Tenure Evaluation File
James M. Belk
July 15, 2015
Contents
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Professional Achievements Since Rehire
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Teaching
2.1 General Approach . . . . . . . . . . . . .
2.2 Curricular Development . . . . . . . . .
2.2.1 Math 352: Differential Geometry
2.2.2 Math 108: Secret Codes . . . . .
2.3 Future Plans . . . . . . . . . . . . . . . .
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Research
3.1 Introduction . . . . . . . . . .
3.2 Thompson’s Groups . . . . . .
3.3 The Conjugacy Problem . . .
3.4 The Brin-Thompson Group 2V
3.5 Automata Groups . . . . . . .
3.6 Rearrangements of Fractals . .
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4
My Publications and Preprints
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Service
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6
Appendix: Other Materials
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1
1
Professional Achievements Since Rehire
Publications
• The paper “Conjugacy and dynamics in Thompson’s groups” (with Francesco Matucci)
was published in Geometriae Dedicata.
• The preprint “A Thompson group for the Basilica” (with Bradley Forrest) was accepted to Groups, Geometry, and Dynamics.
• The preprint “Rövers simple group is of type F∞ ” (with Francesco Matucci) was
written and accepted to Publicacions Matemàtiques.
• The paper “Implementation of a solution to the conjugacy problem in Thompson’s
group F” (with Nabil Hossain, Francesco Matucci, and Robert McGrail) was written
and published in ACM Communications in Computer Algebra.
• The research announcement “Deciding conjugacy in Thompsons group F in linear
time” (with Nabil Hossain, Francesco Matucci, and Robert McGrail) was written and
published in the 15th International Symposium on Symbolic and Numeric Algorithms
for Scientific Computing (SYNASC), 2013.
• The paper “CSPs and connectedness: P/NP dichotomy for idempotent, right quasigroups” (with Robert McGrail, Solomon Garber, Japheth Wood, and Benjamin Fish)
was written and published in the 16th International Symposium on Symbolic and
Numeric Algorithms for Scientific Computing (SYNASC), 2014.
Submissions
• The preprint “Some undecidability results for asynchronous transducers and the BrinThompson group 2V ” (with Collin Bleak) was written and submitted to Transactions
of the American Mathematical Society.
• The preprint “The word problem for finitely presented quandles is undecidable” (with
Robert McGrail) was written and submitted to WOLLIC 2015 Proceedings (a special
issue of Mathematical Structures in Computer Science).
Invited Talks
• “Rearrangement Groups of Fractals”, Geometry and Topology Seminar, Binghamton
University, April 2015
• “Dynamics and Decidability in Brins Group 2V ”, Invited address, Workshop on the
Extended Family of R. Thompsons Groups, University of St. Andrews, May 2014
• “Thompson-Like Groups for Self-Similar Structures”, Invited address, Workshop on
the Extended Family of R. Thompsons Groups, University of St. Andrews, May 2014
2
• “Turing Machines, Automata, and the Brin-Thompson Group 2V ”, Algebra/Topology
Seminar, SUNY Albany, November 2013
• “Turing Machines, Automata, and the Brin-Thompson Group 2V ”, Oberseminar
Gruppen und Geometrie, Universität Bielefeld, October 2013
• “Turing Machines, Automata, and the Brin-Thompson Group 2V ”, Groupes de Travail de L’équipe Topologie Dynamique, Départment de Mathématiques d’Orsay, Université Paris-Sud, October 2013
• “Turing Machines, Automata, and the Brin-Thompson Group 2V ”, Pure Mathematics Colloquium, University of St. Andrews, October 2013
• “Thompson-Like Groups Acting on Julia Sets”, Analysis Research Group, University
of St. Andrews, October 2013
• “Turing Machines, Automata, and the Brin-Thompson Group 2V ”, Geometry and
Topology Seminar, Binghamton University, September 2013
• “Thompson-Like Groups Acting on Julia Sets”, Groups St. Andrews 2013, University of St. Andrews, August 2013
• “Symmetries of Julia Sets”, Mathematics Seminar, Hamilton College, April 2013
• “A Thompson Group for the Basilica”, AMS Special Session on Arithmetic Dynamics and Galois Theory, 2013 Spring Eastern Sectional Meeting at Boston College,
April 2013
3
2
2.1
Teaching
General Approach
I described my general approach to teaching three years ago as part of my pre-tenure evaluation document, and overall my teaching philosophy and strategies have not changed. I
continue to develop my own approach toward courses that I teach, emphasizing the main
themes and ideas of a subject. I hold significant office hours every week, where students
work through difficult homework problems in groups, seeking help whenever they become
stuck. I also incorporate technology into my courses whenever possible, with the students making significant use of Mathematica in my Dynamical Systems course, and writing
homework solutions in LATEX in upper-level proofs-based courses.
My teaching style has always been relatively informal and spontaneous, and I try to be
as flexible as possible about the emphasis and pace of a course in an effort to cater to the
needs of the students. For example, when I taught Point-Set Topology for the second time
in the spring of 2015, I found that many of the students needed more help with proofs than I
had offered in the fall of 2010. As a result, I devoted significant class time to proof-writing,
with the result that I covered significantly less material than I had the first time through the
course. However, I think most of the students benefited from the increased emphasis on
proof-writing much more than they would have from seeing additional topics in topology.
Of course, I am always working to improve my teaching. For instance, one aspect of
mathematics teaching that I have come to appreciate more over the last few years is the idea
the students should learn how to check their own solutions to problems, especially in lowerlevel courses. Students tend to underappreciate the extent to which mathematical truths fit
together and support each other, and need to develop the habit of verifying their conclusions
through independent supporting evidence and reasoning. For a simple example, whenever
I compute a cross product in the middle of a complicated calculation, I usually take the dot
products of the result with the original vectors to make sure that they are indeed perpendicular. It is important to encourage students to perform such checks, both because it reduces
errors and because it encourages the students to appreciate the fundamental unity of mathematical thought. In the last few years, I have begun to emphasize such checks explicitly
while teaching, and I have adopted an announced policy of deducting more credit on homework, quizzes and exams for minor errors that students could have detected. In addition,
I am planning to experiment with having students in lower-level courses occasionally provide a written argument after completing a long problem describing the evidence they have
that their conclusions are correct.
One challenge that I continue to face in my teaching is providing students with timely
feedback on their work, including submitting grades and crite sheets in a timely fashion.
I sometimes find the amount of grading in upper-level math courses overwhelming, especially in courses with 15 or more students. I have become much more aware of this problem since it was brought up during my pre-tenure evaluation, and I have made significant
4
progress towards correcting my habits, though there is certainly still room for improvement.
But in particular, the frequency with which my grades are turned in late has decreased, and
I always make sure to include full crite sheet comments with both midterm and final grades.
I am still working on time management strategies to avoid falling behind in grading over
the course of the semester, and I expect that my timeliness will continue to improve over
the next few years.
