Max Butler
Home university:
University of Waterloo
Supervisor:
P. Haxell
Project:
This
summer
I
have
been
working
on
the
satisfiability
problem
by
turning
it
into
a
graph
theory
problem.
We
then
tried
to
apply
and
improve
upon
different
algorithms
from
graph
theory
to
this
very
specialized
case.
The
goal
is
to
find
a
set
of
independant
vertices
on
the
graph.
We
have
made
some
progress,
guaranteeing
the
existence
of
a
partial
solution.
Most
of
the
work
is
about
reformulating
the
problem
in
different
ways
so
we
can
attack
it
differently,
or
testing
out
a
new
way
of
looking
at
it.
Sometimes
the
new
approaches
don't
work,
other
times
they
yield
a
small
improvement.
I
meet
with
my
superviser
once
a
week,
to
talk
about
the
progess
I
am
making
and
to
discuss
some
new
ideas,
and
then
I
go
work
on
it
until
the
next
week.
Most
of
my
time
is
spent
working
on
the
ideas
my
superviser
has
given
me,
but
sometimes
I'll
try
some
other
approach
of
my
own
if
that
doesn't
lead
anywhere.
Overall
the
Undergraduate
Research
Assistant
(URA)
program
is
very
fun,
and
I
am
very
glad
to
have
learned
what
real
research
is
really
like.
Specifically,
I
learned
how
the
process
of
doing
research
is
much
slower
and
more
long-term
than
doing
math
problems
from
the
context
of
a
homework
problem.
Jon Dietrich
Home university:
UWaterloo
Supervisor:
W.H. Cunningham
Project:
The
URA
position
is
the
perfect
experience
for
anyone
interested
in
continuing
their
mathematical
studies
in
graduate
school
and
beyond.
The
job
promotes
independent
research
and
thought
to
a
degree
which
many
undergraduates
never
have
the
chance
to
experience.
My
own
research
has
been
directed
towards
understanding
the
change
in
dimension
of
the
even
permutations
when
certain
edges
of
the
complete
digraph
are
removed.
In
certain
cases
the
dimension
is
lower
than
it
'should'
be.
The
question
is
why
this
occurs.
And
not
only
is
the
answer
to
this
question
unknown,
to
my
knowledge
no
one
before
me
has
ever
even
worked
on
the
problem!
It
has
been
an
absolute
joy
and
privilege
to
work
in
the
program
on
questions
to
which
there
simply
are
no
answers
yet.
The
URA
program
is
highly
recommended
to
anyone
serious
and
passionate
about
learning
and
their
education.
Ralph Furmaniak
Home university:
UWaterloo
Supervisor:
L. Kauffman
Project:
Normal
lectures
and
assignments
are
great,
but
I
personally
prefer
to
be
able
to
use
what
I
learn
towards
solving
current
problems,
and
to
have
a
bit
more
ndependence
and
creativity.
This
is
why
the
undergraduate
URA
in
Combinatorics
and
Optimization
(C
and
O)
appealed
to
me.
It
has
been
a
great
experience,
allowing
me
to
meet
many
people
and
work
on
various
interesting
problems.
Not
too
many
people
seem
to
know
about
this
opportunity,
or
have
some
misunderstanding
of
it.
For
example,
one
classmate
I
spoke
to
earlier
in
the
year
completely
ruled
it
out
back
then,
thinking
that
she
was
not
yet
able.
Yet
I
have
seen
many
different
types
of
people
here,
including
her
and
a
few
other
students
who
just
finished
first
year.
If
you
do
love
math,
and
are
up
to
the
challenge
and
especially
if
you
are
considering
grad
studies,
this
is
in
my
opinion
the
best
way
to
spend
your
summer.
Just
be
prepared
to
have
to
learn
many
new
things
along
the
way.
Andrew Hoefel
Home university:
UWaterloo
Supervisor:
J. Verstraete
Project:
I
worked
with
Professor
Verstraete
on
various
problems
in
extremal
graph
theory.
Early
in
the
summer
we
spent
time
investigating
the
maximum
number
of
edges
that
a
bipartite
graph
with
no
2k-cycle
can
contain.
The
asymptotic
relationship
between
the
number
of
edges
and
vertices
in
these
graphs
are
known
for
a
few
values
of
k.
We
built
numerous
constructions
of
such
graphs,
using
Cayley
graphs
and
other
techniques,
to
find
lower
bounds
while
considering
discrete
Fourier
analysis
and
probabilistic
methods
to
find
upper
bounds.
Later
in
the
term
we
discussed
similar
problems
involving
maximal
sets
that
do
not
satisfy
certain
categories
of
linear
integer
equations.
The
summer
URA
program
provides
a
warm
atmosphere
for
a
great
first
experience
in
research.
Update 2008:
Graduate student, Dalhousie University
Mohammed Omar
Home university:
UWaterloo
Supervisor:
L.B. Richmond
Project:
This
term
has
been
an
investigation
of
the
asymptotics
of
combinatorial
structures
with
small
largest
component.
Simple
applications
include
permutations
decomposed
into
disjoint
cycles
and
polynomials
over
a
finite
field
factored
into
irreducibles.
It
was
conjectured
that
the
asymptotics
are
closely
related
to
the
Dickman
function,
a
differential
difference
equation
arising
in
many
number
theoretic
settings.
This
conjecture
has
now
been
proven.
