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Since 2004 I have been publishing papers cowritten with with undergraduate students. I didn't set out to become an undergraduate research mentor; it just sortof happened. The facts are simply that (1) I'm perfectly willing to work with motivated students with any level of preparation and (2) nearly all my students have been undergrads. It follows that I do a lot of projects that involve undergraduate research students.
Having been a McNair Scholar as an undergrad myself, I replied that undergrad students can definitely contribute to research. The student, Gabriel Murillo, asked if I had any project ideas he could work with me on. I said I'd see what I could come up with, and within a few days we had a project idea -- continuing the computation of isomorphism classes of finite Alexander quandles using a theorem from my dissertation. We worked on the project over the summer, completed the required computations, and wrote up a paper which we posted to arxiv.org and submitted for publication. It has now appeared the Journal of Knot Theory and its Ramifications .
Natasha chose the project idea of using quandle difference invariants to detect nonclassicality in virtual knots and links. We wrote some Maple programs using the method for computing the quandle counting invariant symbolically from a quandle matrix developed in my paper with Richard and Todd together with a program for generating all 4-crossing Gauss codes, and we found that some 85% or so of non-evenly intersticed 4-crossing Gauss codes with nontrivial counting invariant values for the six smallest connected quandles have nonclassicality detected by quandle difference invariants. Our paper appeared in Topology Proceedings following our presentation of our work at the 2006 Spring Topology and Dynamics conference in North Carolina.
Rohit opted to return to his native India and pursue a career in medicine, a loss for mathematics but a gain for medicine. Anthony chose to work on an open-ended kind of project about studying Latin quandles, i.e. finite quandles whose operation matrices form Latin squares. Such quandles are also distributive quasigroups, and we read a few papers from the quasigroup literature but have not yet been able to prove our main conjecture, namely that all Latin quandles are Alexander. We know from our computer searches that the conjecture is true at least up to order 8. Unfortunately, we didn't feel that we had enough new material to justify a paper, and when Anthony graduated, we put the project on hold.
The winter quarter of my third and final year at UCR saw several more requests from students for project ideas. John and Natasha both wanted to do a second project, and two new students, Conrad Creel and Daisy Lam, asked me for project ideas. Conrad was the first student to ask me for a project without having taken a class with me, though not the last. Daisy had taken my linear algebra class the previous summer. John and I enlisted the help of Jim Dolan, a member of UCR's occasional quandles group (other members included John Baez, Xiao-Song Lin, Alissa Crans and Derek Wise) who was studying a connection between the roots of the Jones polynomial and the fundamental Alexander quandle of a knot. We embarked on a project to define an extension of the Kauffman bracket polynomial using a virtual crossing as a kind of smoothing. We were able to define the invariant but unable to determine whether the new invariant was really new, and we ultimately decided to table the paper.
For my second paper with Natasha, we decided to look at the counting invariants associated to quandles with trivial orbits. We found that we could use the counting invariants associated to a specific family of quandles to recover the linking number of a two-component link. This is of interest since, as a complete invariant of knots and unsplit links in S3 up to reflection, the knot quandle should determine nearly all of the other invariants. In particular, understanding how various knot and link invariants arise from the knot quandle has the potential to tell us a lot about how these invariants are related to each other. Our paper has appeared in the Journal of Knot Theory and its Ramifications.
After reaching the three year limit as a VAP at UCR, I returned to Whittier College for a year. There I met two more students who wanted to do undergraduate research.
That spring, when the tenure-track position I had accepted at California State University, Dominguez Hills was revoked due to budget issues, I was quickly offered a VAP position for one year at Claremont McKenna College, which I was happy to accept.
Early in my visiting year at CMC, a student from neighboring Scripps College, Johanna Hennig, dropped by my office to ask me to be her senior thesis advisor. We decided to look at an enhancement of the quandle and rack counting counting invariants using finite groups we call "column groups". Our paper is currently in peer review. While taking a year off before graduate school, Jose suggested we start work on a second paper. We selected a couple of papers to read and started meeting semi-regularly. After our initial idea started to fizzle, we hit on the idea of using "shadow colorings" to extend our previous work on virtual Yang-Baxter cohomology. Our second paper is currently in preparation. Tim's senior thesis project involved surface biquandles, the algebraic structure determined by dividing knotted surfaces in 4-space into semi-sheets at the singular set and getting axioms determined by the Roseman moves. Tim was sufficiently independent and I was sufficiently busy that we felt Tim should write his work up as a solo paper. Tim's paper is currently in preparation. Tim and Johanna both gave excellent talks at the 2009 Pacific Coast Undergraduate Mathematics Conference at the University of California at Riverside. Wesley, Jose, Jessica and I came along to lend support as well as to see the other talks. As the spring semester wore on and I hadn't had a single job interview, I started to prepare for an eigth year as a visiting professor. Then one morning while preparing my lectures, my department chair stopped by to tell me I should be hearing from the Dean. Much to my surprise, Claremont McKenna College had decided to create a new tenure line in order to offer me a tenure-track position. I could not have been happier to accept.
