Finding Gemstones: on the quest for mathematical beauty and truth
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On Raising Your Hand

A few weeks ago I attended the AWM (Association of Women in Mathematics) Research Symposium in Houston, TX. I gave a talk in my special session, speaking on queer supercrystals for the first time, to a room full of female mathematicians.

I was a bit disappointed when, at the end of my talk, no one raised their hand to ask any questions.  It’s usually the classic sign of an uninteresting or inappropriately aimed talk, so I figured that maybe I had to revisit my slides and make them more accessible for the next time I spoke on the subject.

Afterwards, however, several of the women in my session came up to me privately to ask specific questions about my research.  When I told my husband about this after the conference, he pointed out that perhaps they just were the kind of people to prefer asking questions one-on-one rather than raising their hands during or after the lecture.

“Did anyone in your session ask questions after the other talks?’’ he asked me, testing his theory.

I thought about it, and was surprised when I realized the answer.  “Woah, I think you’re right,” I said.  “I asked at least one question after nearly every talk.  But I think I was the only one.  Once in a while one other woman would ask something too.  But the rest kept their hands down and went up to the speaker during the break to ask their questions.”

Upon further reflection, I realized that this was even true during the plenary talks.  During an absolutely fantastic lecture by Chelsea Walton, I was intrigued by something she said.  She mentioned that the automorphism group of the noncommutative ring \[\mathbb{C}\langle x,y\rangle/(xy-qyx)\] is $\mathbb{C}^{\times} \times \mathbb{C}^{\times}$ for all $q\neq \pm 1$, but the answer is different at $q=1$ and $q=-1$.  I knew that many of the standard $q$-analogs arise naturally in computations in this particular ring, such as the $q$-numbers \[[n]_q=1+q+q^2+\cdots +q^{n-1}.\] So, I wondered if the exceptions at $q=1$ and $q=-1$ were happening because $q$ was a root of unity, making some of the $q$-numbers be zero.  So maybe she was considering $q$ as a real parameter?  I raised my hand to ask.

“Is $q$ real or complex in this setting?”

“It’s complex,” Chelsea answered.  “Any nonzero complex parameter $q$.”

“Really?” I asked. “And there are no exceptions at other roots of unity?”

“Nope!” she replied with a smile, getting excited now.  “Just at $1$ and $-1$.  The roots of unity get in your way when looking at the representation theory.  But for the automorphism group, there are only two exceptional values for $q$.”  Fascinating!

No one else asked any mathematical questions during or after that talk.

Now, I have the utmost faith in womankind.  And I would normally have chalked the lack of questions and outspokenness up to it being a less mathematically cohesive conference than most, because the participants were selected from only a small percentage of mathematicians (those that happened to be female).  But it reminded me of another time, several years ago, that I had been surprised to discover the same phenomenon among a group of women in mathematics.

One summer I was visiting the Duluth REU, a fantastic research program for undergraduates run by Joe Gallian in the beautiful and remote city of Duluth, Minnesota.  As a former student at the program myself, I visited for a couple of weeks to hang out and talk math with the students.  I attended all the weekly student talks, and as usual, participated heavily, raising my hand to ask questions and give suggestions.

The day before I left, Joe took me aside.  “I wanted to thank you for visiting,” he said.  “Before you came, the women never raised their hand during the other students’ talks.  But after they saw you doing it, suddenly all of them are participating and raising their hands!”

I was floored.  I didn’t know that being a woman had anything to do with asking questions.

I have always felt a little out of place at AWM meetings.  They are inevitably host to many conversations about the struggles faced by women in competitive male-dominant settings, which I have never really related to on a personal level.  I love the hyper-competitive setting of academia.  I live for competition; I thrive in it.  And it never occurs to me to hold back from raising my hand, especially when I’m genuinely curious about why $q$ can be a complex root of unity without breaking the computation.

But, clearly, many women are in the habit of holding back, staying in the shadows, asking their questions in a one-on-one setting and not drawing attention to themselves.  And I wonder how much this phenomenon plays a role in the gender imbalance and bias in mathematics.

At the reception before the dinner at the AWM conference, I spotted Chelsea.  She was, unsurprisingly, quite popular, constantly engaged in conversation with several people at once.  I eventually made my way into a conversation in a group setting with her in it, and I introduced myself.

