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\centerline{PURE Math 2012 -- Ideas for Research Projects}
\centerline{{\it First Draft} -- April 28, 2012}
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The following are my initial ideas for the research projects the 
students at PUREMath 2012 will work on.  I am expecting to have
4 groups of 3 students each formed from the {\it Residents} 
at the program working on these for the last 4.5 - 5 weeks
of the program.  There are another 2 or 3 ideas I have in various
states of development at this point.  
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{\it Schedule and Logistics}
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[Fill in later:  How will groups be chosen?  How will topics be
assigned, or will the groups get to chose?  
Who will be the primary contact people for which groups?
What will the daily schedule during the research phase look like?
What will deadlines for various components be?  What will be the
expectations for
progress reports, final technical reports, presentations, etc.]
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{\it A good starting point for context and ideas -- all projects}
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\item{[S]}  Saari, Donald G. Central Conﬁgurations--A Problem for the Twenty-first Century, 
{\tt math.uci.edu/~dsaari/BAMA-pap.pdf}

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{\it Idea 1 -- The ``Planar Restricted 4-body Problem''}
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{\it Comments:}  Getting really good numerically-computed results
here is a nice computational challenge, although much of this 
has been done before in different forms and quite a bit has been
claimed on the basis of numerical evidence.  Formal proofs of any 
sort would be even better, though they are even harder 
to come by in this problem.
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Although the central configurations in the 3-body problem have
been essentially {\it completely} known for many years, the same cannot
really be said even now for the 4-body problem.  There are 
a number of open problems concerning the possible central 
configurations with various specific collections of masses, or with
particular geometries.  For this project, you would investigate
one variation that has attracted some interest in work of
J. Kulevich, G. Roberts and C. Smith ([KRS]), and J. Barros and E. Leandro ([BL]).  If 
three of the bodies have much, much larger mass than the fourth
(think of the system consisting of an inhabited planet with two typical
moons, and a much, much smaller satellite launched from the planet), 
then we can essentially treat the fourth mass as {\it zero} and look at 
possible central configurations in this situation.  If all four bodies have 
center of mass in a single plane, this is called the ``planar restricted
4-body problem'' in the sources below.  

If the three large masses are normalized (e.g. by setting one of 
them to $1$, or by making the sum equal to $1$), then it is 
an interesting problem to try to determine exactly how many central configurations
there are for each combination of masses, where the changeovers or 
{\it bifurcations} occur, and exactly what occurs at the bifurcations.

For this project the goals would be to 
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\item{(a)}  First, get a feeling for what is known in this area by looking
at the references below.   
\item{(b)}  Do some experimentation in {\tt Sage} to get the ``lay of the land.''  How
many solutions do there seem to be for different combinations of the large masses?
What if two of them are equal?
\item{(c)}  Are there actually just three different possiblities for the 
numbers of central configurations?  Suppose the three large masses are normalized to
$(1,m_2,m_3)$.  In the first quadrant of the $(m_2,m_3)$ plane, which regions correspond
to which numbers?  
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{\bf References}
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\item{[BL]}  Barros, Jean F.; Leandro, Eduardo S. G. The set of degenerate central configurations in the planar restricted four-body problem. {\it SIAM J. Math. Anal.} {\bf 43} (2011), no. 2, 
634--661.
\item{[KRS]} Kulevich, Julianne L.; Roberts, Gareth E.; Smith, Christopher J.
Finiteness in the planar restricted four-body problem.
{\it Qual. Theory Dyn. Syst.} {\bf 8} (2009), no. 2, 357--370. 
\item{[L]}  Leandro, Eduardo S. G. On the central configurations of the planar 
restricted four-body problem. 
{\it J. Differential Equations} {\bf 226} (2006), no. 1, 323–351. 
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{\it Idea 2 -- Finiteness Questions with Negative Masses}
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{\it Comment:}  This is probably the most open-ended
of these topics -- it's an area where I have looked at
a lot of things that {\it have not} worked to produce
examples of the kind described below.  Nevertheless

Perhaps the major unsolved problem about central configurations 
is whether the number of such configurations for a given 
collection of masses $m_i > 0$ is always finite.  The answer
is known to be yes for the $n = 3, 4$-body problems, but 
this is not quite known for $n = 5$, and essentially 
completely open in general for $n \ge 6$.  One cautionary 
example is given by the article [R] below, which shows
that in the planar 5-body problem, if one of the 
masses is allowed to be negative, then the finiteness
fails -- there is an infinite (one real parameter) family
of central configurations in that case for the same collection
of masses.  This example seems quite ``generalizable,'' but 
to my knowledge, no one has found any additional examples of
such families (allowing some negative masses).  It would 
be quite interesting to know whether there are any!

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The goals of this project would be to 
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\item{(a)}  Start by verifying Roberts' example using the
Albouy-Chenciner equations (that's not the approach 
he used), then 
\item{(b)}  Explore other related configurations to see
whether similar phenomena occur with $n > 5$ masses (some
allowed to be negative.  ''Anything goes'' here since
you are essentially looking for a new example that
no one has seen before.  But some ideas would be to 
look at cases where you have more than one ``ring'' 
of masses about a central mass, look at other cases
where the configuration can ``flex'' to an extent 
leaving some of the distances fixed, etc.   
\item{(c)}  On the other hand, might Roberts' example
be the {\it only one} of this type?  That seems unlikely
to me, but it's certainly possible.  If you 
could understand how to make the phrase ``of this type''
more precise and prove that, it would be a major result.  
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{\bf References}
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\item[R] Roberts, G.E., A continuum of relative equilibria in the ﬁve-body problem, 
{\it Physica D}, {\bf  127} (1999), 141--145.

