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\begin{document}
\title{Can Biology lead to New Theorems?}
\author{Bernd Sturmfels}
\address{Department of Mathematics, Univ.~of California,
Berkeley CA 94720, USA}
\email{bernd@math.berkeley.edu}
\begin{abstract}
This article argues for an affirmative
answer to the question in the title.
In future interactions between mathematics and
biology, both fields will contribute to each other,
and, in particular, research in the life sciences will inspire
new theorems in ``pure'' mathematics.
This point is illustrated by a snapshot of four
recent contributions from biology to
geometry, combinatorics and algebra.
\end{abstract}
\parskip=2.5pt
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\maketitle
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Much has been written about the importance of
mathematics for research in the life sciences in the
21st century. Universities
are eager to start initiatives aimed at promoting the
interaction between the two fields, and the
federally funded mathematics institutes
(AIM, IMA, IPAM, MBI, MSRI, SAMSI)
are outdoing each other in offering programs
and workshops at the interface of mathematics
and the life sciences.
The Clay Mathematics
Institute has had its share of such programs. For instance, in the
summer of 2005, two leading experts, Charles Peskin and
Simon Levin, served as Clay Senior Scholars
in the {\em Mathematical Biology} program
at the IAS/Park City Mathematics Institute (PCMI),
and in November 2005, Lior Pachter, Seth Sullivant and the author organized a
workshop on {\em Algebraic Statistics and Computational Biology}
at the Clay Mathematics Institute in Cambridge.
Yet, as these ubiquitous initiatives and programs unfold, many mathematicians remain
unconvinced, and some secretly hope that this ``biology fad''
will simply go away soon. They have not seen any
substantive impact of quantitative biology in their area of expertise,
and they rightfully ask: {\bf where are the new theorems?}
In light of these persistent doubts,
some long-term observers wonder whether anything has really changed in the
twenty years since Gian-Carlo Rota
wrote his widely quoted sentence,
{\em ``The lack of real contact between mathematics and
biology is either a tragedy, a scandal, or a challenge, it is hard to
decide which''} \cite[page 2]{Rota}. Of course,
Rota was well aware of the long history of
mathematics helping biology, such as the
development of population genetics by Fisher, Hardy,
Wright and others in the early 1900's.
Nonetheless, Rota concluded that there was no ``real contact''.
But, quite recently, other voices have been heard.
Some scholars have begun to argue that
``real contact'' means being equal partners, and that
meaningful intellectual contributions can, in fact, flow in both directions.
This optimistic vision is expressed succinctly in the title
of J.E.~Cohen's article \cite{Coh}:
{\em ``Mathematics is biology's next microscope, only better; biology
is mathematics' next physics, only better''}.
Physics remains the gold standard for mathematicians, as
there has been ``real contact'' and mutual respect
over a considerable period of time.
Historically, mathematics has made many contributions to
physics, and in the last twenty years there has been a payback
beyond expectations. Many of the most exciting developments
in current mathematics are a direct outgrowth of research in theoretical
physics. Today's geometry and topology are unthinkable
without string theory, mirror symmetry and quantum field theory.
It is ``obvious'' that physics can lead to new theorems.
Any colloquium organizer in a mathematics department who is
concerned about low attendence can reliably fill the room by
scheduling a leading physicist to speak.
The June 2005 public lecture on {\em Physmatics} by
Clay Senior Scholar Eric Zaslow sums up the situation as follows:
{\em ``The interplay between mathematics and physics has, in recent years, become so profound that the lines have been blurred. The two disciplines, long complementary, have begun a deep and fundamental relationship...''}.
Will biology ever be mathematics' next physics?
In the future, will a theoretical biologist
ever win a Fields medal? As unlikely as these possibilities seem,
we do not know the answer to these questions.
However, my recent interactions with computational biologists
have convinced me that there is more potential in this regard than many mathematicians
may be aware of. In what follows I wish to present a
personal answer to the legitimate question:
{\bf where are the new theorems?}
I shall present four theorems which were
inspired by biology. These theorems are in
algebra, geometry and combinatorics, my own areas
of expertise. I leave it to others to discuss biology-inspired
results in dynamical systems and partial differential equations.