2.2
Curricular Development
Since my pre-tenure review, I have developed one entirely new course (Math 108: Secret
Codes), and I have almost completely revised the curriculum for another (Math 352: Differential Geometry). Here I discuss my approach towards these courses and some of the
challenges I faced in their development.
2.2.1
Math 352: Differential Geometry
When I first taught Math 352 in the fall of 2011, I ran into significant unexpected difficulties over the course of the semester, and I was not quite satisfied with the final result. The
main problem was that, having not taught linear algebra or multivariable calculus at Bard
beforehand, my expectations regarding the students’ knowledge and proficiency with vectors and vector calculus was very different from the reality. This resulted in a poor choice
of textbook and severe pacing issues for the first half of the course. Once I realized the
problem, it was a distinct challenge to continue moving forward while filling in the gaps
in the students’ understanding and helping them to follow calculations in the textbook, and
only in the last few weeks did the course really seem to come together for both me and the
students. I finished the semester with the feeling that differential geometry plays a vital
role in our curriculum—the students certainly need an upper-level course that reinforces
their understanding of vectors and vector calculus—while at the same time unsure how to
offer a course that would meet the needs of the students.
When I volunteered to teach the course again in the fall of 2014, I was determined
to pitch the course at a much more appropriate level from the beginning. Having taught
both Linear Algebra with Ordinary Differential Equations (Math 213) and Vector Calculus
(Math 241) the previous semester, I had a much clearer sense of what the students would
understand coming in, as well as which aspects of the course I wanted to emphasize.
First, I wanted the course to have a much more applied focus, with stronger connections to physics. Over the last few years, it has become apparent that there is significant
demand at Bard for upper-level applied math courses, and the Mathematics Program has
often struggled to satisfy this demand. But differential geometry, though often taught as a
theoretical subject, has myriad applications, and I wanted to take the time to discuss such
applications throughout the course.
Second, I wanted to emphasize the computational aspects of mathematics throughout
5
the course. It has become increasingly clear over the last few years that even our best students sometimes struggle with mathematical computations, partially because many of our
advanced courses tend to emphasize theory over practice. This poor computational ability
sometimes holds students back during their senior projects, and has also proven problematic for graduate school bound students who take the Math Subject GRE. But differential
geometry is the ideal subject to reinforce students’ computational facility, especially with
vectors, as well as their overall understanding of calculus.
To accommodate these goals, I began by rewriting the course description to emphasize
applications, and I dropped Math 261 (Proofs & Fundamentals) as a prerequisite. I also
chose a new textbook that seemed much more readable for students who struggle with long
calculations. When registration came around, I was pleased to see several physics majors
enroll in the course.
I was relatively satisfied with how the course turned out. After spending several weeks
on plane curves, emphasizing parameterizations of curves defined geometrically and connections with calculus and differential equations, we spent a short time on curves in three
dimensions and curvature before moving on to parameterizations of surfaces, followed by
a short unit on parameterizations of manifolds. Throughout the first nine weeks of the
course, the emphasis was consistently on learning to transform geometry into algebra and
calculus. For example, I assigned a large number of homework problems that presented
students with a picture or animation defining a curve or surface—often with a brief written
explanation—and asked them to use the given geometry to construct an explicit parameterization1 . Only in the last six weeks did we proceed to a whirlwind tour through the classical
differential geometry of surfaces.
There are a few hitches remaining. Although the course was quite successful at achieving its main pedagogical goals, it is now quite different from a standard differential geometry course, which has two negative effects. First, though the course itself covers some of the
same material as a standard differential geometry course, the emphasis is different, and it is
difficult to find a textbook that focuses on these aspects of the subject. Second, because the
course spends so much time developing a concrete understanding of curves and surfaces, it
is hard to get through the main theorems of the differential geometry of surfaces in the last
few weeks. In the long run, I am hoping to write extensive notes for the course as a supplement to the lectures that will solve both of these problems. These notes would provide
more examples of parameterizations and explicit computations during the first part of the
course, and a quick, streamlined version of the differential geometry of surfaces during the
second part. I hope to make significant progress on writing such notes the next time that I
teach the course. I am also hoping to lower the overall difficulty of the course, making it
closer to the level of an introductory applied 300-level course such as Dynamical Systems,
and ideally attracting a wider audience of students interested in applied math.
1 You
can see these animations and pictures by clicking on the following links: 1a, 1b, 1c, 3, 4a, 4b, 5a,
5b, 5c, 6a, 6b, 6c. See also the homework assignments 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
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2.2.2
Math 108: Secret Codes
During the spring of 2015, I designed and taught Math 108: Secret Codes, a non-majors
course intended for students who wish to fulfill the MATC requirement. The course requires only basic algebra as a prerequisite, and indeed I did not accept any students into the
course who had passed Part II (the precalculus portion) of the Math Placement Diagnostic.
Eight of the nineteen students were graduating seniors who had put off taking an MATC
course until their last semester at Bard, and ten of the nineteen students had originally
failed Part I of the Math Placement Diagnostic, but had subsequently become Q-eligible by
completing the Algebra Workshop (BLC 150).
This is not an audience of students that I had much experience with, and I was very
careful in the design and implementation of the course to remain flexible in my expectations
for the students. Indeed, after interacting with the students for a couple of weeks in office
hours, I realized that I would need to cover certain material more slowly than I had been
anticipating. I had taught a similar course twice previously at Cornell the year after I earned
my Ph.D., but that audience was really quite different, with several science majors in the
course and very few students who had not taken precalculus. Though I used some of the
same materials and the same textbook as the Cornell course, many of the assignments were
brand new, and my whole approach in the first few weeks was designed to ease the students
into the material as gently as possible.
My philosophy in teaching a non-majors course is that, since it is likely to be the students’ only math or computer science course in college, it makes little sense to focus on
developing computational skills or subject-specific expertise as a pedagogical goal. Instead,
the primary goal should be for the students to engage with mathematical ideas, ideally in
a context that allows the students to appreciate both the beauty and applicability of these
ideas more thoroughly than they may have in high school math courses. Cryptology is an
excellent forum for this: though the mathematics involved is relatively simple, it is connected to deep ideas from statistics, number theory, group theory, and computer science. It
is also manifestly applicable, in the sense that solving a math problem lets you decipher a
message that had previously been unreadable. Finally, cryptology has important and fascinating connections to history, politics, and the arts, which means that students can engage
with the subject from the standpoint of their own interests.
I also feel strongly that, since a non-majors MATC course is likely to be the students’
only exposure to math or computer science, it is best if the course can incorporate ideas
from both of these subjects. As a result, a major goal in Math 108 was for the students
to learn to use computers to crack secret codes. Computers are quite helpful for code
breaking: the first truly programmable computer, known as Colossus, was designed and
built by British code breakers during World War II to help crack the German Lorenz cipher.