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{\it Idea 3 -- Central Configurations of Vortices}
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The Albouy-Chenciner equations for central configurations of masses
under the Newtonian gravitation law can be generalized in many ways.  One
that also has real physical interest concerns
the problem of central configurations for any number $n \ge 3$ of ``Helmholtz vortices.'' 
Consider a perfect fluid with an infinite horizontal surface and a constant 
thickness.  Some states of the fluid where small whirlpools are present
are well described by
giving the positions and the vorticities (a measure of ``strength'' of 
the rotation) of a finite number of ``point vortices.''   (Some 
famous experiments by A. M. Mayer simulate Helmholtz vortices by a device where 
$N$ identical magnets are ﬂoating on a surface of water.  Equations similar to 
those developed by Helmholtz are quite frequent in physical models.)
There exist relative equilibria, i.e. configurations
analogous to planar central configurations in the Newtonian problem that remain unchanged up
to rotation and translation.  For instance, collinear configuration of $N$ vortices exist
for all $N$.  

In algebraic terms, the equations describing configurations of vortices
have the same form as the Albouy-Chenciner equations, but in which 
the $S_{ij} = {1\over r_{ij}^3} - 1$ are replaced by the somewhat
simpler 
$S_{ij} = {1\over r_{ij}^2} - 1$.  The $m_i$ must also be interpreted
as the vorticities, which can can be positive or negative real numbers.

For this project, the main goals would be to 
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\item{(a)}  Work through the 3-vortex problem using our algebraic
tools and {\tt Sage} and understand the similarities and differences with
the Newtonian problem.
\item{(b)}  Investigate some special cases of the 4-vortex problem.
For instance, to what extent do the results of Cors and Roberts (see [CR] below)
about co-circular configurations
extend to the vortex case?  Are there parallel statements about the possible geometries
of vortex configurations?

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{\bf References}
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\item{[AF]}  Albouy, A.; Fu, Y. Euler configurations and quasi-polynomial systems. 
{\it Regul. Chaotic Dyn.} {\bf 12} (2007), no. 1, 39--55. 
\item{[CR]}  Cors, Josep; Roberts, G. Four-Body Co-Circular Central Configurations,
{\it Nonlinearity}, {\bf 25}, no. 2, 343-370. 

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{\it Idea 4 -- Real Root Counting and Central Configurations}  
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{\it Comment:}  I think this is probably the most theoretically (algebraically) 
demanding project idea in this list, and hence probably suitable only for 
a really strong group, although there might also be enough to do here to 
split up the aspects and have two groups work in partial collaboration.

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The recent work of M. Hampton and R. Moeckel (see [HM] in references)
on planar central configurations (relative equilibria) in the 4-body problem 
applied a theoretical ``big gun'' from contemporary algebraic/polyhedral
geometry {\it ``BKK theory''}.  What they showed using this 
was that the Albouy-Chenciner equations for the 4-body problem have only finitely
many {\it complex} solutions with $r_{ij} \ne 0$ for all $1 \le i,j\le 4$, 
for all positive collections masses $m_i$.  M. Hampton and A. Jensen have continued 
this approach for the 5-body problem as well, although the results
to date are less definitive.  

Even though this technique can be effective for
proving finiteness, the real interest in celestial mechanics is 
not in the collection of all complex solutions of these equations.  
Instead, what we want ideally is a way to count the number of
{\it real, positive solutions} (recall that the $r_{ij}$ represent mutual
distances in the physical interpretation of the system) and then 
determine each of them to any desired accuracy.  Now the ``BKK theory'' 
applies to give an upper bound on the number of positive real solutions, of course, 
since ${\bf R_{>0}} \subset {\bf C}$.  But there are typically many
more nonreal solutions than real solutions.  In addition, some real solutions
may contain values $r_{ij} \le 0$, so those are not physically relevant either.
Finally, it would be
very desirable to count real solutions with $r_{ij} > 0$ by {\it symbolic, exact methods}
that do not rely on sensitive numerical root-finding. 

One possibility that has not really been explored yet in any of the 
work in celestial mechanics is
to apply some fairly well-developed techniques for real root-counting
that can be seen as multivariate analogs
of the Sturm systems that we applied in the labs in week 3.  
If we can compute a Gr\"obner basis (with respect to any monomial 
order) for a {\it zero-dimensional} ideal $I$ -- and that is an unfortunately
big ``if'' for some of the systems of interest here --  then there
are symbolic techniques based on linear algebra in the quotient ring
$A = {\bf Q}[x_1,\ldots,x_s]/I$ that can be used to determine
the number of real solutions in any region defined by polynomial
inequalities (such as $x_i > 0$ for all $i$).  

The main goals of this project topic would be:
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\item{(a)} to learn about these symbolic real root-counting methods,
\item{(b)} to work out exactly how they could be applied in general
to count the number of positive, real solutions of a system of polynomial
equations, code them for use in {\tt Sage}, and finally 
\item{(c)} to experiment on some cases of interest in celestial mechanics
(starting with the 3-body problem from the labs in week 3) and (if feasible) going
to some more complicated cases as well.
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{\bf References}
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\item{[CLO2]}  Cox, D., Little, J., and O'Shea, D.  Using
Algebraic Geometry, 2nd ed., Springer 2005 (especially Chapter 2 and references
there).
\item{[HM]} Hampton, M. and Moeckel, R.  Finiteness of relative equilibria 
of the four-body problem. {\it Invent. Math.} {\bf 163} (2006), no. 2, 289--312.
\bye