Before embarking on the technical part of this article,
the following disclaimer must be made: the mathematics
presented below is just a tiny first step. The objects and results are
certainly not as deep and important as those in
Zaslow's lecture on Physmatics. But then,
Rome was not built in a day.
We start our technical discussion with a contribution made
by evolutionary biology to the study of metric spaces.
This is part of a larger theory developed by
Andreas Dress and his collaborators \cite{BD, DHM, DT}.
A {\em finite metric space} is a symmetric $n \times n$-matrix
$D = (d_{ij})$ whose entries
are non-negative
$(d_{ij} = d_{ij} \geq 0 )$,
zero on the diagonal
$(d_{ii} = 0)$, and satisfy the
triangle inequalities
$(d_{ik} \leq d_{ij} + d_{jk})$.
Each metric space $D$ on $\{1,2,\ldots,n\}$ is a point in
$\R^{\binom{n}{2}}$. The set of all such metrics
is a full-dimensional convex polyhedral cone
in $\R^{\binom{n}{2}}$, known as the {\em metric cone} \cite{DL}.
With every point $D$ in the metric cone one associates the convex polyhedron
$$
P_D \,\, = \,\, \bigl\{ \,x \in \R^n \,: \, x_i + x_j\,\, \geq \,\,d_{ij}\,\,\,
\hbox{for all $i,j$} \,\bigr\}.
$$
If $D_1, \ldots, D_k$ are metric spaces
then $D_1+ \cdots + D_k$ is a metric space as well, and
$$
P_{D_1 + D_2 + \cdots + D_k} \, \supseteq\,
P_{D_1} + P_{D_2} + \cdots + P_{D_k} .
$$
If this inclusion of polyhedra is an equality then
we say that the sum $D_1 + D_2 + \cdots + D_k$ is {\em coherent}.
A {\em split} is a pair $(\alpha,\beta)$ of disjoint non-empty
subsets of $ \{1,\ldots,n\} $ such that
$\,\alpha \cup \beta \,= \,\{1,\ldots,n\}$.
Each split $(\alpha,\beta)$ defines a {\em split metric}
$D^{\alpha,\beta}$ as follows:
$$
\hbox{
$D^{\alpha,\beta}_{ij} = 0$ \ if \ $\{i,j\} \subseteq \alpha $
or $\{i,j\} \subseteq \beta $, \ \ \ and
$\, D^{\alpha,\beta}_{ij} = 1 \,$ otherwise.}$$
The polyhedron $\,P_{D^{\alpha,\beta}}\,$, which represents a split metric $D^{\alpha,\beta}$, has
precisely one bounded edge, and its two vertices are the zero-one incidence
vectors of $\alpha$ and $\beta$.
A metric $D$ is called {\em split-prime} if
it cannot be decomposed into
a coherent sum of a positive multiple of
a split metric and another metric.
The smallest example of a split-prime metric
has $n=5$, and it is given by the distances among the nodes in the
complete bipartite graph $K_{2,3}$.
\begin{theorem}
\label{BandeltDress}
{\bf (Dress-Bandelt Split Decomposition \cite{BD})} \ \
Every finite metric space $D$ admits a unique coherent decomposition
$\, D \, = \, D_1 + \cdots + D_k \,+ \,D' \,$,
where $D_1,\ldots,D_k$ are linearly
independent split metrics and $D'$ is a split-prime metric.
\end{theorem}
This theorem is useful for evolutionary biology because it offers a
polyhedral framework for phylogenetic reconstruction.
Suppose we are given $n$ taxa, for instance
the genomes of $n$ organisms, and we take
$D$ be a matrix of distances among these taxa.
In typical applications, $d_{ij}$ would be the Jukes-Cantor distance
\cite[\S 4.4]{ASCB} derived from a pairwise alignment of
genome $i$ and genome $j$.
Then we consider the polyhedral complex
$\,{\rm Bd}(P_D)\,$ whose cells are the
bounded faces of the polyhedron $P_D$.
This is a contractible complex known as
the {\em tight span} \cite{DHM}
of the metric space $D$. The metric $D$ is a {\em tree metric}
if and only if the tight span ${\rm Bd}(P_D)$ is one-dimensional,
and, in this case, the one-dimensional contractible
complex ${\rm Bd}(P_D)$ is precisely the
{\em phylogenetic tree} which represents the metric $D$.