Though it wouldn’t be practical for the students to design and build their own computer,
they can learn how to design and implement simple computer programs in Microsoft Excel
that help them to decode secret messages. To facilitate this, I scheduled the course to meet
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in a computer lab, so the students could work together during class to learn to use Excel
effectively. Many of the homework problems throughout the semester were Excel-based2 ,
as was the in-class midterm3 .
Of course, there are a few ways that I hope to improve this course next time I teach
it. First, because I spent more time than I anticipated covering the beginning material
and helping the students learn to use Excel, the unit on RSA near the end was a little
rushed, and most of the students did not seem to achieve a full understanding of public key
cryptography. This problem should be solvable by choosing some topics in the first half
to omit, leaving a little extra time near the end. Second, I wish that I had found a way to
incorporate a discussion of the public policy issues surrounding modern encryption more
fully into the course. This is difficult, because public policy issues are not well-covered in
the textbook4 , so I will need to think about how to design such a unit the next time I teach
the course.
2.3
Future Plans
Over the next few years, I hope to teach several upper-level courses in the mathematics
program that I have not yet taught, such as Probability (Math 328), Differential Equations
(Math 311), Complex Analysis (Math 362), Mathematical Logic (Math 405), or perhaps a
400-level algebra course. If possible, I would also like to continue the curricular development of Differential Geometry (Math 352) and Secret Codes (Math 108), as well as Real
Analysis II (Math 461). I also hope to be able to teach one or more courses outside the
Math Program, though this has been difficult in recent years due to staffing issues. I continue to have an interest in teaching FYSEM, and I would also love to teach a course in
either Physics or Computer Science.
Finally, it has always been a long-term goal of mine to write textbooks, and this would
certainly be my first priority as a tenured professor. My strategy would be to write notes for
courses that I a teaching, with the goal of developing these notes into a book over time. I
already have 76 pages of notes written on sequences and series, which are currently covered
in our Vector Calculus (Math 241) course, and my colleague Ethan Bloch has expressed an
interest in collaborating on a Vector Calculus book. I also have 106 pages of notes written
for Real Analysis II, covering roughly the second half of the course, and I would hope to
double this the next time I teach it.
2 See
homeworks 1, 2, 5, 7, and 10.
blank copy of the Excel-based midterm can be found at math.bard.edu/belk/math108/midterm.xls
4 The textbook was Invitation to Cryptology by T. Barr. It is fairly good overall, and is essentially the only
book available for a cryptology course directed at non-majors.
3A
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3
3.1
Research
Introduction
My primary research area is geometric group theory. This subject lies between algebra
and geometry, and has connections to such diverse subjects as combinatorics, differential
geometry, algebraic topology, cryptology, dynamical systems, probability, number theory,
and mathematical logic. I am particularly interested in connections between group theory,
dynamical systems, fractal geometry, and automata theory.
Group theory can be described as the mathematical study of symmetry, where symmetry
is construed broadly to mean any transformation of a mathematical object that preserves
its structure. The groups that I study describe the symmetries of geometric objects that
possess a self-similar structure, showing the same patterns at a variety of scales. In addition
to simple examples such as lines, which have the same structure at every scale, the class
of self-similar objects includes many fractals—complicated shapes that arise in the theory
of dynamical systems. Because such an object exhibits the same structure at a variety
of scales, it may have transformations that change the size of different parts of it while
maintaining the overall structure.
For example, Figure 1 shows one possible transformation of a fractal known as the
Basilica Julia set. The Julia set is divided into two parts, shown in blue and pink. During
the transformation, the blue portion expands and the pink portion contracts, resulting in the
same fractal that we started with. (From a mathematical point of view, this is an example
of a piecewise-conformal homeomorphism.) The Basilica Julia set has many such transformations, and the collection of all of them form the Basilica Thompson group TB . This
group was the subject of a 2012 paper that I wrote with my colleague Bradley Forrest [5].
Figure 2 shows a transformation of a fractal known as the Cantor square. Again, the
Cantor square is divided into parts, shown in four different colors, and these parts are scaled
and reassembled to form the same fractal that we started with. The collection of all such
transformations is known as the Brin-Thompson group 2V , and was defined by Matthew
Brin in 2004 [22]. In a recent preprint with Collin Bleak [3], we prove a surprising result
about this group, namely that there is no algorithm to determine whether an element of 2V
has finite order.
A
B
®
A
B
Figure 1: A symmetry of the Basilica Julia set. The blue portion (labeled A) expands to
cover the center, while the pink portion (labeled B) contracts and moves to the right.
9
−→
Figure 2: A transformation of the Cantor square. The red and yellow portions contract
vertically, while the blue portion expands horizontally.
Figure 3 shows a certain transformation of a line segment, the simplest possible selfsimilar shape. Mathematically, this transformation is a function f made of three linear
pieces, where f (0) = 0, f (1/4) = 1/2, f (1/2) = 3/4, and f (1) = 1. The set of all
such transformations, subject to certain restrictions on the sizes of the pieces, is known as
Thompson’s group F. This is one of three famous groups discovered by Richard J. Thompson in the 1960’s. The other two, known as T and V , represent symmetries of a circle and a
Cantor set, respectively. Thompson’s groups and their relatives (such as the groups TB and
2V discussed above) are considered very interesting within geometric group theory, with
hundreds of papers published on them in the last two decades.
Though Thompson’s groups have been familiar to geometric groups theorists for nearly
thirty years, it is only in the last ten years that they have come to be seen as three representatives of a much larger family of related groups, and the idea that many generalized
Thompson groups are related to self-similar structures has been emerging slowly. I became
interested in generalizations of these groups in 2010 or so, and they have since become the
main theme of my research. The current goal in the field is to understand the full scope
of the family of Thompson-like groups, with the hope of eventually developing a general
theory of their algebraic, geometric, and dynamical properties.
This is not the direction that I had expected my research to take. My Ph.D. work
was on Thompson’s groups F, T , and V specifically [10, 9, 8], as was my early work with
F. Matucci on conjugacy [7]. During my postdoc at Texas A&M, my goal was to learn about
the theory of iterated monodromy groups and their relation to complex dynamics and fractal geometry, which the group theorists there (including R. Grigorchuk, V. Nekrashevych,
and Z. Sunik) are experts in, with the goal of doing research in that field. I succeeded in
A
0
B
14
C
12
A
−→
1
0
B
12
C
34
1
Figure 3: A transformation of a line segment. The red portion (labeled A) expands by a
factor of two, and the green portion (labeled C) contracts by a factor of two.
10
this endeavor, publishing in 2010 a paper with Sarah Koch [6] (an expert in complex dynamics) in which we computed the wreath recursion for the iterated monodromy group of
a postcritically finite map on CP2 , the first direct calculation of its kind.
