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Researching and understanding the process of proving the Poincaré conjecture
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Researching and understanding the process of proving the Poincaré conjecture
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Description
Identifier
Thesis
1978
Author
Nelson, Julie M.
Title
Researching
and
understanding
the
process
of
proving
the
Poincaré
conjecture
Publisher
Central Connecticut State University
Date of Publication
2008
Resource Type
Master's Thesis
Abstract
What
began
as a
competition
to
appease
a
member
of
royalty's
birthday
wishes
,
continued
on to not
only
introduce
an
entirely
new
area
of
mathematical
studies
but also to
leave
one
particular
question
to
remain
without
a
complete
and
substantiated
answer
for a
span
of
centuries
. On a
much
smaller
,
simpler
scale
, the
consideration
of the
classification
of
surfaces
can
help
to
model
the
overall
goal
within
the
Poincar
Conjecture
. By
surfaces
we
mean
any
twodimensional
figure
which
can
cover
a
threedimensional
figure
. These
concepts
can
be
simply
manipulated
and
proven
to
stem
from the
combinations
of
three
basic
twodimensional
figures
. The
Poincar
Conjecture
takes
this
proposition
to a
new
level
and
seeks
to
find
a
contradiction
to a
threedimensional
body
having
the
same
physical
characteristics
of what will be
considered
a
3sphere
while
not
being
able
to be
molded
into the
3sphere
itself
. The
complexity
of the
Poincar
Conjecture
lies
in the
fact
that
we
are
unable
to
manipulate
such
threedimensional
bodies
because
in
order
to
do
so
we
must
imagine
ourselves
in a
fourdimensional
world
.
It
is
this
one
element
which
proves
to be
most
challenging
to those
who
have
made
attempts
to
answer
the
unanswerable
since
without
the
knowledge
of what
actions
can
and
cannot
be
performed
in the
fourth
dimension
. This
paper
is
an
entirely
basic
outline
of the
process
which
was
involved
to
prove
the
Poincar
Conjecture
.
It
begins
with the
classification
of
surfaces
in
order
to have the
overall
understanding
of the
basic
representations
and
manipulations
that are
used
throughout
the
process
. This also
helps
to
indicate
the
structure
that
many
mathematicians
sought
to
follow
within
their
approach
due
to the
fact
that
it
seemed
the
most
logical
format
.
After
the
classification
of
surfaces
, the
paper
moves
on to
consider
various
topological
invariants
, as
well
as the
mathematicians
whose
work
supplemented
one
another
in
order
to
arrive
at
new
branches
of
mathematics
. With
each
new
piece
of
information
,
more
mathematicians
were
able
to
find
intricate
areas
of
relevance
to the
problem
at
hand
.
Within
this
section
, the
Euler
characteristic
is
introduced
,
along
with
Thurston
, a
mathematician
who
was
able
to
explain
how
being
in the
fourth
dimension
could
be
considered
. He was
able
to
take
prior
knowledge
,
along
with his
own
visualizations
and
formulate
some
of the
most
unique
and
inspiring
work
on the
Poincar
Conjecture
,
along
with the
area
of
topology
altogether
. For the
remainder
of the
paper
, the
focus
is
on the
actual
attempts
to
prove
and
disprove
the
Poincar
Conjecture
using
the
knowledge
discussed
previously
in the
paper
.
Finally
we
are
able
to
revel
in the
general
concept
of the
proof
which
was
brought
together
in
2003
by
Grigory
Perelman
,
who
was
able
to
bring
together
the
work
of
many
mathematicians
and
fill
in the
gaps
which
they had been
missing
.
Every
piece
of his
proof
required
more
proof
and he had the
reasoning
to
back
it
up
. To
see
how
everything
was
able
to
fit
together
so
beautifully
is
breathtaking
,
which
is
precisely
why
this
paper
was
able
to
come
together
in the
end
.
Subject
Poincaré conjecture
Department
Department of Mathematical Sciences
Advisor
McGowan, Jeffrey K
Type
Text
Digital Format
application/pdf
Language
eng
OCLC number
713733989
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