The space of phylogenetic trees on $n$ taxa was introduced
by Billera, Holmes and Vogtmann \cite{BHV}.
Since every tree metric uniquely determines its tree, this
space is a subset of the metric cone.
It can be characterized as follows:
\bigskip
\noindent {\bf Corollary.} {\em
The space of trees of \cite{BHV}
equals the following subset of the metric cone:
$$ {\rm Trees}_n \quad = \quad
\bigl\{ \,D \in \R^{\binom{n}{2}} \,: \,
D \,\hbox{is a metric \ and} \,\,\,\,
\hbox{dim} \,{\rm Bd}(P_D) \,\leq \,1 \,\bigr\}.$$}
\smallskip
\begin{figure}[h] % figure placement: here, top, bottom, or page
\centering
\includegraphics[width=3.5in]{graph.eps}
\caption{The space of phylogenetic
trees on five taxa is a seven-dimensional polyhedral fan
inside the ten-dimensional
metric cone. It has the combinatorial structure of the Petersen graph,
depicted here. The fan ${\rm Trees}_5$ consists of
$15$ maximal cones, one
for each edge of the graph, which represent the trivalent
trees. They meet along $10$ six-dimensional cones, one for each
vertex of the graph.}
\label{fig:example1}
\end{figure}
%\vskip 1cm
If the metric $D$ arises from real data then
it is unlikely to lie exactly in the space of trees.
Standard methods used by biologists, such as the
neighbor joining algorithm, compute a
suitable projection of $D$ onto ${\rm Trees}_n$.
From a mathematical point of view, however, it is
desirable to replace the concept of a tree
by a higher-dimensional object that faithfully
represents the data. The tight span ${\rm Bd}(P_D)$
is the universal object of this kind. It can be computed
using the
software {\tt POLYMAKE}. Figure 2 shows the tight
span of a metric on six taxa. This metric was derived from an
alignment of DNA sequences of six bees. For details
and an introduction to {\tt POLYMAKE} we refer to \cite{J}.
We note that, for larger data sets, the tight span is often too big. This
is where Theorem \ref{BandeltDress} enters the scene:
what one does is remove the {\em splits residue} $D'$ from the
data $D$. The remaining split-decomposable metric
$D_1 + \cdots + D_k$ can be computed efficiently
with the software {\tt SPLITSTREE} due to Huson and Bryant \cite{HB}.
It is represented by a {\em phylogenetic network}.
%\vskip 0.5cm
\begin{figure}[h] % figure placement: here, top, bottom, or page
\centering
\includegraphics[width=4.in]{bees.eps}
\caption{The tight span of a six-point metric space
derived from aligned DNA sequences of six species of bees. We thank
Michael Joswig and Thilo Schr\"oder for drawing this diagram
and allowing us to include it. See \cite{J} for a detailed description.
}
\label{fig:example2}
\end{figure}
%\vskip 0.5cm
Andreas Dress now serves as director of the
Institute for Computational Biology
in Shanghai ({\tt www.icb.ac.cn}), a joint
Chinese-German venture. He presented his theory at the
November 2005 workshop
at the Clay Mathematics Institute in Cambridge.
In his invited lecture at the 1998 ICM in
Z\"urich, Dress suggested that the
{\em ``the tree of life is an affine building''} \cite{DT}.
Affine buildings are highly symmetric infinite simplicial complexes
which play an important role in several areas of mathematics,
including group theory, representation theory, topology
and harmonic analysis.
The insight that phylogenetic trees,
and possible higher-dimensional generalizations thereof,
are intimately related to affine buildings is an important one.
The author of this article agrees enthusiastically
with Dress' point of view, as it is consistent with recent advances
at the interface of phylogenetics and tropical geometry.
An interpretation of tree space as a Grassmannian
in tropical algebraic geometry was given in \cite{SS}:
Figure 1 really depicts a Grassmannian together with
its tautological vector bundle.
It is within this circle of ideas that the next
theorem was found, three years ago, by
Lior Pachter and Clay Research Fellow David Speyer \cite{PS}.