However, around that same time Bradley Forrest and I discovered the Basilica Thompson group discussed above, which raised the possibility of a connection between dynamical
systems, fractal geometry, and generalizations of the Thompson groups. This reignited my
interest in the field, and I have since devoted myself to the growing field of generalized
Thompson groups. During my junior sabbatical, I began collaborations with several other
researchers who share the same interest, and I am now involved in active collaborations
with Francesco Matucci, Collin Bleak, Conchita Martı́nez-Pérez, Brita Nucinkis, Matthew
Zaremsky, and Stefan Witzel, all of which are likely to continue in the years ahead. I have
submitted three papers in the last three years on generalized Thompson groups [5, 4, 3],
and I expect to submit two more in the next few months [2, 1]. My collaborators and I also
have a large number of preliminary results, which should lead to at least three more papers
in the next two years or so.
In addition to this work, I have been fortunate to collaborate over the last few years with
my colleague R. McGrail at Bard college on two projects related to quandles and left-selfdistributive algebras [11, 13]. These structures have been subject to considerable attention
in both universal algebra and knot theory, and I am glad that my expertise in group theory
has proven helpful for this research. I have also had the opportunity to collaborate with
several Bard undergraduates, including Nabil Hossain, Jasper Weinrich-Burd, and William
Smith. I now have two publications within computer science with Prof. McGrail and Nabil
Hossain based on my earlier work with Francesco Matucci [14, 12], and I am expecting
that Jasper Weinrich-Burd and William Smith will be involved in a future publication with
myself and Bradley Forrest.
What follows is a more technical explanation of my research directed at an audience of
mathematicians.
3.2
Thompson’s Groups
A dyadic subdivision of the interval [0, 1] is any subdivision obtained by repeatedly cutting
intervals in half. For example the subdivision
0
14
38
12
34
1
is dyadic, since it can be constructed by first cutting the interval [0, 1] in half, then cutting each of the resulting subintervals [0, 1/2] and [1/2, 1] in half, and finally cutting the
subinterval [1/4, 1/2] in half.
Given two dyadic subdivisions with the same number of subintervals, the corresponding dyadic rearrangement of [0, 1] is the piecewise-linear homeomorphism obtained by
mapping each interval of the first subdivision linearly to the corresponding interval of
11
the second subdivision. For example, Figure 3 shows a dyadic rearrangement that maps
[0, 1/4] linearly to [0, 1/2], maps [1/4, 1/2] linearly to [1/2, 3/4], and maps [1/2, 1] linearly to [3/4, 1]. The set of all dyadic rearrangements of the interval [0, 1] forms a group
under composition; this is Thompson’s group F.
Thompson’s group F is one of three groups discovered by Richard J. Thompson in the
1960’s. The others are T , which acts on the unit circle by piecewise-linear homeomorphisms, and V , which acts on the standard middle-thirds Cantor set by piecewise-linear
homeomorphisms. See [25] for a general introduction to these groups.
Thompson’s groups F, T , and V are considered quite interesting in geometric group
theory because of their unique mix of properties. In particular:
• F, T , and V are finitely generated and finitely presented. Indeed, all three groups
have type F∞ , meaning that each can be realized as the fundamental group of an
aspherical CW-complex with finitely many cells in each dimension [24].
• T and V are simple, and F has simple commutator subgroup. T and V are among
only a few known examples of infinite, finitely presented simple groups.
• Each of F, T and V acts properly by isometries on a CAT(0) cubical complex [29] [30].
In addition to these properties, much of the interest in Thompson’s groups has been driven
by the question of whether Thompson’s group F is amenable. This question was posed by
Ross Geoghegan in the early 1980’s, and remains open more than 30 years later, despite
numerous purported solutions.
My Ph.D. thesis [8] developed several different ways of representing elements of F,
most notably forest diagrams and strand diagrams. Forest diagrams provide a simple geometric way of understanding the action of the finite generating set {x0 , x1 }, and have since
been used by several other researchers [16, 39, 44, 27]. In joint work with my advisor Kenneth Brown, I used forest diagrams to explore geometric properties of the Cayley graph of
F, including providing an explicit formula for the metric, estimates of growth, and the best
known upper bound on the Cheeger constant [10]. Along with Kai-Uwe Bux, I also proved
that metric balls in the Cayley graph are not minimally almost convex [9], a weak convexity
condition introduced by M. Elder and S. Hermiller [28]. This result was later generalized
by Claire Wladis to a related family of groups [53].
In addition, my thesis introduced strand diagrams for elements of F, which are similar
to braids, except that instead of twisting the strands of a strand diagram can split apart
and merge together. (These strand diagrams are similar to the earlier “transistor pictures”
of Guba and Sapir [36].) Using strand diagrams, I constructed a classifying space for F
analogous to configuration spaces for braid groups. This space is the subject of a recent
preprint by L. Sabalka and M. Zaremsky [50].
12
3.3
The Conjugacy Problem
The conjugacy problem is one of three fundamental algorithmic problems for finitely presented groups identified by Max Dehn in his famous 1911 paper [26]. For a given finitely
presented group, the conjugacy problem asks for an algorithm to determine whether two
given words in the generators represent conjugate elements of the group. This problem is
not solvable for all finitely presented groups [47], but it is known to be solvable for almost
all groups of interest in geometric group theory.
In 2005, Francesco Matucci and I began an investigation of the conjugacy problem in
Thompson’s groups. By extending the strand diagrams for F described in my Ph.D. thesis
to T and V , we managed to prove the following theorem.
Theorem (Belk & Mattucci, 2007 [7]). The conjugacy problem is solvable in Thompson’s
groups F, T , and V .
Our solution is simple and geometric, and works in roughly the same way for all three
groups. We were not the first to tackle these problems: the conjugacy problem for F
was previously solved by Guba and Sapir [36], while the conjugacy problem for V was
solved by Higman [40], and then again by Salazar-Diaz [51]. However, our solution to the
conjugacy problem in T was new, as was the unified approach.
In 2013, I began a collaboration with R. McGrail (a computer science professor at Bard
College) and Nabil Hossain (an undergraduate student) to implement this algorithm and
analyze the running time. We prove that the conjugacy problem for F can be solved in linear
time[14][12]. A Java implementation of this algorithm is available online5 . This result
raises the question of how quickly conjugacy can be checked in T and V . Prof. McGrail
and I hope to collaborate on this question in the future, either together with Nabil Hossain
(who is now enrolled in a computer science Ph.D. program) or with the help of another
undergraduate student.
In the meantime, Prof. Matucci and I have continued our investigation into the conjugacy problem for Thompson-like groups. We have two specific groups in mind whose
conjugacy problems might be interesting to solve.
Question. Does the Brin-Thompson group 2V have solvable conjugacy problem?
Here 2V is the higher-dimensional Thompson group defined by Brin [22], which acts
by homeomorphisms on the square Cantor set. Prof. Matucci and I began working on this
problem in 2008, and the ideas stemming from this effort have led to two preprints [1]
[3] (discussed further in Section 4 below). Surprisingly, it seems likely that conjugacy
problem in 2V is not solvable, though we do not yet have a proof of this. If this is true,
it raises the possibility that 2V may be useful for group-based public-key cryptography
[15, 42]. Prof. McGrail, myself, and Cyril Kuhns (an undergraduate student) are planning
to explore the feasibility of cryptography in 2V this coming year.
