working on it, too, and asked Noether for help. 

That she did help is an understatement. Noether’s expertise in symmetry led her to discover that the symmetries of a physical system are inextricably linked to physical quantities that are conserved, such as energy. These ideas became known as Noether’s theorem (E. Noether Nachr. d. Ges. d. Wiss. zu Göttingen, Math.-phys. Kl. 1918, 235–257; 1918).

 As well as answering a conundrum in general relativity, this theorem became a guiding principle for the discovery of new physical laws. For example, researchers soon realized that the conservation of net electric charge — which can neither be created nor destroyed — is intimately related to the rotational symmetry of a plane around a point. The impact was profound: those who created the standard model of particle physics, and the researchers who attempt to extend it, think in terms of Noether’s symmetries.

All of this is to say that mentioning Emmy Noether now is timely. Her work 100 years ago was outstanding and groundbreaking, and scientists today still find it influential. Moreover, the struggles and discrimination she overcame throughout her life may serve as a lesson for our times: that someone can overcome those obstacles, but wouldn't our world be better off if they didn't have to?

Mathematical Inclusion Principle: Maryam Mirzakhani

This is a post in an ongoing series which features people, events, and organizations that demonstrate and/or support the importance of inclusivity and diversity in the mathematical sciences. I will share these posts with the students in courses I teach, and I encourage you readers to do the same. The series title may be considered a "play on words" with the method from combinatorics known as the "inclusion-exclusion principle".

Maryam Mirzakhani born May 12, 1977; Tehran, Iran died July 14, 2017; Stanford, California

The story of Maryam Mirzakhani's life is at turns inspiring and heartbreaking. She was born in the capital of Iran and achieved academic success at an early age, becoming the first female Iranian student to earn a gold medal at the International Mathematical Olympiad and the first Iranian student to obtain a perfect score at that competition [1]. However, her sixth grade teacher "discouraged her interest in mathematics, noting that she was not particularly talented, not at the top of the class" [2]. She earned a bachelor's degree in mathematics from Sharif University of Technology in Tehran and subsequently emigrated to the United States, obtaining a PhD from Harvard University.

Everyone who writes about Mirzakhani notes her talent, passion, and commitment to both her mathematical work and the communication of that work. For example, an article in the New York Times Magazine's series "The Lives They Lived" [3] and an article in Quanta Magazine [4] both describe and portray her penchant for visualization, drawing, and "doodling", perhaps an informal act that the lay public may not associate with high-level, abstract mathematical research. Her three-year-old daughter would remark, "Oh, Mommy is painting again!", and Mirzakhani herself remarked that "the process of drawing something helps you somehow to stay connected".

The Fields Medal is sometimes referred to lazily in the media as the "Nobel Prize of Mathematics" but this is inaccurate for a few reasons: it is only given out every four years, it has an (arbitrary) age cap, and it hardly represents the diversity of mathematical research and the people who do that work. (See Michael Barany's recent comment in Nature [5] for further information.) In any event, Maryam Mirzakhani was the first (and still only) female recipient of this award, so bestowed in 2014, seventy-eight years after the award's inauguration. The most recent class received their medals in August 2018, and Caucher Birkar earned Iran that nation's second such prize. However, the historical list of recipients is heavily dominated by North American and European nations, and white males, overall [6]. This is hardly the image of mathematics that we in the community wish to convey, but it is a fact. 

Unbeknownst to many, Maryam Mirzakhani was battling breast cancer at the time this award was announced. Her closest friends and colleagues were aware of her struggles and even helped to keep the press from making this illness the main story at the Fields Medal ceremony [7]. She passed away in July 2017 at the age of 40, at the time a professor of mathematics at Stanford University [8]. It's hard to overstate the outpouring of support from the mathematical community upon her death [9]. She will never be forgotten. 

Why do I write this now? Why is this the first post in a series about inclusivity? In today's political climate, it should be obvious, but I will state it anyway. Maryam Mirzakhani is an exemplary American. She found a talent and pursued it, against all odds. Mathematics was her passion and she traveled across the globe to continue her studies and share her work with others. However, an executive order passed by President Donald Trump, and recently upheld (in part) by the Supreme Court of the United States, would have made it difficult for someone like Maryam Mirzakhani to emigrate to America, simply because of her country of origin [10]. Never mind her intellect, her passion, her humanity, her obvious abilities to contribute to the vibrancy of America's scientific community. Never mind all that. The current administration would not bother to consider these attributes merely because she comes from a "Muslim-majority country" [11].

Now, you may say: "How do you know the administration would deny her visa? The ban doesn't apply to all immigrants from those countries." To this, I say: "You're missing the point." The executive order was obviously designed based on those countries being Muslim-majority and nothing else, so why would there have been any reason to allow someone like Mirzakhani to enter the country? At what point will we admit that it is utterly ridiculous that someone can say, "You can't prove I'm racist", and feel like they mean it, despite the fact that they are so obviously racist?

