Research Statement - Mathematics and Computer Science

Sarah Wright Research Statement
My main research interests involve the C ∗ -algebras associated to topological k-graphs. Overall,
this work lies in the broad area of functional analysis and operator algebras. The results of my thesis
have a combinatorial and graph-theoretic feel, while still depending on basic analytical ideas. My
future work in the area of graph algebras involves a more in depth study of the structure of these
graph algebras and relies heavily on some deeper analytic results in operator algebras and groupoid
theory. In addition to this research program, I enjoy working on problems about which I can tell
students, discuss with a wide audience, and in which I can eventually involve undergraduates. Along
these lines, I am currently working on a project to geometrically realize, in R 2 , the complex roots
of polynomials with rational coefficients.
1 Graph C ∗ -Algebras
The study of C ∗ -algebras is an area of specialization within functional analysis. The field of operator
algebras began as a way to study quantum mechanics; C ∗ -algebras provide a mathematical model
of states and observables. A first example of a C∗-algebra is
A = f : R → C
f is continuous and linear,
and lim f(x) = 0
x→±∞
.
In general, any algebra of bounded linear operators on a Hilbert space is a C ∗ -algebra. While a more
algebraic and abstract definition has been developed and is used more often, an important result
of Gelfand and Naimark says that all C ∗ -algebras can be represented as a subalgebra of B(H), the
bounded operators on the Hilbert space H.
As with other abstract mathematical structures, we appreciate versions that we can actually
visualize. In this vein the idea of working with graphs and C ∗ -algebras started. The theory of
graph C ∗ -algebras began with the work of Cuntz and Kreiger [3] and the later work of Enomoto and
Watatani [5]. Since then, there have been many contributions by a variety of researchers resulting
in an extensive collection of literature. The main idea is to associate a C ∗ -algebra to a graph and
use the combinatorics of the graph to answer questions on the structure of the C ∗ -algebra.
An example of this relationship can be seen in the graph criteria equivalent to the simplicity of
the C ∗ -algebra. Simple C ∗ -algebras are those with no non-trivial closed ideals, and therefore are the
building blocks of other C ∗ -algebras. As with the classification of other mathematical structures, we
work to break down large complicated C ∗ -algebras into their smaller simple parts. To appreciate the
theorem relating simplicity of the C ∗ -algebra to the structure of the graph, we need to understand
a few terms as they appear in the graphs below. A cycle (or loop) is a path that starts and ends at
the same point and has no self intersections. An entry is an edge which is not part of the cycle, but
whose range (arrow head) vertex is part of the cycle. A directed graph is considered cofinal if for
every infinite path µ in the graph (or a path that cannot be extended farther backwards because of
a source in the graph) and every vertex v there is another path starting at v and ending somewhere
along µ.
The first directed graph of Figure 1(a) has a cycle of length 5. The sixth edge points out of the
cycle, therefore it is not an entry. The graph in Figure 1(b) has infinitely many cycles of varying
lengths, but we can see each has an entry. The graph in Figure 1(c) is usually referred to as En
and has n-cycles made of one edge each. Any of these cycles is an entry to any other cycle, when
n ≥ 2. Checking cofinality is not as quick, but still only requires looking at the picture of the graph.

Sarah Wright - Research Statement 2
In Figure 1(b), we can see that there is a path from vm to vn if m > n and a path from v1 to any
other vertex. These two observations together give a path from any vertex to any other vertex, so
the graph is cofinal.
v1
v2
(a) Cycle Without Entry
(b) A cofinal graph
(c) The Graph En
Figure 1: Three Examples of Directed Graphs
The existence of entries for each cycle and the cofinality of the graph relate to the structure of the
C ∗ -algebra by the following theorem of Kumjian, Pask, Raeburn, and Renault. The requirement
of a graph to be row-finite (each vertex receives only finitely many edges) was later dealt with by
Drinen and Tomforde yielding a similar theorem in [4].
Theorem. [15, Theorem 4.14] Suppose E is a row-finite graph in which every cycle has an entry.
Then C ∗ (E) is simple if and only if E is cofinal.
In particular, this theorem tells us the C ∗ -algebra associated to the graph En, Figure 1(c), which is
the Cuntz algebra On, is simple when n ≥ 2. As the Cuntz Algebra is well-studied, its simplicity is
not new upon the proof of this theorem, but the ease with which the theorem is applied demonstrates
its importance and beauty. There are further theorems in the field that give a description of the
ideal structure of a graph-algebra, see [2, 8]. Similar to the theorem above, the results give a way
to nearly “read-off” the ideal structure of the algebra from properties of the graph.