Let $T$ be a phylogenetic tree with leaves labeled
by $[n] = \{1,2,\ldots,n\}$, and with a non-negative
length associated to each edge of $T$.
Then we define a real-valued function $\delta^{T,m}$ on the
$m$-element subsets $I$ of $[n]$ as follows: the number
$\delta^{T,m}(I)$ is the sum of the lengths
of all edges in the subtree spanned by $I$.
For $m=2$ we recover the tree metric $\,D_T = \delta^{T,2}$.
We call $\,\delta^{T,m} : \binom{[n]}{m} \rightarrow \R\,$
the {\em subtree weight function}.
\begin{theorem} {\bf (Pachter-Speyer Reconstruction
from Subtree Weights \cite{PS}) \ }
Suppose that $n \geq 2m-1$.
Every phylogenetic tree on $n$ taxa is uniquely determined
by its subtree weight function.
More precisely, $\delta^{T,m}$ determines the tree metric
$\delta^{T,2}$.
\end{theorem}
The punchline of this theorem is a statistical one.
The aim of replacing $m=2$ by larger values of $m$ is that
$\delta^{T,m}$ can be estimated from data in a more
reliable manner. Practical advantages of this method
were shown in \cite{LYP}.
Phylogenetics has spawned several different
research directions in current mathematics,
especially in combinatorics and probability.
For more information, we recommend the book by Semple and Steel \cite{SSteel},
and the special semester on Phylogenetics which will take place
in Fall 2007 at the Newton Institute in Cambridge, England.
Algebraists, geometers and topologists may also enjoy
a glimpse of {\em phylogenetic algebraic geometry} \cite{ERSS}.
Here the idea is that statistical models of biological sequence
evolution can be interpreted as algebraic varieties in
spaces of tensors. This approach has led to a range of recent
developments which are of interest to algebraists;
see \cite{AR, LM, SSul} and the references given there.
As an illustration, we present a recent theorem due to
Buczynska and Wisniewski \cite{BW}. The abstract of
their preprint leaves no doubt that this is an unusual paper as far as
mathematical biology goes:
{\em ``We investigate projective varieties which are geometric models of binary symmetric phylogenetic 3-valent trees. We prove that these varieties have Gorenstein terminal singularities (with small resolution) and they are Fano varieties of index 4....''}.
The varieties studied here are all
embedded in the projective space
$\,\P^{2^{n-1}-1} = \P(\C^2 \otimes \C^2 \otimes \cdots \otimes \C^2)$
whose coordinates $x_I$ are indexed by subsets $I$
of $\{1,\ldots,n\}$ whose cardinality $|I|$ is even.
We fix a trivalent tree $T$ whose leaves are labeled by $1,\ldots,n$.
Each of the $2n-3$ edges $e$ of the tree $T$ is identified with a
projective line $\P^1$ with homogeneous coordinates $(u_e:v_e)$.
For any even subset $I$ of the leaves of $T$ there exists a unique
set $\,{\rm Paths}(I)\,$ of disjoint paths,
consisting of edges of $T$, whose end points are
the leaves in $I$.
This observation gives rise to a birational morphism
$$ \phi_T \,:\, (\P^1)^{2n-3} \rightarrow \P^{2^{n-1}-1}
\quad \mbox{defined by} \quad
x_I \,\,\, = \,
\prod_{e \in {\rm Paths}(I)} \!\!\! u_e \, \cdot \!\!
\prod_{e \not\in {\rm Paths}(I)} \!\!\! v_e.$$
The closure of the image of $\phi_T$ is a projective
toric variety which we denote by $X_T$.
\begin{theorem} {\bf (Buczynska-Wisniewski Flat Family of Trees \cite{BW}) }
All toric varieties $X_T$ are the same connected component of the Hilbert scheme
of projective schemes, as $T$ ranges over all combinatorial types of trivalent trees
with $n+1$ leaves.
Combinatorially, this means that the convex polytopes associated with these
toric varieties all share the same Ehrhart polynomial
(a formula for this Ehrhart polynomial is given in \cite[\S 3.4]{BW}).