“Hi, I just wanted to say I really enjoyed your talk!  I was the one asking you whether $q$ was real.”

Her expression suddenly shifted from ‘oh-no-not-another-random-person-I-have-to-meet’ to a warm, smiling face of recognition.  “Oh!  I liked your question!” she exclaimed.  The conversation immediately turned to math, and she was nice enough to walk me through enough computations to convince me that $q=\pm 1$ were special cases in computing the automorphism group of the noncommutative ring.  (See Page 2 of this post for the full computation!)

The entire experience got me thinking.  It was because I raised my hand that Chelsea recognized me, that she was happy to talk to me and mathematics was communicated.  It was because I raised my hand that I got the question out in the open so that other participants could think about it as well.  It was because I raised my hand that women were doing mathematics together.  And perhaps it is because I raise my hand that I have no problem interacting in a male-dominant environment.  After all, they raise their hands all the time.

It is tempting to want to ask the men in mathematics to take a step back and let the women have the limelight once in a while.  But I don’t think that’s the answer in this case.  Men should keep raising their hands.  It’s part of how mathematics gets done.  It helps to communicate ideas more efficiently, to the whole room at once rather than only in private one-on-one settings.  It draws visibility to the interesting aspects of a talk that other participants may not have thought of.

What we really need is for women to come out of the shadows.  So, to my fellow women in mathematics: I’m calling on all of us to ask all our questions, to engage with the seminar room, to not hold back in those immensely valuable times when we are confused.  And raise our hands!

To cap off this post with a mathematical gemstone, below is the full computation of the automorphism group of the graded noncommutative ring mentioned on the previous page.

Any automorphism of the noncommutative $\mathbb{C}$-algebra \[R_q:=\mathbb{C}\langle x,y\rangle/(xy-qyx)\] that preserves the grading by degree can be thought of as an element of $\mathrm{GL}_2(\mathbb{C})$, since the degree-one generators $x$ and $y$ must map to linear combinations of $x$ and $y$, and these images determine the map. In other words, we can represent the automorphism that sends $x$ to $ax+by$ and $y$ to $cx+dy$ by the $2\times 2$ matrix \[\left(\begin{array}{cc} a & b \\ c & d\end{array}\right).\]

For a map of this form to extend to an automorphism, it is necessary and sufficient that the ideal generator $xy-qyx$ maps to a scalar multiple of itself. In this case we have \[\begin{align*} xy-qyx &\mapsto (ax+by)(cx+dy)-q(cx+dy)(ax+by) \\ &= (1-q)acx^2+(ad-qbc)xy+(bc-qad)yx+(1-q)bdy^2 \end{align*}\]

If $q=1$, this simplifies to $(ad-bc)(xy-yx)$, which lies in the desired ideal. Thus the group of degree-preserving automorphisms is all of $\mathrm{GL}_2(\mathbb{C})$ in this case.

If $q\neq 1$, since the $x^2$ and $y^2$ terms must vanish, we have either $a=0$ or $c=0$, and either $b=0$ or $d=0$. If $a=0$ and $b=0$ simultaneously then the $xy$ coefficient would be $0$, and similarly for $c$ and $d$. Thus we must have either $a=d=0$ or $b=c=0$.

In the case that $a=d=0$, the above simplifies to $bc(-qxy+yx)$, and in the case that $b=c=0$, the it simplifies to $ad(xy-qyx)$. The latter is clearly a scalar multiple of $xy-qyx$, but the former is so only if $q=-1$. Hence if $q\neq \pm 1$, the automorphism group is isomorphic to $\mathbb{C}^\times \times \mathbb{C}^\times$, consisting of the matrices of the form \[\left(\begin{array}{cc} a & 0 \\ 0 & d\end{array}\right)\] with $a,d\neq 0$. And if $q=-1$, we have a second copy of $\mathbb{C}^\times \times \mathbb{C}^\times$, from the matrices of the form \[\left(\begin{array}{cc} 0 & b \\ c & 0\end{array}\right)\] with $b,c\neq 0$.

The upshot is that because this ring is generated in degree $1$ with a single quadratic homogeneous relation, the higher degree $q$-numbers do not appear in the computation, and only $q=\pm 1$ are special values. A nice little gemstone!