The study of graph algebras had been particularly fruitful in its provision of a rich class of easily
accessible examples of C ∗ -algebras with various properties. To this end, the idea of a graph has been
generalized in many ways to provide even more examples. The long list of generalizations includes
the labeled graphs of Bates and Pask where edges of the graph are labelled from an alphabet set, the
k-graphs of Kumjian and Pask, where rather than paths of length ℓ ∈ N we work with k-dimensional
paths of shape n ∈ N k , and the topological graphs of Katsura, where instead of discrete sets of
vertices and edges we give these spaces interesting topologies. For definitions of these structures see
[1], [12], and [11] respectively. Most recently the work of Yeend provides a generalization of both
higher-rank graphs and topological graphs, with the unifying theory of topological k-graphs [21].
2 Aperiodicity Conditions and Related Results
The importance of cycles having or not having entries is the idea of being able to “back out of a
cycle” and build an aperiodic infinite path. (The convention to construct paths backward comes
from the methods used to construct the associated C ∗ -algebra.) The relationship between the
periodic paths (or lack there of) in the graph and the simplicity of the associated C ∗ -algebra is
an important result for all types of graphs. The C ∗ -algebras of directed graphs, k-graphs, and
topological k-graphs were all originally constructed using groupoid methods (a well-studied method
of constructing a C ∗ -algebra that generalizes the algebra generated by a group). More elementary
methods for construction have been established since these initial works. As the infinite paths of
the graph play such a central role in the groupoid construction of the C ∗ -algebra, in the case of
higher-rank graphs, the original aperiodicity condition, called Condition (A), was stated in terms
v3
v4
v5
v6
en
.
e3
e2
e1
v

Sarah Wright - Research Statement 3
of these infinite paths [12]. Lewin, Robertson, and Sims have developed equivalent conditions to
aperiodicity [18, 13] that have proven easier to verify in concrete examples and more useful tools in
proofs. In the following theorem, I extend these results to topological k-graphs.
Theorem. [20, Theorem 3.1] Let (Λ, d) be a compactly aligned topological k-graph with no sources.
Then the following conditions are equivalent.
(A) For any open set V ⊂ Λ 0 there exists an aperiodic path x ∈ V ∂Λ.
(B) For any open set V ⊂ Λ 0 and any pair m = n ∈ N k there exists a path λV,m,n ∈ V Λ such that
d(λ) ≥ m ∨ n and
λ(m, m + d(λ) − (m ∨ n)) = λ(n, n + d(λ) − (m ∨ n)). (⋆)
(C) For any distinct pair of open sets X, Y ⊂ Λ such that s(X) = s(Y ) and both s|X and s|Y are
homeomorphisms there exists τ ∈ s(X)Λ such that MCE(Xτ, Y τ) = ∅.
Condition (A) is Yeend’s aperiodicity condition concerning infinite paths. The second condition
deals with specific sections of finite paths. In a directed graph it would require that for any vertex
v and any m = n ∈ N we could find a path ending at v where the m th and n th edges are distinct.
Condition (C) deals with the possibility of extending distinct paths such that they are equivalent.
Considering each condition of the above theorem in the discrete k-graph case, we derive a corollary
giving the equivalence of Kumjian and Pask’s [12], Robinson and Sims’ [18], and Lewin and Sims’
[13] definitions of aperiodicity. This is a previously known fact, but reducing the proof of the theorem
gives a direct proof, which is a nice (and new) result.
To demonstrate the importance of these formulations of aperiodicity I have constructed a new
class of examples of topological k-graphs, which I call twisted product topological k-graphs.
α 1 0
α 2 α
0
2 1
α 3 α
0
3 α
1
3 2
v0 v1 v2 v3
β 1 0 β 2 0
β 2 β
1
3 0
β 3 1
β 3 2
· · · · · ·
vn
α n n-1
.
α n α
0
n 1
β n 0
β n 1
.
β n n-1
vn+1
Figure 2: We visualize k-graphs by drawing
their 1-skeletons. The vertices are those of
the k-graph, the edges are the paths of the kgraph
of shape ei ∈ N k which are colored in kdifferent
colors to show their shape. To know
the entire structure of the k-graph we also need
to give factorization rules. See [15, 16].