5 http://asclab.org/asc/nhossain/conjugacyF
13
−→
Figure 4: The baker’s map in 2V . Both colored sections of the Cantor square expand
horizontally and contract vertically during the homeomorphism.
Question. Does Röver’s group V G have solvable conjugacy problem?
Röver’s group V G is the group generated by Thompson’s group V and the first Grigorchuk group G , which was investigated by C. Röver in [48, 49]. Röver proved that V G is
finitely presented and simple, and is isomorphic to the abstract commensurator of the first
Grigorchuk group. These results have since been extended to a whole class of groups by
V. Nekrashevych [45, 46], who considers arbitrary groups obtained by combining Thompson’s group V with a self-similar group. In a recent preprint, F. Matucci and I prove:
Theorem (Belk & Matucci, 2014). Röver’s group V G has type F∞ .
The proof involves constructing a complex on which Röver’s group acts properly discontinuously and then analyzing the connectivity of the descending links, using a combination of Brown’s criterion [24] and Bestvina-Brady Morse theory [20] to obtain the desired
finiteness property.
Prof. Matucci and I became quite familiar with Röver’s group during this research,
and we plan to continue our exploration of this group. The conjugacy problem in the
first Grigorchuk group is solvable [38], so it ought to be possible to combine this with
our solution to the conjugacy problem in V to obtain a solution to the conjugacy problem
in V G .
3.4
The Brin-Thompson Group 2V
The Brin-Thompson group 2V is the second in the sequence of higher-dimensional Thompson groups nV defined by Brin in [22], the first being Thompson’s group V . The group 2V
acts by piecewise-affine maps on the Cantor square C × C, where C is the middle-thirds
Cantor set in R. Specifically, a dyadic subdivision of the Cantor square is any subdivision
obtained by repeatedly cutting rectangles in half either horizontally or vertically. Given
two dyadic subdivisions of the Cantor square together with a bijection between the pieces,
the corresponding dyadic rearrangement of the Cantor square is the piecewise-affine map
that maps each rectangle in the first subdivision to the corresponding rectangle in the second subdivision via a dilation and translation. The group of all such dyadic rearrangements
is 2V .
14
Brin proved [22, 23] that 2V is finitely presented and simple. Moreover, Brin observed
that elements of 2V can have interesting dynamics. Specifically, 2V contains the baker’s
map shown in Figure 4, which has the same dynamics as the full two-sided shift on two
symbols. Subsequent to Brin’s work, D. Kochloukova, C. Martı́nez-Pérez, and B. Nucinkis
proved that 2V and 3V have type F∞ [37], a result that was later simplified and generalized
to all nV by M. Fluch, M. Marschler, S. Witzel, and M. Zaremsky [31].
In 2012, Francesco Matucci and I began to explore the dynamics of elements of 2V ,
with the hope of solving the conjugacy problem. This collaboration later expanded to include Conchita Martı́nez-Pérez, and Brita Nucinkis, as well as Collin Bleak. We identified
an important class of elements of 2V with similar properties to the baker’s map.
Definition. An element f ∈ 2V is hyperbolic if there exists a power f n that expands each
rectangle horizontally by a factor greater than one and contracts each rectangle vertically
by a factor less than one.
For example, the baker’s map shown in Figure 4 is hyperbolic, since the map itself
already stretches horizontally and compresses vertically. It is not hard to show that any
element of 2V conjugate to a hyperbolic element is itself hyperbolic, although the required
exponent n may change.
Theorem (Belk, Matucci, Martı́nez-Pérez & Nucinkis, 2015 [1]). Every hyperbolic element of 2V is topologically conjugate to a subshift of finite type. Moreover, every subshift
of finite type is topologically conjugate to some hyperbolic element of 2V .
Here topological conjugacy refers to the natural notion of isomorphism for dynamical
systems, i.e. two dynamical systems are topologically conjugate if there exists a homeomorphism of their state spaces conjugating the first map to the second. The proof of this
theorem involves recovering some version of the theory of hyperbolic dynamics in the context of the Cantor square. Specifically, we construct an analogue of a “Markov partition”
for each hyperbolic element of 2V , and use this to construct the topological conjugacy
with a subshift of finite type. This result on the dynamics allows us to prove the following
characterization of centralizers for hyperbolic elements.
Theorem (Belk, Matucci, Martı́nez-Pérez & Nucinkis, 2015 [1]). If f ∈ 2V is hyperbolic,
then the centralizer of f is virtually free abelian of finite rank. In particular, if f is topologically mixing, then C2V ( f ) is virtually cyclic.
The proof of this theorem involves constructing analogues of stable and unstable manifolds for fixed points of hyperbolic elements of 2V , and analyzing where they intersect.
The conjugacy problem for hyperbolic elements in 2V seems to be particularly interesting. It appears to be a refinement of the equivalence of subshifts problem in dynamical
systems, which asks whether there exists an algorithm to determine whether two subshifts
of finite type are topologically conjugate. This is a famous open problem, and it would be
quite interesting to establish a definite relationship between them.
15
Question. Can the equivalence of subshifts problem be reduced to the conjugacy problem
for hyperbolic elements of 2V ?
In addition to these results on hyperbolic elements, my colleague Collin Bleak and I
have found another class of elements of 2V with particularly interesting dynamics. For the
following theorem, recall that a Turing machine is a computer with a finite set of internal
states that operates by reading and writing symbols on a single infinite tape. A Turing
machine is complete if it cannot halt, and reversible if it can be operated backwards in
time, if each state of the Turing machine and tape determines the previous state. Such a
machine can be thought of as a dynamical system operating on the set of pairs (q, τ), where
q denotes the internal state of the machine and τ represents the contents of the tape.
Theorem (Belk & Bleak, 2014 [3]). Every complete, reversible Turing machine is topologically conjugate to an element of 2V .
J. Kari and N. Oliver have shown [41] that there is no algorithm to determine whether
a complete, reversible Turing machine is periodic, i.e. whether the machine always returns
to its initial state (q, τ) after a fixed finite number of steps. Using this theorem, Prof. Bleak
and I were able to prove the following.
Theorem (Belk & Bleak, 2014 [3]). The group 2V has unsolvable torsion problem. That
is, there does not exist an algorithm to determine whether a given element of 2V has finite
order.
The existence of a group with solvable word problem and unsolvable torsion problem
was previously shown by G. Arzhantseva, J.F. Lafont, and A. Minasyan [18], but 2V is
the first “naturally occurring” group with this property. The unsolvability of the torsion
problem in 2V suggests that the conjugacy problem should also be unsolvable, but we do
not know how to prove this.