In July of 2009, I gave an invited address at the UnKnot conference at Dennison Univeristy in Ohio, accompanied by Jose, Jessica and Scott. All three students gave outstanding talks about our various projects.
As the curtain rises on my first year as a tenure-track assistant professor at CMC, I look forward to completing my current projects with students as well as doing new projects with my new and current student collaborators. Want to add your name to this list? Drop by my office hours! |
It all started during my first year as a VAP at the University of
California at Riverside. One afternoon, a former student from my Vector
Calc II class from the previous quarter dropped by my office hours to
say "hi" and ask me about undergraduate research. He wanted to know,
was it really possible for an undergraduate to contribute to research?
Toward the end of that summer, I ran into a former student from my
topology class, Benita Ho, at the gym. When I mentioned that I was
finishing a paper I was writing with a student, she asked if I had
any other project ideas. I said I'd see what I could come up with,
and after a few days I contacted her with a project idea involving
representing finite quandles symbolically as matrices. We met
informally about once per week during the fall quarter of my
second year at UCR, and we posted our
Another student from my topology class, Chau-Yim (Jason) Wong,
dropped by my office to say hello early that fall quarter, and when
I mentioned the paper I had just completed and the project I was
working on, Jason asked if I had any other ideas. Indeed I
did -- the matrix representation of finite quandles Benita and
I had developed made it clear how quandles can be broken down into
subquandles, and we decided to study this type of decomposition
further. Ultimately, we wrote a
The spring quarter of my second year at UCR, Gabe was taking my topology
class and wanted to do
a second project with me; we settled on trying to find a method for
determining whether a finite quandle is isomorphic to an Alexander
quandle. One day Gabe was waiting outside my office working on the
project when Anthony Thompson, a classmate from my topology
class, stopped to ask what he was doing. Gabe explained the project
and Anthony decided to stick around for our meeting. Indeed, Anthony
later told me he stayed up all night thinking about the problem, even
missing a few classes the next day as a result. We devised and implemented
an algorithm which finds all Alexander presentations of a finite
quandle from its matrix representation. The resulting
That spring quarter, another former student, Todd Macedo, asked me about
doing a project. Todd was a computer science major, so I put him to
work on a distributed algorithm version of the finite quandle computer
search program from my paper with Benita. We enlisted the aid of my
friend Richard Henderson of Red Hat Software, who offered to run our
search on his personal network. After an initial run of several hours
on several processors, Richard modified our algorithm such that the
n=6 case completed in under 2 seconds on a single processor, and
was able to get the n=7 and 8 cases, while the n=9 case
is still out of range even with a large network. Our
Meanwhile, a freshman from my calculus class that spring, Natasha Harrell,
started showing up to my research meetings with Anthony and Gabe. Soon
enough, she asked me for a project idea. Around the same time, another former
student from a vector calc class, John Vo, asked me about doing a project.
By this time I had started keeping a running list of project ideas I thought
were suitable for joint work with undergrads. Anthony wanted to do another
project, as did Rohit Jain, another former calculus student. I sent the four
of them a list of four project ideas to choose from. 
John chose the project of extending the symbolic matrix representation of
quandles from my paper with Benita to biquandles, a generalization of
quandles. We were able to classify all biquandles with two, three and
four elements using our method as well as write Maple software for computing
the biquandle counting invariants for any knot or link given its Gauss
code. One of the four element biquandles we found detects the non-triviality
of all of the Kishino knots, not an easy thing to do. Our
Daisy and I tried out several research ideas before finally settling on
the idea of trying to extend my classification theorem for finite Alexander
quandles to finite Alexander biquandles. We were able to show that two
Alexander biquandles are isomorphic iff their (1-st) submodules are
isomorphic and they have sets of coset representatives satisfying certain
extra criteria. These extra criteria are always satisfied if the Alexander
biquandles are actually Alexander quandles, but need not be for general
Alexander biquandles. The resulting
Conrad, who is a software developer as well as a student of
mathematics, proposed working with me on a symbolic computation project
relating to knot theory.
