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First, a bit of a primer on some mathematical termsmight be in order. Addition is an example of a “commu-tative” operation, because a + b equals b + a. Subtractionis an example of a “non-commutative” operation, be-cause generally, a – b does not equal b – a.

Mathematics, in general, is the study of patterns and,frequently, such patterns are described via systems ofequations, Vancliff explains. For instance, systems ofpolynomial-style equations and their solutions play acritical role in almost every scientific field, including el-ementary-particle physics, quantum mechanics, robot-ics, crystallography and more.

“Often, the solutions cannot be found by experimen-tation, and often they are not numbers but are functionsand so, in general, they do not commute,” said Vancliff,a UT Arlington professor of mathematics. “Non-com-mutative algebra has application to fields such asphysics and chemistry. The science of seeking methods

that find all solutions to a system of polynomial-styleequations in non-commuting variables is non-commu-tative algebra.

“The problem of solving a system of equations innon-commutative algebra may be translated to one in-volving an algebra over a field, and the representationtheory (or module theory) of that algebra. My researchis in the subarea of non-commutative algebraic geome-try, which is about using geometric methods to under-stand the algebra and its representation theory thatarise in this way.”

The main idea to finding solutions to a system ofpolynomial-style equations is to associate an algebraicobject, called a ring, which encodes all the properties ofthe original equations, Vancliff said. Associated to thisring are modules, which encode all the properties of thesolutions to the equations. So, in order to find all of thesolutions, one should find all the modules for the asso-

ciated ring. In many of the applications, the rings tendto share certain properties satisfied by commuting poly-nomials. Such rings are called AS-regular algebras andare the main focus of Vancliff’s research.

“One of the goals of the study of AS-regular algebrasand their modules is to use geometric techniques to findcertain modules of the AS-regular algebra, and then touse those modules to find the modules that give the so-lutions to the original system of equations,” she said.“My underlying goal throughout my research career hasbeen to improve on these geometric techniques. Beingan algebraist, I don’t work on the physics that generatesthe equations that need to be solved; nor do I work onthe equations themselves.

“Typically, mathematical physicists translate thequantum physics into algebraic problems, and then analgebraist picks up the problem at that stage. In my case,I work on techniques that solve types of equations, in

Brandon Wade

Vancliff discusses a problem with students inher Graduate Algebraic Geometry class last fall.

he subject of non-commutative algebraic geometry is likely to be baffling or even slightly intimidating to thoseunfamiliar with advanced concepts in algebra. For Michaela Vancliff, it’s the subject she has devoted her careerto studying.Vancliff has been at the forefront of research in non-commutative algebraic geometry since it firstcame on the scene in the late 1980s.

So, what is it, exactly?

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Michaela Vancliff thriveswhile focusing her researchand teaching on a branch of algebra that even manyother mathematicians find to be mystifying. By Greg Pederson

43Maverick Science 2013-14

the hopes that such techniques might not only solveequations of current interest, but also any equations ofthe same type that might arise in physics in the future.”

Among Vancliff’s projects which could benefit re-search in physics is one with Thomas Cassidy, professorand math department chair at Bucknell University,which focuses on generalizing the notion of Clifford al-gebra and using geometry to motivate their proposedgeneralization. Clifford algebras, which offer a directway to model geometric objects and their transforma-tions, have numerous applications in physics.

“Our work together involves a variation on classicalClifford algebras. Clifford algebras have many applica-tions in theoretical physics, and we have been stretchingthese ideas to encompass a broader family of algebraicstructures, with the hope of finding applications inquantum physics,” Cassidy said. “Michaela is a remark-ably tenacious researcher. Mathematical research isoften characterized by sudden realizations or insights,but those insights can only come after prolonged andintense study. Michaela has the mathematical drive andvision to delve into very abstract concepts and find hid-den connections.”

Vancliff is also working with UT Arlington associateprofessor of math Dimitar Grantcharov, an expert inrepresentation theory, on a long-term project aimed atdescribing a recently defined ring, called a graded skewClifford algebra, in terms of a Lie bracket and Poissongeometry, and using that information to classify certainmodules of the ring. Vancliff and Cassidy were the firstones to propose the idea of the graded skew Clifford al-gebra, in 2010.

“Dr. Vancliff is a very talented mathematician andshe is highly dedicated to research, teaching and serv-ice,” said Jianzhong Su, professor and chair of the mathdepartment. “The field she works in is quite abstract,even for other mathematicians, but this kind of mathe-matics is reflective to some of the deeper insights inmodern physics. Dr. Vancliff is one of the leaders in thisresearch field, and the importance of her work is widelyrecognized by the scientific community.”

n analytical mind was seeminglyhard-wired into Vancliff’s DNA.She was born in England, north-east of London in Essex County,and her father was an electricalengineer. He instilled in her a de-sire to understand how thingswork. Initially, she was interested

in physics, but in high school she was inspired by one ofher math teachers, who felt strongly that Vancliff shouldpursue a mathematics degree. The teacher remained afriend and mentor until his passing a few years ago.

“In hindsight, I believe I was interested in structuralpatterns that make stuff work, more than in the physicalinner workings, and that interest translated into a de-sire to understand the mathematics that explains howstuff works,” she said. “From the entire cosmos downto the tiniest flower petal, mathematics is behind thescenes making it all work, and I wanted to understandit all.”

After finishing secondary school, Vancliff enrolled atthe University of Warwick, in Coventry, England whereshe earned a B.S. in Mathematics in 1986. Her degreewas in pure mathematics, but she also took courses inquantum physics, special relativity and general relativ-ity. While at Warwick, one of the math faculty memberssuggested that she pursue a graduate degree in theUnited States, at the University of Washington.

