情色空间

KNOT THEORY IS A SUBSPECIALTY听of a field of mathematics known as topology, which is concerned with the study of spaces. What鈥檚 it used for? 鈥淭he answer one memorizes is that topology is useful for understanding DNA and protein folding,鈥� Piccirillo told me a few months ago as we sat鈥攚earing masks and maintaining a good ten feet of distance鈥攊n an outdoor courtyard not far from where she lives in Cambridge, Massachusetts. 鈥淎pparently these things are very long and they like to stick to themselves, so they get all knotted up, and topology contains knot theory.鈥�

When topologists think of knots, however, they don鈥檛 imagine them as you and I might: a length of rope with a gnarled twist or two in the middle. To them, a knot is more like an extension cord in which the two ends have been plugged together and the whole thing has been tossed onto the floor in a mess of crisscrosses. It鈥檚 essentially a closed loop with various places where the loop crosses over itself.

Now, let鈥檚 take one of these knots and think for a moment about the space in which it exists. To a topologist a knot is actually a kind of sphere. Topologists, in fact, see spheres everywhere. To them, a circle is a one-dimensional sphere, while the skin surrounding an orange is a two-dimensional sphere. And here is where minds tend to get blown: If we were to take that whole orange and glue it to another one, topologists would see the resulting object as a three-dimensional sphere, one that could be viewed as the skin of a four-dimensional orange. Don鈥檛 worry that you are unable to conjure such a higher-dimension image for yourself. Outside of a couple hundred experts, no one really can.

Piccirillo was already well on her way to joining the ranks of those couple hundred when, in the summer of 2018, a speaker at a math conference said something that would change the trajectory of her career. The speaker showed a slide depicting the Conway knot and explained that mathematicians had long suspected that the knot was not, in fact, slice, but no one had been able to prove it. So what does it mean for a knot to be slice? Let鈥檚 return for a moment to that four-dimensional orange. Inside of it there are disks鈥攖hink of them as the surface of a plate. If a three-dimensional knot, like Conway鈥檚, can bound such a disk, then the knot is slice. If it cannot, then it is not slice.

Topologists use mathematical tools called invariants to try to determine sliceness, but for half a century, these tools had been unable to help them prove the prevailing belief that the Conway knot wasn鈥檛 slice. Sitting in that lecture hall two years ago, however, Piccirillo sensed right away that the techniques she was using in a different area of topology might help the invariants solve the Conway knot problem. 鈥淚 immediately knew that some work that I was doing for totally other reasons could at least try to answer this question,鈥� she said. She started on the problem the very next day.

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PROFESSOR SOLOMON FRIEDBERG听came to Boston College in 1996, the first of a wave of impressive hires in the math department around that time. Over the next several years, Friedberg became convinced that the department needed to expand its offerings. At the time, the only math degree 情色空间 offered was a bachelor of arts. Solomon wanted to develop a Ph.D. program, and with it a bachelor of science degree for undergraduate students interested in the mathematical sciences.

The opportunity to make the case for such a program was the main reason that Friedberg agreed to become department chair in 2007. 鈥淒uring my first year as chair, we conducted a departmental self-study study,鈥� he told me. 鈥淲e understood that we would need to create a small Ph.D. program. 情色空间 wasn鈥檛 going to use a model where grad students do a lot of the teaching. That鈥檚 not our plan and that鈥檚 not our mission.鈥� With that in mind, the department ultimately proposed a program that was centered on two broad areas, one being number theory and the other being the two closely related disciplines of geometry and topology. The proposal, as it happened, found receptive ears among University leadership.