This time I suggested we write software to compute Yang-Baxter cocycles
of finite biquandles and the knot and link invariants they define, using
the symbolic matrix representation of finite biquandles from my paper
with John Vo. The cool thing about this is that the software works in an
algebra-agnostic way -- you don't actually have to know formulas for the
biquandle operations, just the biquandle matrix. Our
For the final summer that I spent at UCR, I had one more student who asked
for a project, Esteban Adam Navas. Like Conrad, Adam had not taken any
classes from me, though he did sit in one one of my calculus lectures after
being invited by a friend who was in my class. For this project we decided
to study a type of finite quandle defined in terms of a symplectic form
on a finite vector space, which we called "symplectic quandles" (though
we've since learned that these are also called "quandles of transvections").
We were able to prove some results about these quandles and use their extra
structure to enhance the quandle counting invariant. Our
Jacquelyn Rische was a student from my abstract algebra II class who had
some previous research experience -- she had done an REU project on number
theory and error-correcting codes the previous summer, for which she won a
prize at the Joint Meetings in 2007. While discussing my recent work with
Adam, we decided to try to generalize the symplectic quandle definition to
finite biquandles. After an initial computer search to find the appropriate
form for the operations, we were able to prove a number of results about
these bilinear biquandles as we call them (since the form in general
is bilinear and not always symplectic) and we were able to define biquandle
versions of the symplectic quandle invariants from my paper with Adam. Our
Jose Ceniceros was a student from my
combinatorics class at Whittier with excellent taste in music. We originally
set out to extend my classification theorem for finite Alexander quandles to
finite Alexander virtual quandles, but when this turned out to be easy, we
decided to try my backup plan of extending Yang-Baxter cocycle invariants to
virtual biquandles. We successfully defined an infinite family of invariants
of virtual knots and links using pairs of compatible cocycles from the
Yang-Baxter cohomology and a new "S-cohomology" theory. These invariants
reduce to the ordinary Yang-Baxter cocycle invariants for classical knots
but provide extra information about virtual knots and links. Our
Following my second year at Whittier College, I spent one year as a
visiting assistant professor at Pomona College. As a long-time member of the
Claremont Topology seminar, I was no stranger to Pomona College. After inviting
students from my classes to come to my seminar talk, I started working with
Ryan Wieghard, a freshman in my linear algebra class, on a project on
finite Coxeter racks and their enhancements of the rack counting invariants.
Our 
Toward the end of my second and final semester at Pomona College,
a student from my Topology class, Tim Carrell, asked me to be his senior
thesis advisor, a role I found myself informally playing for Jose as well.
I agreed and we set out to do some summer research as a warm-up for Tim's
upcoming thesis research this fall. We were able to settle a number of
conjectures as well as define a new family of link invariants using
generalized rack polynomials. Our paper is currently in peer review.
That fall, my honors calculus III course included CMC student Jessica
Ceniceros, who had taken my calculus I class at Pomona the previous year
and who had attended a few of my meetings with Jose, who happens to be
her brother. Jessica and I decided to start a project of our own that
fall; alas, my like my work with Daisy, we ended up going through two
complete project ideas before hitting on one that looked likely to
work out in a practical way -- the first turned out to be equivalent to
previous work, while the second invariant gave us trivial values on the
knots smalls enough for our software to compute quickly. The new idea involves
enhancement of the rack counting invariants using (t,s)-racks. Our
paper is currently is preparation.
Another student from my honors calculus III class, Wesley Chang,
asked to do a project. I knew Wesley was a high school student, but it
wasn't until our second or third research meeting that I realized he was
a junior in high school! Starting in the spring, we considered a
few ideas before deciding to apply the shadow coloring idea to quandle
and rack based counting invariants. Our paper is currently in preparation.
Not long after that happy day, I received a letter from a student
at the University of Wisconsin, Madison. Scott Pellicane had read several
of my papers (as well as this very webpage!) and wanted to know if I would
do a project with him if came to Claremont for the summer. I said "sure"
and sent a list of possible project ideas. We decided to study the
structure of a biquandle from my paper with Daisy Lam, which I had since
noticed was an example of what Allison Henrich and I had decided to call
"semiquandles." Ultimately Scott uncovered a connection between Latin
semiquandles, finite linear switches and finite Weyl algebras, drawing
on the work of Roger Fenn. Our paper is in preparation.