danceyourphd:representations ofthebraidgroups · dean, forwarded science magazine’s announcement...
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THE GRADUATE STUDENT SECTIONTHE GRADUATE STUDENT SECTION
Dance Your PhD: Representationsof the Braid GroupsNancy ScherichCommunicated by Alexander Diaz-Lopez
EDITOR’S NOTE. See the interview of Scherich byRachelCrowell atwww.ams.org/news?news_id=3826.
I learned about the Dance Your PhD competition manyyears ago, but thought it would be too impossible to turnmath research into dance. This past June, my boyfriend,Dean, forwarded Science magazine’s announcement oftheir 2017 competition to me and encouraged me tomake a submission. He said if ever there was someoneto figure out how to blend math and dance together, itwould be me. I have been a dancer and performer allmy life, and about a year and a half ago I started aerialdance lessons. Dean suggested that I use aerial dance todescribe my work with braids.
At first I was reluctant. But the more I relaxed and gavemyself permission to think outside of my math box, themore the ideas came to me. Sometimes I think all youneed is the right encouragement at the right time. In myvideo,1 I chose to describe braid group representationsand focus on the property of faithfulness, as my researchis highly motivated by the infamous open question offaithfulness for the Burau representation of braid groupsfor 𝑛 = 4.
Much like the braids in one’s hair, a braid is a diagramof tangled strands. The braid group on 𝑛-strands, denoted𝐵𝑛, is a group whose elements are certain equivalenceclasses of all the braids made with 𝑛 strands. The groupoperation is vertical stacking of braids. Alternatively, thebraid group can be presented by 𝑛− 1 generators 𝜎𝑖 and
Nancy Scherich is a fifth year graduate student at UC Santa Bar-bara studying representations of braid groups with Darren Long.Her email address is [email protected]/news/2017/11/announcing-winner-year-s-dance-your-phd-contest
For permission to reprint this article, please contact:[email protected]: http://dx.doi.org/10.1090/noti1657
Figure 1. Scherich was the overall winner across alldisciplines in the 2017 Science magazine Dance YourPhD competition with her creation of and dancing inher video “Representations of the Braid Groups.” Theannouncement says, “It involves linear algebra andmurder.” According to the video, many kernels areannihilated, but “some are extremely clever and hidein the matrices. It remains an open question formathematicians to find these kernels.” The creditsinclude her advisor Darren Long and her mom.
relations:
𝜎𝑖𝜎𝑖+1𝜎𝑖 = 𝜎𝑖+1𝜎𝑖𝜎𝑖+1 and 𝜎𝑖𝜎𝑗 = 𝜎𝑗𝜎𝑖
for 𝑖 in {1, 2,… ,𝑛 − 2} and |𝑖 − 𝑗| ≥ 2, as in Figure 2.A common way to study the braid groups is to look
at their representation theory. The Jones representationsof the braid groups are the representations where thegenerators have two eigenvalues. It turns out that much isknown about these representations, and in fact, they areparameterized by the Young tableaux in the sameway thatthe Young tableaux parameterize the representations ofthe symmetric group. The Burau representation is one ofthe Jones representations. All of the Jones representationshave a variable called 𝑞. The topic of my thesis is to find
420 Notices of the AMS Volume 65, Number 4
THE GRADUATE STUDENT SECTION
Figure 2. The braid group can be described by thetwo pictured types of relations: 𝜎𝑖𝜎𝑖+1𝜎𝑖 = 𝜎𝑖+1𝜎𝑖𝜎𝑖+1and 𝜎𝑖𝜎𝑗 = 𝜎𝑗𝜎𝑖.
careful specializations of 𝑞 to certain algebraic numberswhich then force the representation to map into a lattice.
Image CreditsFigures 1 and 2 courtesy of Nancy Scherich.Photo of Nancy Scherich courtesy of Dean Morales.
ABOUT THE AUTHOR
Nancy Scherich’s interests arelow dimensional topology, planaralgebras, quantum computation,dancing, aerial acrobatics, sewing,and welding.
Nancy Scherich
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April 2018 Notices of the AMS 421