In any event, my point is that Maryam Mirzakhani is a treasure of humanity, that she is now lost, that the current culture of our country (and, indeed, the world) did not appreciate her when she was alive, and that the current politics of our country would likely never have allowed her entry in the first place. I don't want to live in that world, so I will tell everyone I know about Maryam Mirzakhani and her contribution to the history of humanity. She was a stellar human being. She succeeded by fighting obstacles. She was a mathematician. She will be missed.

College: what should it be?

This post is intended for both college students (especially first-year students) and college instructors as we begin another academic year. 

This past summer, I spent a lot of time reflecting on my experiences as a student and teacher. (I've been one, the other, or both for literally my entire adult life.) I spend a little time on this kind of reflection every summer: pondering the previous school year, appreciating what went well, thinking about how to make my courses better, etc. But I spent even more time doing so this past summer, perhaps because I've now been teaching at college full time for 5 years and I needed a mental break for a couple months, and perhaps because I now have an infant daughter (7 months) and I'm feeling especially philosophical about life and learning.

As part of that reflection, I read two books about education, both of which I highly recommend: Educated: A Memoir, by Tara Westover (historian and writer); and College: What It Was, Is, and Should Be, by Andrew Delbanco (Professor of American Studies at Columbia University). I was struck by Dr. Westover's stories in Educated, particularly by how unfathomably different her life has been from mine, apart from being academics by trade. And I was moved by Dr. Delbanco's passionate writing about the history, present, and future of colleges in America and what they can and ought to do for us as a society. 

The following quote is from the introduction to Dr. Delbanco's book:

At its core, a college should be a place where young people find help for navigating the territory between adolescence and adulthood. It should provide guidance, but not coercion, for students trying to cross that treacherous terrain on their way toward self-knowledge. It should help them develop certain qualities of mind and heart requisite for reflective citizenship. Here is my own attempt at reducing these qualities to a list, in no particular order of priority, since they are inseparable from one another.

1. A skeptical discontent with the present, informed by a sense of the past.
2. The ability to make connections among seemingly disparate phenomena.
3. Appreciation of the natural world, enhanced by knowledge of science and the arts.
4. A willingness to imagine experiences from perspectives other than one's own.
5. A sense of ethical responsibility.

These habits of thought and feeling are hard to attain and harder to sustain. They cannot be derived from exclusive study of the humanities, the natural sciences, or the social sciences, and they cannot be fully developed solely by academic study, no matter how well "distributed" or "rounded". It is absurd to imagine them as commodities to be purchased and delivered to student consumers. Ultimately they make themselves known not in grades or examinations but in the way we live our lives.

 Delbanco, A. (2012) College: What It Was, Is, and Should Be. Princeton University Press. p. 3-4

I had this list of qualities in mind throughout the rest of my reading of the book and while I prepared my courses for the fall semester that officially starts tomorrow. These lofty ideals of character and citizenry appeal to me. But I wonder how much I, as a teacher, am able to engage with students on these ideals, let alone encourage them to cultivate habits and behaviors associated with those ideals. Moreover, I wonder whether students are thinking of a similar list, or something else entirely. Take note that nothing in the quote above is remotely related to "employability" or "marketable skills". And yet, this is what I imagine many students are thinking of as they enter college. They know that a college degree is somehow required to "get a good job" and "succeed in life". But what do those goals even mean? And if we teachers don't have those ideas on our list of goals, why should we be teaching college anyway? What's the point?

So, it feels important to share this list publicly as a way to say that the list above truly is important to me, both as a teacher and as a citizen of this world. Those qualities are vastly more important to me than job skills or a grade point average, and I would hope that most other teachers would express a similar opinion. Yes, ultimately, I want my students to "succeed in life", under whatever definition they have individually for "success". But, I truly believe that pursuing the ideal qualities in that list will help lead to success and so much more.

In conclusion: students, that list of qualities is important to me and I will try to keep it in mind all year. I doubt that every class meeting, every assignment, or every conversation will be directly related to that list. It won't feel like it to you, nor to me. But I want that list to percolate through everything we do. I want you to consult the list every once in a while and reflect about your time at college and your life experiences, your behaviors, your personality, your culture. I will do the same. Together, I hope that we can push each other to be better people and to cultivate the ideals and habits represented in that list. In other words, I hope we can collectively pursue education in the broadest sense of the term.

And fellow teachers, I hope you will also keep that list in mind as you prepare and teach your courses, as you mentor students, as you give advice and grades and letters of recommendation, as you serve on college committees and do research and ... everything else that we do. Take time now and then to reflect on why you teach. Think about not only what you hope your students will learn from your courses, but also what you hope to learn from teaching those courses and working with your students.

Best wishes to all for a happy, productive, meaningful, and educational semester!