*Credit: Lucy Reading-Ikkanda/Simons Foundation; Models by Alba*

# Mathematician Bianca Viray Invites Everyone to the Table

Conversations with Bianca Viray, who is described as “precise” and “intentional” by peers, usually include long pauses as she searches for exactly the right words. Those silences happen whether she’s describing a complicated mathematical argument, the necessity for change in the mathematical community, or details from her life.

“She makes things seem very natural, but things don’t really happen by accident when Bianca is around,” says mathematician Anthony Várilly-Alvarado, one of Viray’s most frequent collaborators, of this intentionality, which also shows up in her math. “She’s extremely good at highlighting ideas which get to the heart of what is happening: you choose the right example or you rephrase an old theory in just the right way to see the new application. By recasting old results in new ways, she has pushed very cool ideas in arithmetic geometry.”

Those mathematical ideas, in particular her novel methods of dealing with a tool called the Brauer-Manin obstruction, led to her being named a 2020 Simons Fellow in Mathematics and a 2021 Fellow of the American Mathematics Society. Viray, who is still relatively early in her career, received a nearly half-million-dollar National Science Foundation CAREER grant in 2016 when she was just 32. This followed a National Security Agency Young Investigator grant in 2014, which in turn was awarded on the heels of an NSF postdoctoral fellowship from 2010-2014.

But she’s not just known for her research. Viray is also an active organizer in the mathematical community, especially as a proponent against racial and gender discrimination and sexual harassment in the field. In 2022, she was named an Association for Women in Mathematics Fellow.

“Sometimes it feels like there’s a narrative of ‘you do great math or you do other things’: If you do any ‘other things,’ that means you don’t do great math,” says Viray, a professor of mathematics at the University of Washington (UW), reflecting on her career. “I think that’s false.”

There is a growing bloc of mathematicians who believe that doing “other things” actually leads to more “great math” rather than stymie it. Brendan Hassett, a professor at Brown University and one of Viray’s long-time colleagues, says that she bucks the stereotype of a math professor as a “male mathematician sitting alone proving some 100-year-old theorem in complete isolation.”

“That archetype is not the heart and soul of what drives progress in the field,” says Hassett, who co-organized a workshop and co-edited a book with Viray. Hassett still remembers Viray’s 2019 talk on modular curves from the annual meeting of the Simons Collaboration on Arithmetic Geometry, Number Theory and Computation, which inspired many attendees to investigate “isolated points,” a new concept she defined with her co-authors. “Despite the mythology around the isolated genius, building community and encouraging interactions is really crucial for the health of mathematics.”

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**An organizer from the start**

From the beginning, Viray’s course seemed destined for both research greatness and challenging the status quo in the community. Her Ph.D. advisor, Bjorn Poonen, wrote in a recommendation letter during her second year of graduate school in 2006 that “there is every indication that she will develop into a mathematical leader, not only as a researcher, but also as an organizer.”

Long before she knew what math research was, Viray knew the importance of both organizing and feeling like you belong. Born in the Philippines to an English mother and Filipino father, Viray started life in the U.S. as an undocumented immigrant.

After getting her green card as a teenager, Viray became more acutely aware of the racism her father faced, and how differently she was treated based on which parent she was with. For instance, she watched her brown father regularly get chosen for “random checks” in security lines while she, who could pass as white, and her white mother always whizzed through.

Racism wasn’t the only inequity young Viray noticed. After receiving a full scholarship offer to the honors college at the University of Maryland, Viray majored in math and made a habit of counting the number of women students in her classes. She could count on one hand the women in a course with over 100 pupils. Though she stood out in the male-dominated math and computer science majors, Viray made more friends and found a sense of belonging in the Filipino Community Association, in which she took on leadership and organizing roles.

After graduating, she headed to the University of California, Berkeley for graduate school, where she quickly became an officer in the math grad student association. For years before she came, moving graduate offices would stretch “forever”. First-year students couldn’t set up shop for months into the semester. This had a destabilizing effect on their transitions to graduate school because they lacked a set place for their books, computers or quiet work — no place to belong. Viray’s first order of business was to institute a “moving day” for graduate students, so first years could get on track quickly, a change that was much appreciated by her fellow students. The desk story exemplifies Viray’s strength in subtly causing small changes that can greatly affect others. The same is true of her research, which has guided the careers of many younger mathematicians.