“He was very familiar with the university and the cityof Seattle, and he felt that I would excel in that environ-ment,” Vancliff said. “Since I very much enjoyed livingin Seattle and being a student at the University of Wash-ington, I guess he was right.”

She taught high school math in London for a yearafter graduating from Warwick, then moved to Seattleand began work on a Ph.D. in mathematics with the in-tention to focus on applied math. During her first year,while taking a mandatory graduate algebra class, shewas introduced to the notion of a module over a ring.

“I immediately recognized it as a generalization ofthe idea of matrices acting on a vector space, which is apervasive topic throughout all of the applied sciences,”Vancliff said. “So a light bulb lit up in my brain, and Ifell in love with algebra.”

Her interests were mainly in physics applications, soVancliff was drawn to study modules over non-commu-tative rings. At that time, the study of modules overcommutative rings had been ongoing for decades dueto the use of geometric techniques. In a bit of perfecttiming, Vancliff began her graduate studies just as a newmovement had started in the world of algebra thatpushed the study of non-commutative rings and theirmodules via geometric techniques. This new subject be-came known as non-commutative algebraic geometry.

“This was an entirely new subject, and I was fortu-

nate to enter it at its inception,” she said. “There weremany open problems ripe for the picking and many thatwere accessible to junior researchers such as myself.This meant that I was able to make groundbreakingcontributions to the subject while I was still a student,simply because the subject was so new. I found it to bevery exciting.”

Vancliff says that she was fortunate that her Ph.D.advisor at the University of Washington, S.P. Smith,took his responsibilities very seriously.

“Not only does he have a rare gift for being able toexplain mathematics and its intricate beauty, but he alsodevoted time to teaching me how to write research pub-lications and funded my participation at conferencesand workshops,” Vancliff said. “He actively encouragedme to network and interact with famous mathemati-cians, and those opportunities proved to be invaluableto me throughout my career.”

Vancliff earned her Ph.D. in 1993, and then workedfor two years as a visiting assistant professor at the Uni-versity of Southern California in Los Angeles. Fromthere, she moved to Belgium and worked for a year as aresearcher at the University of Antwerp, before spend-ing two years at the University of Oregon in Eugene. In1997, Vancliff had a conversation with a colleague inOregon which led to a significant career decision. The

colleague had family living in Fort Worth and was fa-miliar with the North Texas region.

“He felt very strongly that I would be happy workingat UT Arlington and living in the DFW area, so he en-couraged me to apply to UT Arlington,” Vancliff said. “Iinvestigated UT Arlington online and found a vibrant,growing university.”

When she interviewed in 1998, she found that thedepartment was strong in applied mathematics and lessso in algebra, which Vancliff took as a “positive chal-lenge”. She joined the faculty as an assistant professorstarting that fall.

“The department has changed much in the past 15years, and its research has grown in strength, with manyof the current faculty earning research grants,” she said.Her own research has been continuously funded sinceher arrival.

ancliff is working with her grad-uate students on several proj-ects. She and third-year doctoralstudent Richard Chandler arelooking at the point schemes andline schemes of a family of alge-bras and trying to understandthe algebras' underlying struc-

ture. Chandler first met Vancliff as an undergraduate inher Abstract Algebra class in 2010. He earned a B.A. inMathematics with a teaching certification in May 2011and entered the Ph.D. program that fall. He wants to gointo academia after earning his doctoral degree.

“Dr. Vancliff is an amazing mentor,” Chandler said.“She has very high standards for all of her students, butthey are always reasonable. She doesn’t expect perfec-tion, but she does expect that you put 100 percent intoall aspects of your work.”

Padmini Veerapen studied under Vancliff andearned a Ph.D. in May 2013. She’s now an assistant pro-fessor of math at Tennessee Technological University inCookeville, Tenn.

“Dr. Vancliff emphasized a level of detail and thor-oughness during my years under her supervision that isallowing me now to successfully handle all my respon-sibilities as a faculty member,” Veerapen said.

In addition to her research and teaching, Vancliff isthe organizer of the long-running DFW AlgebraicGeometry, Algebra and Number Theory (AGANT) sem-inar series, which brings together researchers and stu-dents from academia and industry in the Metroplex andbeyond and features national and international speak-ers. She also created the department’s Graduate Forum,which helps junior Ph.D. students by letting them talkwith faculty mentors and senior doctoral students.

“It is very fulfilling and satisfying to share my knowl-edge with my students and see them enjoy the materialas much as I do,” Vancliff said. “When they see the con-nections that I see and share with me their delight infinding new connections, I can see how much they havegrown mathematically, and that is a joy to witness. It isparticularly exciting to see them continue a researchpath after graduation, especially in academia where theycan continue this sharing of knowledge with the nextgeneration of students.”

Vancliff says that helping and sharing her knowledgewith others, working as part of a team, and using thetechnical expertise she has acquired to solve problemsare all rewarding aspects of her job as a researcher andeducator.

“I very much enjoy that my work at UTA entails allthese components, both individually and in combina-tion,” she said. “I also find that my success at earningresearch grants renews my energy, not only in the re-search arena, but in all aspects of my job. I consider my-self very fortunate to be able to work in my chosencareer, and in the supportive environment of UTA.” n

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VV“Michaela is a remarkablytenacious researcher. [She] has the mathematical driveand vision to delve into veryabstract concepts and findhidden connections.”

– omas CassidyBucknell University professor of mathematics