3.5
Automata Groups
In 2000, R. Grigorchuk, V. Nekrashevych, and V. Sushchanskii defined the class of automata groups, whose elements can be represented as finite Mealy automata [34]. This
class includes all self-similar groups, whose elements can be represented as synchronous
automata (i.e. automata that output exactly one symbol at each step), and the Thompson
groups F, T , and V , whose elements can be represented as asynchronous automata. The
use of these automata is widespread in the theory of self-similar groups, but the automata
viewpoint has been slow to gain traction within the Thompson’s group community.
In our recent preprint [3], Collin Bleak and I prove that the Brin-Thompson group 2V
can be represented as a group of Mealy automata. Given our result on the torsion problem,
this implies the following.
16
Theorem (Belk & Bleak, 2014 [3]). There does not exist an algorithm to determine whether
a given reversible Mealy automaton operating on an alphabet Σ defines a bijection of Σ∞
of finite order.
This result settles the finiteness question for automata groups posed in [34], i.e. given
a finite set of reversible automata, is there an algorithm to determine whether the group
that they generate is finite? According to the above theorem, this is impossible even for
the cyclic group generated by a single automaton. This question and its variants have
received much attention in the literature. For example, Gillibert [33] proved that the finiteness problem for the semigroup generated by a finite set of (not necessarily reversible)
automata is undecidable. Other partial undecidability results have been proven by Akhvai,
Klimann, Lombardy, Mairesse and Picantin [17], Klimann [43], and Bondarenko, Bondarenko, Sidke, and Zapata [21]. Though our result settles the question for arbitrary automata, it is still open in the special case of synchronous automata.
Collin Bleak, Francesco Matucci, and I are currently collaborating on a project whose
goal is to prove that a large number of well-known groups can be represented as groups
of asynchronous automata. As a general rule, it seems that whenever a group acts on a
space with a self-similar structure in a way that preserves the structure, this results in a
representation of the group as a group of asynchronous automata. This includes virtually
all groups in the Thompson family that act as homeomorphisms, as well as many other
groups. For example, we have proven:
Theorem (Belk, Bleak & Matucci, 2015). Every hyperbolic triangle group can be represented as a group of finite-state asynchronous automata.
It follows that surface groups can also be represented this way. Roughly speaking, this
representation arises from the self-similar structure on the boundary of the hyperbolic plane
derived from the triangular tiling. In particular, every hyperbolic triangle group embeds
naturally into a Thompson-like group of homeomorphisms of the circle that is based on
this self-similar structure.
This theorem leads to many natural questions. For example, can every hyperbolic group
be realized as a group of asynchronous automata? Can every Coxeter group? What about
SL(3, Z), using its action on the corresponding building? Our current goal is to formalize
exactly what conditions are needed for a group acting on a self-similar space to guarantee
that it has a representation of this form. Once this is accomplished, it should be possible to
investigate whether these large families of groups act on their boundaries in an appropriately self-similar way.
3.6
Rearrangements of Fractals
Inspired by the work of L. Bartholdi, V. Nekrashevych, and others connecting complex dynamics, fractal geometry, and the theory of self-similar groups, B. Forrest and I have been
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(b)
Figure 5: (a) Filled Julia set and Julia set for z2 − 1. (b) Invariant lamination for z2 − 1.
investigating connections between complex dynamics, fractal geometry, and Thompsonlike groups. So far, our primary success has been to construct a large family of previously
unknown Thompson-like groups that act on fractal spaces, but our hope is to construct a
theory rich enough that applications may be found in complex dynamics or fractal geometry.
We begin by reviewing the definition of a Julia set. Given a polynomial function
f : C → C, recall that the orbit of a point p ∈ C under f is the sequence p, f (p), f ( f (p)), . . ..
The filled Julia set for f is the set
K f = {p ∈ C | the orbit of p under f is bounded}.
The filled Julia set is a compact subset of the complex plane, and its boundary J f is known
as the Julia set for f . For example, Figure 5(a) shows the filled Julia set and the Julia set
for the function z2 − 1. This Julia set is known as the Basilica.
Topologically, the Basilica Julia set can be obtained from a circle by identifying certain
pairs of points. Figure 5(b) shows the invariant lamination for the Basilica. This lamination consists of a circle [0, 1]/{0, 1} together with a single hyperbolic arc between each
pair of points that should be identified. Any homeomorphism of the circle that preserves
this lamination descends to a homeomorphism of the Basilica Julia set.
The Basilica Thompson group TB consists of all piecewise-linear homeomorphisms h
of the circle satisfying the following conditions:
1. All the slopes of h are powers of 2.
2. All of the breakpoints of h and h−1 are at angles of the form
18
kπ
, where k, n ∈ Z.
2n · 3
2
9
10
8
1
3
4
7
6
5
Figure 6: A cellular partition of the Basilica Julia set.
3. h preserves the invariant lamination for the Basilica.
An element of the Basilica Thompson group TB was shown previously in Figure 1. This
element maps the interval [1/3, 2/3] on the circle linearly to [1/6, 5/6], and maps the complementary interval [2/3, 1/3] linearly to [5/6, 1/6].
In our 2012 preprint [5], B. Forrest and I prove the following theorem about TB .
Theorem. The Basilica Thompson group TB is finitely generated and virtually simple.
We also show that TB is contained in Thompson’s group T , and also contains a copy of
T , but is not virtually isomorphic to T . Although TB is finitely generated, S. Witzel and
M. Zaremsky have recently proven that TB is not finitely presented [52].
There is a slightly different description of TB that makes it much more analogous to
Thompson’s groups. Specifically, define a cell in the Basilica
Julia set J to be any compo√
−n
nent of the complement J − f (p), where p = (−1 + 5)/2 is the left-hand fixed point.
A cellular partition of J is any subdivision of J into finitely many of these cells. For example, Figure 6 shows a cellular partition of the Basilica into ten different cells of various
sizes. Then a rearrangement of the Basilica is any homeomorphism that maps conformally between corresponding pairs of cells in two cellular partitions. The group of all such
rearrangements is the Basilica Thompson group TB .
In the last three years, B. Forrest and I have been working to generalize the definition
of TB to a much larger family of Thompson-like groups acting on fractal sets. The results
of this work will appear in the preprint [2].
Our current formalism is based on the notion of an edge replacement system, which
consists of a base graph G0 together with a rule for a rule for replacing directed edges of
G0 with larger directed graphs. By applying the replacement rule repeatedly to G0 , one
obtains a sequence of graphs that limit to a compact metric spaces, which we refer to as the
limit space. Each edge of G0 and each of the intermediate graphs corresponds to a subset
of the limit space, which we refer to as a cell. A ceullular partition of the limit space
is any subdivision into finitely many cells, and a rearrangement of the limit space is a
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6
2
2
6
4
1
5
8
1
4
5
8
−→
3
7
7
3
Figure 7: A rearrangement of the Vicsek fractal. Each numbered cell on the left maps to
the corresponding cell on the right under the homeomorphism.
homeomorphism that maps canonically between the cells of two cellular partitions. The rearrangements of such a space always form a group, which we refer to as the rearrangement
group.