The idea is to start with a discrete k-graph Λ and a topological space X. We replace each vertex in
Λ with a copy of X and twist the sources of the edges according to a continuous functor τ. Starting
with the k-graph whose 1-skeleton is pictured in Figure 2, replacing the vertices with a copy of
the circle T, and twisting the edges by a factor of n in the n th grouping gives and example of a
topological 2-graph in which Condition (C) is fairly straightforward to check, where Condition (A) is
nearly impossible [20]. The previous example demonstrates the benefits of Condition (C) in checking
aperiodicity for a particular topological k-graph, but it is also useful condition in proving other more
general results. In [17], the authors use Condition (C) in the proof of a Cuntz-Krieger Uniqueness
Theorem for topological k-graphs.
3 On-Going and Current Work
3.1 Aperiodicity and Twisted Product Graphs
It is worth noting that my work in [20] covers only topological k-graphs without sources. This was a
condition required in the early work on directed graphs, topological graphs, and k-graphs alike that

Sarah Wright - Research Statement 4
was eventually replaced with less restrictive hypotheses. I am currently working to develop similar
conditions and results for topological k-graphs with sources.
Many interesting questions arise out of the development of these twisted topological k-graphs.
The first and (I feel) most obvious question to ask is, “What structural similarities does C ∗ (Λ×τ X)
inherit from the discrete k-graph (Λ, d)?” In his thesis work developing the basic theory of topological
k-graphs, Yeend defines and describes similar constructions for constructing new topological graphs
from old ones. He is able to show specific properties of the graph and the associated C ∗ -algebra
that are retained through these constructions.
I am particularly interested in the levels of aperiodicity that are retained or lost in this twisted
product. Can we start with a periodic discrete k-graph and twist it in a particular way to make
it aperiodic? Can the reverse be done? The fact that Conditions (B) and (C) of the theorem
above reduce aperiodicity to a question of finite paths rather than infinite makes the questions of
aperiodicity in the twisted product graphs easier to answer.
3.2 Labeled Higher-Rank Graphs
Yeend combined two previously studied generalizations of graphs to define topological k-graphs.
Beyond the fact that this provides one unifying theory of the two fields, it also provides researchers
with an even larger class of C ∗ -algebras that can be visualized. The work of Bates and Pask on
labeled graphs and their associated C ∗ -algebras simultaneously generalize the ultragraph algebras
of Tomforde [19] and the shift space C ∗ -algebras of Matsumoto [14]. Currently, there is no work
that combines labeled graphs with higher-rank graphs. The work of [7] gives fundamental results
concerning the structure of the 1-skeleton and set of commuting squares of a higher-rank graph.
This gives a set of objects generalizing the edge set of a directed graph, that we can label with an
alphabet set. At the most recent Great Plains Operator Theory Conference in Houston, Sam Webster
presented the results of [7], and Paulette Willis and I started began considering and discussing the
idea of labeling higher-rank graphs. I am currently starting work with Pask, Webster and Willis to
formalize these results into a theory of labeled k-graph algebras.
3.3 The Structure of the Primitive Ideal Space
At the same GPOTS conference in June, Sooran Kang presented her work with David Pask on the
primitive ideal space of a row-finite k-graph [10]. A member of the audience asked if she could say
when this space was Hausdorff. She was not able to give an answer to this question, but pointed
out that there was a similar open question in the field of directed graph-algebras. Geoff Goehle,
has results answering such questions on groupoid C ∗ -algebras. As graph-algebras were originally
constructed using a groupoid model, it is reasonable to think we have almost everything we needed
to answer these questions for graph-algebras of various types. Using the work of Goehle, Clark,
and Williams, we are working to translate groupoid C ∗ -algebra results into conditions on the graph,
whether it be a directed graph, k-graph, or a structure even more general.
3.4 Complex Roots of Polynomials
An idea any calculus student can understand is that we can know the real roots of a polynomial
function or solutions to a polynomial equation by seeing where the graph of the function crosses
the x-axis. But, what about those roots that are not real? Are there geometric or graphical
interpretations of imaginary and complex roots?

Sarah Wright - Research Statement
5
It turns out the answer to that is also yes. The work of Gehman
[6] and Irwin and Wright [9] deal with constructions of complex roots
of polynomials of various degrees. For example, the cubic of Figure 3
has one real root at R, but we know from the Fundamental Theorem
of Algebra that there must be two more. The line in Figure 3 is
constructed to intersect the real root at R and be tangent to the curve
at the point T . We call the slope of this tangent line β and let α be
Figure 3: A cubic polyno-
the abscissa of the point T , then the work of Irwin and Wright states
mial
that the two complex roots of the cubic are α ± βi.