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**Learning to learn from mistakes**

Viray’s field is called arithmetic geometry, which deals with algebraic varieties: shapes that are ‘cut out’ by systems of polynomial equations. Arithmetic geometers want to know if these varieties have rational points, or solutions that can be written as fractions. For instance, the Pythagorean theorem gives infinitely many rational points for the variety *x*^{2 }+ *y*^{2 }= *n* when *n *is a perfect square, including (3/2, 2, 5/2) and (5, 12, 13).

But other values of *n *may or may not have rational points: *n* = 2 gives infinitely many rational points, while *n* = 3 yields no rational solutions. One convenient way to tell if certain varieties have rational points is to use the Hasse principle, formulated in 1924 by the German number theorist Helmut Hasse. It says that a rational point exists if you can: 1) find a solution in the real numbers (which include irrational numbers such as π) and 2) find a solution in all of the *p*-adics, number systems based on the primes.

However, the Hasse principle doesn’t always work, and from the 1940s to the 1960s, mathematicians constructed several examples that had no rational points, but which did have a real solution and solutions in all the *p-*adics. In the 1970s, the Russian algebraic geometer Yuri Manin explained all those exceptions away with what’s now called the Brauer-Manin obstruction using what’s called the Brauer group of a variety (which Viray calls a “slippery and mysterious object”). In 1999, the Russian mathematician Alexei Skorobogatov finally found an example that fails the Hasse principle where the failure wasn’t explained by the Brauer-Manin obstruction.

Six years after Skorobogatov’s discovery, Viray entered graduate school. Even though it had taken nearly 30 years to find that first example — and most graduate students try to start with a tractable problem — Viray tasked herself with an ambitious project: She wanted to prove that another class of varieties, called Enriques surfaces, failed the Hasse principle and couldn’t be explained with the Brauer-Manin obstruction.

Viray’s perseverant nature well-equipped her for the esoteric problem. Várilly-Alvarado, who met Viray during her first week of graduate school and has since published four papers with her, says, “She has the strongest stomach for hardcore calculations that I’ve just ever seen in a human being.”

But calculations by hand necessarily come with mistakes; dealing with those mistakes can make or break a mathematician. That’s exactly why Viray caught Poonen’s eye during her first year at Berkeley when she pointed out a mistake he had made in a course he was substitute teaching.

“I continued to be impressed not only by her interesting and clearly posed questions about the material, but by her ability to even discover errors in the [text]book,” says Poonen, who is now a professor at the Massachusetts Institute of Technology. “Like everyone, she would sometimes make mistakes herself, but I could see early on that she was exceptional in terms of being able to correct her own errors and learn from them.”

Viray needed that skill for her thesis problem. Elements in groups can be classified as either “algebraic” or “transcendental”: in Enriques surfaces, the Brauer group contains a single transcendental element. Transcendental elements make calculations much trickier, and the initial problem proved too difficult for a single graduate student. However, Viray did succeed in finding examples of Enriques surfaces that fail the Hasse principle. These examples lacked *algebraic* Brauer-Manin obstructions, a particular sub-case of the obstructions that are easier to work with than the more complex *transcendental* Brauer-Manin obstructions. That work, while less ambitious than her original project, was still impressive enough to earn her multiple job offers, including the one she accepted at Brown University.

Before her dissertation defense, Poonen pointed out a small error in a calculation, and Viray cheerfully set to it on a plane ride. But by the time she landed, she was “bawling” as she realized that her entire theorem was false. It turned out that her examples were in fact explained by the algebraic Brauer-Manin obstruction. She ended up reluctantly writing up a subset of her existing work for her thesis.

Despite the excitement of her soon-to-begin postdoc position at Brown, Viray’s uneven experience as a graduate student contributed to feeling that she was at her nadir as a research mathematician. “I just had no ability to work on it on my own. … I felt like I couldn’t do any math,” she says. That summer, she took another stab at fixing the small error from her thesis work and ended up writing her first joint paper with Várilly-Alvarado. “Working with Tony helped me find the joy in math again. Without that, my confidence was so shot that I probably would have left math.”