For example, Figure 7 shows a rearrangement of the Vicsek fractal, which is the limit
space of a certain simple edge replacement system. The domain and range are both partitioned into eight cells, and each cell in the domain maps to the corresponding cell in the
range in the canonical way.
Theorem (Belk & Forrest, 2015 [2]). Every rearrangement group acts properly by isometries on a CAT(0) cubical complex.
This theorem generalizes the results of Farley [29, 30] for Thompson’s groups F, T ,
and V . By analyzing the descending links in the CAT(0) complex for the Vicsek fractal,
we also prove:
Theorem (Belk & Forrest, 2015 [2]). The rearrangement group of the Vicsek fractal has
type F∞ .
In addition to these results, Prof. Forrest and I have also developed an algorithm to produce an edge replacement system whose limit space is homeomorphic to the Julia set for
any postcritically finite polynomial, and we have also generalized the notion of rearrangement groups to arbitrary graph diagram groups, which are similar to the diagram groups
of Guba and Sapir [36], but involve a graph rewriting system instead of a string rewriting
system. Both of these results will appear in later preprints.
In addition to this work, I have recently become involved in a new collaboration with
S. Witzel and M. Zaremsky regarding a certain specific family of graph diagram groups.
These appear to be natural multidimensional generalizations of Thompson’s group F, and
we have been denoting them nF. We have managed to prove that 2F has type F∞ , and are
working to extend this result to the entire family of groups.
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4
My Publications and Preprints
Group Theory Publications and Preprints
[1] J. Belk, F. Matucci, C. Martı́nez-Pérez, and B. Nucinkis, “Hyperbolic dynamics and
centralizers in the Brin-Thompson group 2V .” Preprint in preparation.
[2] J. Belk and B. Forrest, “Rearrangement groups of fractals.” Preprint in preparation.
[3] J. Belk and C. Bleak, “Some undecidability results for asynchronous transducers and
the Brin-Thompson group 2V.” Preprint (2014). Submitted to Transactions of the
American Mathematical Society. arXiv:math/1405.0982.
[4] J. Belk and F. Matucci, “Röver’s simple group is of type F∞ .” Preprint (2014). Accepted to Publicacions Matemàtiques. arXiv:math/1312.2282.
[5] J. Belk and B. Forrest, “A Thompson group for the Basilica.” Preprint (2012). Accepted to Groups, Geometry, and Dynamics, arXiv:math/1201.4225.
[6] J. Belk and S. Koch, “Iterated monodromy for a two-dimensional map.” In the Tradition of Ahlfors-Bers, V: The Triennial Ahlfors-Bers Colloquium, May 8-11, 2008, Rutgers University, Newark, New Jersey, vol. 510, American Mathematical Soc., 2010.
[7] J. Belk and F. Matucci, “Conjugacy and dynamics in Thompson’s groups.” Geometriae Dedicata 169.1 (2014): 239–261. arXiv:math/0708.4250
[8] J. Belk, “Thompson’s group F.” Ph.D. Thesis, Cornell University, 2004,
arXiv:math/0708.3609.
[9] J. Belk and K. Bux, “Thompson’s group F is maximally nonconvex.” Geometric Methods in Group Theory, 131-146, Contemp. Math., 372, AMS 2005.
arXiv:math/0301141.
[10] J. Belk and K. Brown, “Forest diagrams for elements of Thompson’s group F.” International Journal of Algebra and Computation 15, no. 05–06 (2005): 815–850.
arXiv:math/0305412.
Computer Science Publications and Preprints
[11] J. Belk and R. McGrail, “The word problem for finitely presented quandles is undecidable.” Preprint (2015). Submitted to WOLLIC 2015 Proceedings (special issue of
Mathematical Structures in Computer Science).
[12] J. Belk, N. Hossain, F. Matucci, and R. McGrail, “Implementation of a solution to
the conjugacy problem in Thompson’s group F.” ACM Communications in Computer
Algebra 47, no. 3/4 (2014): 120–121.
21
[13] J. Belk, B. Fish, S. Garber, R. McGrail, and J. Wood, “CSPs and connectedness: P/NP
dichotomy for idempotent, right quasigroups.” Symbolic and Numeric Algorithms for
Scientific Computing (SYNASC), 2014 16th International Symposium on, pp. 367–
374. IEEE, 2014.
[14] J. Belk, N. Hossain, F. Matucci, and R. McGrail, “Deciding conjugacy in Thompson’s
group F in linear time.” In Symbolic and Numeric Algorithms for Scientific Computing
(SYNASC), 2013 15th International Symposium on, pp. 89–96. IEEE, 2013.
22
Other References
[15] I. Anshel, M. Anshel, and D. Goldfeld, “An algebraic method for public-key cryptography.” Mathematical Research Letters 6 (1999): 287–292.
[16] G. Arzhantseva, V. Guba, M. Lustig, and J.P. Praux, “Testing Cayley graph densities.”
In Annales mathmatiques Blaise Pascal, vol. 15, no. 2, pp. 233–286. 2008.
[17] A. Akhavi, I. Klimann, S. Lombardy, J. Mairesse, and M. Picantin, “On the finiteness
problem for automaton (semi) groups.” International Journal of Algebra and Computation 22, no. 06 (2012): 1250052.
[18] G. Arzhantseva, J.F. Lafont, and A. Minasyan, “Isomorphism versus commensurability for a class of finitely presented groups.” Journal of Group Theory 17, no. 2 (2014):
361–378.
[19] L. Bartholdi and V. Nekrashevych, “Thurston Equivalence of Topological Polynomials.” Acta. Math. 197 (2006), no. 1, 1-51.
[20] M. Bestvina and N. Brady, “Morse theory and finiteness properties of groups.” Inventiones mathematicae 129, no. 3 (1997): 445–470.
[21] I. Bondarenko, N. Bondarenko, S. Sidki, and F. Zapata, “On the conjugacy problem
for finite-state automorphisms of regular rooted trees (with an appendix by Raphaël
M. Jungers).” Groups, Geometry, and Dynamics 7, no. 2 (2013): 323–355.
[22] M. Brin, “Higher dimensional Thompson groups.” Geometriae Dedicata 108, no. 1
(2004): 163–192.
[23] M. Brin, “Presentations of higher dimensional Thompson groups.” Journal of Algebra
284, no. 2 (2005): 520–558.
[24] K. Brown, “Finiteness properties of groups.” Journal of Pure and Applied Algebra 44,
no. 1 (1987): 45–75.
[25] J. W. Cannon, W. J. Floyd, and W. R. Parry, “Introductory notes to Richard Thompson’s groups.” L’Enseignement Mathmatique 42 (1996), 215–256.