While the work of those mentioned above does cover some higher degree polynomials, we are
not able to find results that cover polynomials with no real roots. Mitsuo Kobayashi (a graduate
school and Project NExT colleague of mine) and I are hoping to further generalize some of these
results to higher degree polynomials, and to those with only complex roots. Both of our research
areas (he works in analytic number theory) depend on basic complex analysis, but this has never
been the focus of either of our work. We are motivated by our love of talking to everyone about
mathematics as well as the enjoyment we get from listening to our students ideas about solving new
problems. We hope to have results soon and present them not only as generalizations of the results
of Gehman, Irwin, and Wright, but as a next step in the classical constructions of Greek geometry
where one is only allowed a compass and straight edge. If we allow graphs
of polynomials
it is easy
to construct the cube root of two, for example. If we also allow the use of
a protractor to measure
slope, what class of numbers can we add to the list of constructible numbers? We hope
not only to
determine these complex roots, but also investigate the minimal tools needed
to construct them.
References
[1] Teresa Bates and David Pask, C
∗-algebras of labeled graphs, J. Operator Theory 57 (2007),
no. 1, 207–226.
[2] Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, The
C
∗-algebras of row-finite
graphs, New York J. Math. 6 (2000), 307–324 (electronic). MR MR1777234 (2001k:46084)
[3] Joachim Cuntz and Wolfgang Krieger, A class of C∗-algebras and topological Markov chains,
Invent. Math. 56 (1980), no. 3, 251–268. MR MR561974 (82f:46073a)
[4] D. Drinen and M. Tomforde, The C∗-algebras of arbitrary graphs, Rocky Mountain J. Math.
35 (2005), no. 1, 105–135. MR MR2117597 (2006h:46051)
[5] Masatoshi Enomoto and Yasuo Watatani, A graph theory for C∗-algebras, Math. Japon. 25
(1980), no. 4, 435–442. MR MR594544 (83d:46069a)
[6] H. M. Gehman, Complex roots of a polynomial equation, American Mathematical Monthly 28
(1941), no. 4, 237–239.
[7] Robert Hazlewood, Iain Raeburn, Aidan Sims, and Samuel B.G. Webster, On some fundamental
results about higher-rank graphs and their C∗-algebras, preprint, arXiv:1110.2269v1.
[8] Jeong Hee Hong and Wojciech Szymański, The primitive ideal space of the C∗-algebras of
infinite graphs, J. Math. Soc. Japan 56 (2004), no. 1, 45–64. MR 2023453 (2004j:46088)

Sarah Wright - Research Statement 6
[9] Frank Irwin and Wright H. N., Some properties of polynomial curves, The Annals of Mathematics
19 (1917), no. 2, 152–158.
[10] Sooran Kang and David Pask, Aperiodicity and the primitive ideal space of a row-finite k-graph
C ∗ -algebra, preprint, arXiv:1105.1208v2.
[11] Takeshi Katsura, A class of C ∗ -algebras generalizing both graph algebras and homeomorphism
C ∗ -algebras. I. Fundamental results, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4287–4322
(electronic).
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1–20 (electronic). MR MR1745529 (2001b:46102)
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Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 333–350. MR 2670219
[14] Kengo Matsumoto, On C ∗ -algebras associated with subshifts, Internat. J. Math. 8 (1997), no. 3,
357–374. MR 1454478 (98h:46077)
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Edinb. Math. Soc. (2) 46 (2003), no. 1, 99–115. MR MR1961175 (2004f:46068)
[17] Jean Ranault, Aidan Sims, Dana P. Williams, and Trent Yeend, Uniqueness theorems for
topological higher-rank graph C ∗ -algebras, preprint.
[18] David I. Robertson and Aidan Sims, Simplicity of C ∗ -algebras associated to higher-rank graphs,
Bull. Lond. Math. Soc. 39 (2007), no. 2, 337–344. MR MR2323468 (2008g:46099)
[19] Mark Tomforde, A unified approach to Exel-Laca algebras and C ∗ -algebras associated to graphs,
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[20] Sarah Wright, Aperiodicity conditions in topological k-graphs, J. Operator Theory, to appear.
[21] Trent Yeend, Topological higher-rank graphs, their groupoids and operator algebras, Ph.D. thesis,
March 2005.