Further
investigation
into
these
asymptotics
under
more
relaxed
conditions
is
now
being
conducted.
The
URA
program
provided
an
opportunity
to
apply
various
mathematical
tools
and
learn
new
ones.
Combined
with
working
in
an
environment
with
motivated
and
talented
students,
it
proved
to
be
an
enriching
and
rewarding
experience.
I
would
like
to
thank
Prof.
Bruce
Richmond
for
his
efforts
in
supervising.
Jordan Sehn
Home university:
UWaterloo
Supervisor:
J. Cheriyan
Project:
My
term
has
centered
around
finding
a
"good"
approximation
algorithm
for
minimum-cost
3-vertex-connected
graphs.
This
work
lead
to
the
exploration
of
connections
between
edge
and
vertex
connectivity;
as
well
as
related
integer
programs
and
their
IP/LP
gap.
I
am
very
grateful
to
have
geen
given
this
opportunity
and
it
has
been
a
valuable
experience.
I
learned
a
lot
about
connectivity,
graph
theory,
and
optimization
techniques.
My
supervisor
Professor
Cheriyan
was
very
helpful
and
I
thoroughly
enjoyed
working
with
him.
The
summer
URA
program
is
great
and
I
would
definitely
recommend
it.
Jamie Sikora
Home university:
UWaterloo
Supervisor:
B. Guenin
Project:
The
URA
program
has
been
a
very
enlightening
experience.
It
has
given
me
the
chance
to
get
a
glimpse
of
what
graduate
studies
is
like
and
the
chance
to
get
to
know
many
of
the
staff
members
in
the
department.
The
great
part
of
the
program
is
that
it
has
given
me
the
chance
to
see
another
side
of
mathematics,
a
more
constructive
side.
We
had
weekly
meetings
(accompanied
with
professor
Jim
Geelen)
and
threw
around
some
ideas
and
eventually
agreeing
on
some.
Then
my
part
was
to
put
in
all
together,
make
it
work,
simplify
it,
come
up
with
new
ideas,
prove
them,
etc.
Overall,
it
is
a
great
job
and
getting
paid
to
do
graph
theory,
definitely
awesome!
Update
2008: Graduate
student,
C
and
O
Department,
University
of
Waterloo
William Slofstra
Home university:
UWaterloo
Supervisor:
I. Goulden
Project:
I
have
been
working
on
generalizations
of
the
Harer-Zagier
formula,
under
the
supervision
of
Professor
Goulden
and
Professor
Nica.
The
formula
counts
factorizations
of
permutations
with
restrictions
to
certain
conjugacy
classes.
I
have
really
enjoyed
working
on
this
problem
with
my
supervisors,
and
learning
about
new
areas
of
mathematics.
David Tweedle
Home university:
UWaterloo
Supervisor:
H. Wolkowicz
Project:
My
work
with
Professor
Wolkowicz
centred
on
matrix
completion
problems
(specifically
Euclidean
Distance
Matrix
Completion).
We
used
several
different
approaches
including
a
semidefinite
programming
approach
and
a
non-smooth
optimization
approach,
both
using
matlab.
The
URA
program
is
very
good
if
you
are
thinking
about
grad
studies
(like
me)
and
want
to
get
a
taste
for
research.
Also,
the
opportunity
to
develop
a
relationship
with
my
supervisor
and
other
URA's
was
quite
valuable.
Zheng Wang
Home university:
UWaterloo
Supervisor:
A. Menezes/N. Theriault
Project:
I've
been
working
on
solving
Discrete
Log
Problem
for
Genus
four
Hyperelliptic
curves
for
the
past
two
months.
The
Discrete
Logarithm
Problem
(DLP)
is
a
very
hard
problem
and
its
intractablity
is
essential
to
the
security
of
many
public-key
cryptographic
schemes.
The
choice
of
which
group
to
use
in
DLP
is
crucial
to
both
the
efficiency
and
the
security
of
cryptosystem.
And
we
are
working
on
the
groups
from
Jacobian
of
hyperelliptic
curves
of
genus
four,
and
the
best
known
attack
on
the
DLP
is
the
index
calulus
attack.
My
job
is
to
implement
a
fast
attack
on
this
special
DLP.
The
hardest
part
for
me
is
to
fully
understand
the
behavior
of
the
groups
from
the
Jacobian
of
Hyperelliptic
curves.
I
am
very
appreciated
to
have
the
oppotunity
to
learn
such
advanced
materials.
My
opinion
on
the
URA
program
is
that
URA
makes
us
appreciate
how
beautiful
and
useful
math
can
be.
Anyu Zhang
Home university:
UWaterloo
Supervisor:
R. Shioda
Project:
I
have
been
working
with
Professor
Shioda
on
Support
Vector
Machines
over
the
Summer.
The
Support
Vector
Machine
is
a
tool
for
classification
and
an
important
subject
in
Optimization.
Finding
a
unique,
efficient,
and
well-approximated
formulation
to
solve
the
SVM
problem
is
of
interest.
It's
been
a
wonderful
learning
and
working
experience.
Not
only
have
I
learned
the
topic
of
SVMs,
but
I
have
also
learned
many
other
topics
as
well
as
how
to
use
some
important
mathematical
software
relevant
to
SVMs.
A
Summer
URA
is
defintely
a
credible
working
experience
for
people
who
love
Math
and
consider
Graduate
studies
in
the
near
future.