1.5 2
1
0.5
-1.5
-1
0
-0.5
Im
f (z)
Im
0.5
1
Re
-3
1.5
-2
-1
0
1
2
Re
3
4
5
-0.5
-1
-2
-1.5
f (z) = h(z) + g(z)
(( ) ( ( )–2
)
2 = 18 12 11 +– zz + 2 11 +– zz + log 11 +– zz – 52 +
1 8
1 1+z 2 1–z
2
1 + z + log 1 + z + 3 1–z 2 1–z
)
Research While there are many exciting areas of mathematics
and graphs seem to indicate a conjecture’s truthfulness. For
research that are more immediately applicable to
example, in 1958, at the conclusion of a groundbreaking
medicine, industry and government, there are just as
paper in my research niche, the two authors posed an
many amazing and valuable avenues of pure mathematics
open question, now known as the Pólya-Schoenberg
research that advance our knowledge. Moreover, there is
Conjecture. Though many outstanding mathematicians
a great deal of give and take between pure and applied
worked on this conjecture, making progress on special
mathematics, making each useful to the other. My research
cases, it was not until 2003, through an imaginative and
as a pure mathematician is in the field of complex analysis
unexpected approach using a differential equation, that
with a specialization in geometric function theory. My
this conjecture was proved. In 2007, I published a paper
motivation to understand and discover more mathematics
on an extension of these results. One of the hypotheses
is grounded in the search for knowledge itself rather than
of my main theorem involves a geometric condition that I
being driven by an application, a perplexing thought for
believe will be true in a more general setting. While I have
my parents wondering what I was doing all those years in
yet to produce an example that fails to support my belief,
graduate school and how I could make a living from it!
the general result remains elusive and is an ongoing work
Since I study the geometric properties of mappings,
in progress. In 2008-2009, I mentored an honors student
I have the advantage of being able to use mathematical
who investigated specific changes to the differential
software and graphics programs to explore conjectures
equation and the geometry of the resulting solution
and pose new questions, a luxury not available to many
graphs. The work this student conducted in her honors
other mathematicians. However, the time comes to turn
thesis just to understand the problem goes well beyond
off the computer and pick up a pencil to pursue a rigorous
what our typical undergraduate majors learn, and she
argument. This process often requires creativity and can
produced an additional example related to this research
sometimes be unsuccessful no matter how many examples
problem in support of my more general hypothesis.
Diagrams courtesy of Dr. Muir Spring 2013 The University of Scranton
27