That summer marked the first of an annual tradition of weeklong collaboration visits, and yet another attempt at Viray’s tricky initial problem. She and Várilly-Alvarado had hoped to find an example of an Enriques surface without any Brauer-Manin obstructions. Though their efforts came up short (their resulting paper described an example that failed the Hasse principle without an algebraic Brauer-Manin obstruction only, fixing Viray’s initial thesis draft), it was a step in the right direction and the beginning of their fruitful long-term collaboration.

**Pursuing that problem while building community**

During her postdoc, Viray continued to publish papers on Brauer groups and other failures of the Brauer-Manin obstruction, but Enriques surfaces kept gnawing at her. By 2014, Viray got a tenure-track professor position at UW and decided to re-focus on her original aim. Remembering how her experience with Várilly-Alvarado helped her regain her confidence, she decided to invite early-career researchers to collaborate on this problem.

“All the ingredients were there to finally go back and finish off that original goal that we had set in 2010,” says Várilly-Alvarado, now a professor at Rice University. “I think she could have finished it all off by herself, but instead she took the project to a workshop and mentored four people on the techniques and the ideas of all the previous papers. Then, together, they ended up finishing the paper.”

That paper on Enriques surfaces makes use of general techniques that Viray had developed in previous papers: one written with Várilly-Alvarado; two written with Brendan Creutz, her other most frequent collaborator; and one with two postdocs and another professor written at the workshop she co-organized with Hassett. The paper proves that the surface constructed by Viray and Várilly-Alvarado back in the summer of 2010 — which had no algebraic Brauer-Manin obstruction — *also *has no transcendental Brauer-Manin obstructions, and so Enriques surfaces’ failure of the Hasse principle could not be attributed to the common obstruction. This critical finding added another example to the short list begun in 1999 by Skorobogatov.

“It wasn’t just the research; there was a lot of community building at the same time,” Várilly-Alvarado says. “There’s a huge arc that goes over the better part of five or six years; you can see that at every stage, she’s left a trail of people who now know about this stuff, who have gone on to work on arithmetic of surfaces and more general varieties in other contexts.”** **

The workshop that spawned the Enriques-surfaces paper was the third iteration of the trailblazing Women in Numbers (WIN) conference, which Viray first attended as a graduate student. At the time, most math conferences focused on sharing results and networking, but were not a place to actually do math research. WIN was different. It brought together small groups with the explicit aim of publishing a paper with the work done during that week — a format since copied by many other new “Women in…” conferences and other workshops. After attending, leading projects and co-organizing the annual workshop, Viray is now a member of the WIN steering committee.

“She is a role model as someone who was able to navigate being a woman in a subfield that for a long time didn’t have many women, and still do fantastic math,” says Jen Berg, who as a graduate student was a co-author on the Enriques-surfaces paper and is now a professor at Bucknell University. “She’s always been there to provide some encouragement or nuggets of wisdom, or just share a personal story to remind me that there’s a community. We’re not alone.”

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**Pushing for change **

Besides mentoring younger women and disseminating new mathematical ideas, Viray has also been a sounding board and microphone for women’s experiences with harassment.

“She uses her platform to talk about her own experiences, quite bravely, and also to talk about the issue in general and what it means for the community,” says Michelle Manes of the University of Hawaii, Viray’s collaborator and friend. “Because she is such a spectacular, remarkable scientist, she gets big speaking invitations and she leverages that — she’s standing up in front of big crowds and saying things.”

All of the mental energy that women and other people of minoritized identity have to put into their safety takes away from the research they could be doing. That’s why Viray shares her experiences, makes safe spaces for others to speak, and calls out the community for inaction on issues of diversity, equity and inclusion.

Viray augments those personal actions with her institutional positions. Besides chairing the diversity committee in her department at UW, she is one of the youngest members of the AMS Council, a select group of researchers who advise and steer the scientific program of the American Mathematical Society (AMS). In addition to WIN and the workshop organized with Hassett with the similar format, Viray has organized workshops and a course on communicating mathematics effectively, and recently co-founded a new AMS initiative to connect and guide graduate program directors to encourage diversity and equity in their programs.

“I guess for me, the question is, why wouldn’t you care?” Viray says, when asked why she tries so hard to make the math community accessible and welcoming. “I’ve realized there’s been so many times where my ability to do math was not about the math that I knew. It was about the atmosphere that I had, in both good and bad ways. An environment of mutual respect and faith in one another nurtures us to dream and be bold and create inspiring mathematics.”