[26] M. Dehn, “Über unendliche diskontinuierliche Gruppen.” Mathematische Annalen 71
(1911), 116–144.
[27] M. Elder, . Fusy, and A. Rechnitzer, “Counting elements and geodesics in Thompson’s
group F.” Journal of Algebra 324, no. 1 (2010): 102–121.
[28] M. Elder and S. Hermiller, “Minimal almost convexity.” Journal of Group Theory 8,
no. 2 (2005): 239–266.
[29] D. Farley, “Finiteness and CAT(0) properties of diagram groups.” Topology 42 (2003),
1065–1082.
23
[30] D. Farley, “Actions of picture groups on CAT(0) cubical complexes.” Geometriae
Dedicata 110, no. 1 (2005): 221–242.
[31] M. Fluch, M. Marschler, S. Witzel, and M. Zaremsky, “The BrinThompson groups
sV are of type F∞ .” Pacific Journal of Mathematics 266, no. 2 (2013): 283–295.
[32] S.B. Fordham, “Minimal length elements of Thompson’s group F.” Geometriae Dedicata 99, no. 1 (2003): 179–220.
[33] P. Gillibert, “The finiteness problem for automaton semigroups is undecidable.” International Journal of Algebra and Computation 24, no. 01 (2014): 1–9.
[34] R. Grigorchuk, V. Nekrashevich, and V. Sushchanskii, “Automata, dynamical systems
and groups.” Proceedings of the Steklov Institute of Mathematics 231, no. 4 (2000):
128–203.
[35] V. Guba,“On the properties of the Cayley graph of Richard Thompson’s group F.”
International Journal of Algebra and Computation 14, no. 05–06 (2004): 677–702.
[36] V. Guba and M. Sapir, Diagram groups. Vol. 620. American Mathematical Soc., 1997.
[37] D. Kochloukova, C. Martı́nez-Pérez, and B. Nucinkis,“Cohomological finiteness
properties of the BrinThompsonHigman groups 2V and 3V .” Proceedings of the Edinburgh Mathematical Society (Series 2) 56, no. 03 (2013): 777–804.
[38] Y.G. Leonov, “Conjugacy problem in a class of 2-groups.” Mathematical Notes 64,
no. 4 (1998): 496–505.
[39] S. Haagerup, U. Haagerup, and M. Ramirez-Solano, “A computational approach
to the Thompson group F.” International Journal of Algebra and Computation 25,
no. 03 (2015): 381–432.
[40] G. Higman, Finitely presented infinite simple groups. Notes on Pure Math., vol. 8,
Australian National University, Canberra 1974.
[41] J. Kari and N. Ollinger, “Periodicity and immortality in reversible computing.” In
Mathematical Foundations of Computer Science 2008, pp. 419–430. Springer Berlin
Heidelberg, 2008.
[42] K.H. Ko, S.J. Lee, J.H. Cheon, J.W. Han, J.S. Kang, and C. Park, “New publickey cryptosystem using braid groups.” In Advances in cryptology—CRYPTO 2000,
pp. 166–183. Springer Berlin Heidelberg, 2000.
[43] I. Klimann, “The finiteness of a group generated by a 2-letter invertible-reversible
Mealy automaton is decidable.” Leibniz International Proceedings in Informatics
(LIPIcs) 20 (2013): 502–513.
[44] F. Matucci, “Algorithms and classification in groups of piecewise-linear homeomorphisms.” Ph.D. Thesis, Cornell University, 2008, arXiv:math/0807.2871.
24
[45] V. Nekrashevych, “Cuntz-Pimsner algebras of group actions.” Journal of Operator
Theory 52, no. 2 (2004): 223–250.
[46] V. Nekrashevych, “Finitely presented groups associated with expanding maps.”
Preprint (2013). arXiv:math/1312.5654.
[47] P. Novikov, “Unsolvability of the conjugacy problem in group theory.” Izvestiya
Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 18, no. 6 (1954): 485–524.
[48] C. Röver, “Constructing finitely presented simple groups that contain Grigorchuk
groups.” Journal of Algebra 220, no. 1 (1999): 284–313.
[49] C. Röver, “Abstract commensurators of groups acting on rooted trees.” Geometriae
Dedicata 94, no. 1 (2002): 45–61.
[50] L. Sabalka and M. Zaremsky, “On Belk’s classifying space for Thompson’s group F.”
Preprint (2013). arXiv:math/1306.6534.
[51] O. Salazar-Dı́az, “Thompson’s group V from the dynamical viewpoint.” Ph.D. Thesis,
State University of New York at Binghamton, 2006.
[52] S. Witzel and M. Zaremsky, “The Basilica Thompson group is not finitely presented.”
Preprint in preparation.
[53] C. Wladis, “Thompsons group F(n) is not minimally almost convex.” New York
J. Math 13 (2007): 437–481.
25
5
Service
This section summarizes my service to the Bard community since I arrived in 2008.
Service to the Program
• Scheduled mathematics moderation boards (Spring 2015)
• Coordinated mathematics moderation boards (Fall 2014, Spring 2014, Spring 2013)
• Held Math Subject GRE review sessions (Fall 2011, Fall 2012)
• Coordinated the Putnam Exam (Fall 2012)
• Organized the Mathematics/Computer Science/Physics Seminar (2009 – 2011)
• Helped design the Bard Math Placement Diagnostic (Summer 2008)
Service to the College
• Bard College Diversity Committee, 2014 – Present
• Faculty Education and Oversight Committee for Bard College Berlin, 2015 – Present
• Bard College Campus Facilities Committee, 2011 – 2013
• Physics Working Group 2011
• Superadviser during first-year registration (2009, 2010, 2011, 2012)
By far my most significant service to the college since my rehire has been my participation in the faculty Diversity Committee. This committee met 10 times in the 2014–2015
academic year, and worked on three major proposals. First, we submitted to the Executive
Committee proposed revised language regarding the duties of the committee for inclusion
in the faculty handbook. Second, we had a lengthy discussion of the curricular role of
the DIFF requirement, with a view towards how the requirement could be improved. This
resulted in a detailed proposal that was submitted to the DRRC, who have been reconsidering the curricular role of all of the distribution requirements at Bard. Third, we began to
consider Bard’s practices for faculty hiring, and how these could be improved in a way that
would increase Bard’s ability to attract new faculty members with diverse backgrounds.
We started to tackle this last issue near the end of the spring semester, so our discussion did
not result in a concrete proposal, but the committee is planning to continue this work in the
coming year.
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Appendix: Other Materials
Because many of my recent teaching materials take the form of computer animations, Excel
workbooks, and Mathematica notebooks, and class web pages, it is not possible to print out
copies of these teaching materials for my file in a meaningful way. Instead, I will be making
these materials available at the following address:
http://math.bard.edu/belk/teachingmaterials.htm
I have attached a printed copy of each of my recently published papers and submitted
preprints. Links to online versions of these papers can found at the following address:
http://math.bard.edu/belk/research.htm
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