Statement of Purpose
Math first captures
my interest and imagination when my teacher in primary school told us that a
famous American mathematician called Andrew Wiles solved a seemingly easy but very difficult
question
.
This was the first time that math
impressed me
—
a mathematical question coul
d cost
people hundreds of years.
In college, I was attracted by the beauty of
formula in
mathematics
and
enjoyed the competition with each mathematical problem. I
knew I liked math
and
was
gradually
set
ting
the goa
l of being a professional mathematician in my future. I
began to learn
those basic
but still interesting courses and spent my previous 3 years in all these fields
and
was amazed by
Mean Value Theorem when I
first
learnt
from
Calculus. The construction of f
unction to prove that
there exists a
yupsilon
is the most interesting part to solve a problem. I still remember I had been
fascinated
by this for a long time.
F
inally, after finishing the whole part of differentiation
in
Calculus
,
I did a lot of exercise
s
and
made a generalization, which consists of common techniques
for constructing
desired
functions. The generalization report is 7 pages and my calculus professor
remarked that it
was brilliant job for a freshman and it
sh
ould be copied to every student
in
his
calculus classes.
At the end of the
second year of my college life,
I was gradually confirming that I will pursue a
Ph.D in mathematics after graduation and be a professional mathematician in the future.
I
began to
touch on some interesting contributi
ons in
mathematics, especially in
dynamical system and
cryptography.
T
his is just because of a lecture given by Prof. Hu, who introduced us that he was
applying dynamical system in the field of image processing, especially combining with
cryptography. I wa
s
surprised
by this combination and start to contact him to discuss some
relevant ideas. When I
read
some classical literature reviews in this field, I found that in most
papers, disadvantages such as low computation complexity, vulnerability to differenti
al attack and
chosen

plaintext attack cannot be eliminated simultaneously. I found most papers are trying to
utilize one pixel as the smallest unit and I
thought
if we convert the values of pixels into 8

bit
binary sequences and scramble them according to
dynamical system and cryptography rules, we
could get better
permutation and diffusion
result
s
.
Unfortunately, the
outcomes of
simulation experiment did not go as expected. I felt disappoint
ed
about the results since they
were
even
not as good as the prev
ious research. I
then
analyzed all
previous papers
again
and looked up materials about
cryptography
, but found nothing. I thought I
should contact other experts in this field rather than did this alone. I tried to get some help f
rom
Prof. Wang, who was als
o doing
some research in this field. After reading
my algorithm
, he
thought it was great progress and could solve
some of the existing
problems as I mentioned and
encouraged me to continue to find whether the problems was in the theoretical analysis or in
the
experiments. Two months later, I still got nothing and told him I decided to giv
e up. However, in
my heart, I did
n
’
t want to abandon it
.
I believed I had ability and patience to conquer it
. Hence,
another morning when I re

checked all the previous pape
rs, I suddenly found that I could combine
the
classical
pixel method with 8

binary

bit method and it may reach
better
results
than the
research of the past
. I revised my algorithm and analyzed them from theoretical
angle
and
simulated them on a PC. Finally
, it was this combination that helped me reach
a satisfactory result
.
Later, I discussed them with Prof. Wang and Prof. Hu and submitted the final paper
to Chinese
Physics Letters
.
Though my GPA suffered a little that semester given my dedication to resear
ch, I
felt more interested in doing scientific research.
The
experience
of research again
reinforced
my determination
to pursue mathematics as a career.
In the second term of my junior year,
I learnt advanced course of analysis, including Real Analysis
an
d Functional Analysis and
was attracted
by
Prof.
Daoxing Xia
’
s
textbook
of
Functional Analysis
.
Several friends of mine and I then started to organize a discussion
seminar
, setting this book as the
main material.
I thought analysis was classical and elegan
t
. It has more relations with other
directions such as geometry, algebra and physics. Furthermore, Noncommutative Geometry, as a
hot
direction
,
interests me
with its
amazing
combination with
Measure Theory, Algebraic
Topology, Operators Algebra, and P
hysic
s, especially with Quantum Mechanics.
Prof. Connes and
Prof. Yu
are leaders in this field.
Vanderbilt University
is
the center
of Noncommutative Geometry
and Analysis
in the world
.
Prof. Neamtu
’
s contribution in
Approximation
Theory
and Numerical
Analysis
also impresses me. I
believe Vanderbilt University could provide me with excellent
research
conditions so that I can obtain an
academic
position in universities in the future. With the
guidance of Vanderbilt University, I think I will make my own
original
contributions to
mathematics via pursuing a Ph.D here.
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