# Queering Mathematics

First: Trans people are people who deserve to have their identities respected and validated and to live without fear. On this Trans Day of Remembrance #TDoR2020, we mourn those who have died from transphobic violence. As a broader queer community, we remember that this violence disproportionately impacts trans women of color, a group who have also led many of the efforts to secure human rights for us all.

I am proud to have participated with Ron Buckmire, Emily Riehl, Juliette Bruce, Anthony Bonato, and Robin Gaudreau in LGBTQ+Math Day (virtually) at the Fields Institute this week. An event like this contributes to the visibility of queer people and our accomplices/allies, and a major theme of our conversation was the fact that each of us, by virtue of existing, queered spaces by being in them, from the rugby pitch to mathematics conferences. But for this post, I’d like to reflect on the ways that I saw our queer identities influencing our mathematics.

Ron Buckmire kicked off the event with a talk entitled “Different Differences”. He started by telling us about some of his identities, which developed into an observation that our community treats some identities like race and gender as “standard” differences and often ignores others, like his gay and Caribbean identities. Ron reminded us that the data about representation in the higher education mathematics are sparse (and shameful) for the “standard” differences and often completely absent for the “different” differences. But Ron’s title also applies to the overview he gave of his research. Using calculus-based methods, we describe continuous change using a limit of an average rate of change, and this process can be discretized in multiple ways. Discretizing the most famous definition of a derivative leads to what is often called a forward difference, but those of who have taught calculus will be familiar with versions that could be called backward or center differences, corresponding to various ways to draw secant lines to continuous curves near a point.

Ron’s observation was that these three approaches all implicitly assume that the width of the approximations, h, will approach 0 linearly. In what I saw as the first example of queering mathematics in the session, Ron demonstrated how we can reject this assumption (i) by replacing the role of h with more interesting functions that still approach zero on the order of h to get “non-standard differences, which he would like to name Mickems differences, after Ron Mickems, and (ii) by taking non-local discretizations, which felt to me like a radical reorganization of an approach to discrete difference modeling.

Next, Emily Riehl talked about “Contractibility as Uniqueness”, which she explicitly framed as “queering uniqueness”. Emily’s work is in category theory, a subfield that I see as overtly working to reconsider and rebuild the foundations of mathematics, and this talk showed how we might think of that agenda as a queer one. Emily started from the idea of the first fundamental group of a topological space, which is a group made of all loops at a base point in the space with the (associative) operation of composition (or concatenation). If we loosen the restriction that the paths be loops, allowing them to start and end at different points, we lose the ability to compose an arbitrary pair elements, so Emily’s core question became: when it is possible to compose them, how unique is that composition? Is it unique enough for associativity? She built a tetrahedron representing a homotopy that answers this question affirmatively, but she wanted to step back even further. Uniqueness can be quantified as: there exists an x such that, for all y, x=y. Interpreting this from a categorical perspective, this became the sum over all x of the product of all y of the set of proofs that x=y, where sum and product are adjoints of the pull-back, and uniqueness because the observation that this is a contractible space, which can be proved without referencing the base topological space. I am personally drawn in powerfully by this kind of work in category theory, which reimagines the foundations of what had seemed necessary in mathematics and logic in ways that feel analogous to the work of reimagining the messages of what had seemed necessary in society for gender and sexuality.

Third, Juliette Bruce told us about “Computing Syzygies”. This is a classical subfield that owes a lot to Emmy Noether, but Juliette’s overview showed how there is still so much we don’t know about the relationships between monomials! In light of the themes of queering mathematics above, one particular move by Juliette stood out to me. Her research has made progress in understanding syzygies by bringing tools from applied mathematics for computing with enormous matrices to bear on what had previously been viewed as a classical, abstract problem. While it might be a bit of a stretch from the talk, I see this as Juliette rejecting the false binary of abstract and applied mathematics. I was pulled into algebraic geometry as a graduate student because I like the double vision of seeing objects from both algebraic and geometric perspectives, and I’ve long felt that the most exciting mathematics brings this kind of binary-rejecting double vision to approach problems from new directions. And while we didn’t get to hear about Robin’s work during this event, in a later communication they offered a sketch of a similar interpretation of some of their work in virtual knot theory in which some functions require or break a binary.

The final presenter, Anthony Bonato, told us about “Out, Proud, and Combinatorial: A gay mathematician’s journey”. He told us about his efforts to be an out, proud, mathematician as well as the challenging ways environments have resisted these efforts. In his overview of his work, he described efforts that overlap abstract and applied mathematics, similar to Juliette, in complex network analysis. He studies the hidden geometry behind complex networks, such as social networks, showing how complex social networks often reduce to a small number of characteristics (the dimensions in this hidden geometry). I might be reaching to claim this as queering the mathematics, but this hidden geometry certainly reminds me of Ron’s different differences and my own experiences learning to see implicit social structures that my non-queer peers could safely ignore. Anthony also turned the connection I’m making in reverse, perhaps mathing queerness, stating his axioms for mathematics and diversity/equity/inclusion work that remind me of Federico Ardila’s axioms. In both situations, mathematicians are using the disciplinary concept of an axiom to structure their approach to justice.

My own work is also inspired by my experiences as a queer person. I never fit into the implicit expectations for gender or sexuality as a young person, and that misalignment made me hyper aware of the systems that guided and structured human activity in general. In mathematics, this awareness served me well: I spent my time asking why we did what we were doing and how we knew the things we claimed to know. More recently, I have come to see that I was lucky to happen into those habits in mathematics, buttressed by my other privileges. I see how students, especially students of color, who ask similar kinds of critical questions are often driven out of mathematics when people assert that there is nothing to ask, that mathematics is just “pure” truth that they must accept. And even those who aren’t driven away are forced to experience mathematics as a form of authoritarianism. As a teacher educator, I work on helping teachers to critique, perhaps to queer, mathematics so that they in turn can build classrooms that don’t recreate this history of violence in the name of mathematics. In response to a comment I made on the panel, a participant offered the amalgamated word inqueery, on which I’m going to keep reflecting.

Queerness has been my entryway into understanding exclusion, but queering mathematics will not be enough, just as spaces dominated by cisgender gay men can sometimes be racist and toxic in other ways. The game historically called “cops and robbers” is an example of how dangerous ideas related to white supremacy and the police can subtly pervade even queer mathematics spaces.

Thank you for coming with me as I reflected on the ways that Ron asked us to queer our assumptions, Emily to queer our foundations, Juliette to queer our methods, and Anthony to queer our results. These talks will appear shortly on the Fields Institute’s YouTube channel, and I hope watching them through this lens encourages you to help further queer mathematics.

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