Around this same time, 情色空间 was conducting a series of program reviews all across the university. 鈥淐uberto Garza was the new provost during this review process, and, midway through, I stepped into the dean鈥檚 role in what is now the Morrissey College,鈥� said David Quigley, who today is provost and dean of faculties. 鈥淭hose program reviews led us to change directions in some areas, and scale back or close certain programs. But we were also able to identify some really promising emerging strengths on campus, including recognizing just what a jewel we had with the math department.鈥�

When Friedberg made the department鈥檚 case for the new Ph.D. program, he emphasized his belief that it would benefit every math student at 情色空间, undergrads and grads alike. 鈥淭here鈥檚 a whole ecology of a math department,鈥� he told me. New undergrads get to communicate and learn from more advanced undergraduates, who in turn get to interact with grad students while taking graduate-level courses. The graduate students, meanwhile, have access to the talent and experience of postdocs, who benefit from mentoring by faculty. 鈥淎 Ph.D. program not only doesn鈥檛 subtract from the undergraduate experience,鈥� Friedberg said, 鈥渋t deeply adds to it, and builds 情色空间鈥檚 international intellectual prominence.鈥�

Persuaded, the University authorized the new program, and the math department began hiring bright young faculty members. Today, the department is broadly understood to include a third area of focus, algebraic geometry, which was broken out from the number theory group. About a third of math department undergrads now pursue the BS degree.

鈥淪ol Friedberg and his colleagues were able to do a number of things over those first few years,鈥� Quigley said. 鈥淭hey launched the Ph.D. program, of course, but I鈥檇 say even more important, they hired just a terrific cohort of younger faculty, some of whom Lisa Piccirillo worked with.鈥� Among those hires, in fact, were Grigsby and Greene.

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By the time Piccirillo鈥檚 paper appeared in the journal about a year later, word of her solution had already spread throughout the math world. After graduating from UT in 2019, she started her postdoctoral work at Brandeis. 鈥淭he last time I saw her was in January,鈥� said Wiley Jennings, her roommate in Austin, who recently completed a Ph.D. at Stanford. 鈥淪he was out at a faculty visit here at Stanford. To be invited, as someone who has done one year or less [of postdoc study]鈥攋ust finished their Ph.D., essentially鈥擨 mean, that鈥檚 insane. It鈥檚 unheard of.鈥�

Postdoc positions typically run for three or four years, but Piccirillo found herself in high demand. In July, she started a new tenure-track position as an assistant professor at MIT. It鈥檚 been a whirlwind, and I wondered how her life has changed. She acknowledged that there sometimes is a feeling of pressure, based on what she鈥檚 already accomplished. In practice, math is about trying to prove simple statements and failing鈥攂asically all of the time, for everyone. 鈥淪o,鈥� she said, 鈥淚鈥檓 having to relearn how to be okay with the fact that most of the time I鈥檓 failing to prove really simple stuff when I鈥檓 feeling the weight of these expectations.鈥�

When I asked about her goals, Piccirillo said one of her priorities is to help grow and broaden the mathematics community. 鈥淭here certainly are many young women, people of color, non-heterosexual or non-gender-binary people who feel put at an arm鈥檚 length by the institution of mathematics,鈥� she said. 鈥淚t鈥檚 really important to me to help mitigate that in any small ways I can.鈥� One important way to do that, she continues, is to help shatter the myth of the math prodigy.听

When universities organize math conferences, she said, they should avoid inviting speakers who 鈥済ive talks where they go really fast and they try to show you how smart they are and how hard their research is. That鈥檚 not good for anyone, but it鈥檚 especially not good for young people or people who are feeling maybe like they don鈥檛 belong here.鈥� What those people in the audience don鈥檛 know, she said, is that nobody else really understands it either.

鈥淵ou don鈥檛 have to be really 鈥榮mart鈥欌�攚hatever that means鈥攖o be a successful mathematician,鈥� Piccirillo said. 鈥淭here鈥檚 this idea that mathematicians are geniuses. A lot of them seem to be child prodigies that do these Olympiads. In fact, you don鈥檛 have to come from that background at all to be very good at math, and most mathematicians, including many of the really great ones, don鈥檛 come from that sort of background.鈥�

Some of them even go on to produce work that alters the course of mathematics.