\end{theorem}
Earlier work with Seth Sullivant \cite{SSul} had shown that
the homogeneous prime ideal of $X_T$ has a Gr\"obner basis
consisting of quadrics. These quadrics are the $2 \times 2$-minors
of a collection of matrices, two for each edge $e$ of $T$.
After relabeling we may assume that the edge $e$
separates the leaves $1,2,\ldots,i$ from the leaves $i+1,\ldots,n$.
We construct two matrices $M^e_{\rm even}$ and $M^e_{\rm odd}$ each having
$2^{i-1}$ rows and $2^{n-i-1}$ columns.
The rows of $M^e_{\rm even}$ are indexed by subsets
$I \subset \{1,\ldots,i\}$ with $|I|$ even and the columns
are indexed by subsets $J \subset \{i+1,\ldots,n\}$
with $|J|$ even. The entry of $M^e_{\rm even}$
in row $I$ and column $J$ is the unknown $x_{I \cup J}$.
The matrix $M^e_{\rm odd}$ is defined similarly.
Our Gr\"obner basis for the toric variety $X_T$
consists of the $2 \times 2$-minors of the matrices
$M^e_{\rm even}$ and $M^e_{\rm odd}$ where $e$ runs over
all $2n-3$ edges of the tree $T$. In light of Theorem 3,
it would be interesting to decide whether all the $X_T$
lie on the same irreducible component of the Hilbert scheme,
and, if yes, to explore possible connections
between the generic point on that component to
the quadratic equations derived by Keel and Tevelev \cite{KT} for
the moduli space $\bar{M}_{0,n}$.
The toric variety $X_T$ is known to evolutionary biologists as the
{\em Jukes-Cantor model}. For some applications, it is more natural
to study the {\em general Markov model}. This is a non-toric
projective variety in tensor product space which generalizes
secant varieties of Segre varieties \cite{LM}. The state of
the art on the algebraic geometry of these models appears in the work of Elizabeth
Allman and John Rhodes \cite{AR}.
For our last theorem, we leave the field of phylogenetics
and turn to mathematical developments inspired by other
problems in biological sequence analysis.
These problems include {\em gene prediction}, which seeks
to identify genes inside genomes,
and {\em alignment}, which aims to find the biological relationships
between two genomes. See \cite[\S 4]{SIAM} for an introduction
aimed at mathematicians.
Current algorithms for {\it ab initio} gene prediction and alignment are based on methods from
statistical learning theory, and they involve {\em hidden Markov models}
and more general {\em graphical models}.
From the perspective of algebraic statistics \cite{ASCB},
a graphical model is a highly structured polynomial map
from a low-dimensional space of parameters to
a tensor product space, like the $\P^{2^{n-1}-1}$ we encountered
in Theorem 3.
It is from this algebraic representation of graphical models
that the following theorem was derived:
\begin{theorem} {\bf (Elizalde-Woods' Few Inference Functions) \cite{Eli, EW})}
Consider a graphical model $G$ with $d$ parameters,
where $d$ is fixed,
and let $E$ be the number of edges of $G$.
Then the number of inference functions
of the model is at most $O(E^{d(d-1)})$.
\end{theorem}
We need to explain what an inference function is
and what this theorem means. A graphical model
is given by a polynomial map $\, p : \R^d \rightarrow \R^N$ where
$d$ is fixed and each coordinate $p_i$ is a polynomial
of degree $O(E)$ in $d$ unknown parameters. The polynomial $p_i$ represents the
probability of making the $i$-th observation $\# i$, out of a total
of $N$ possible observations.
The number $N$ is allowed
to grow, and in biological applications it can be very large, for instance
$N = 4^{1,000,000}$, the number of DNA sequences with
one million base pairs.
The monomials in $p_i$ correspond
to the possible {\em explanations} of this observation,
where the monomial of largest numerical value will be
the most likely explanation. Let ${\rm Exp}$ be the set
of all possible explanations for all the $N$ observations.
For a fixed generic choice of parameters $\theta \in \R^d$,
we obtain a well-defined function
$$ \phi_\theta \, : \, \{1,2,\ldots,N\} \rightarrow {\rm Exp} $$
which assigns to each observation its most likely explanation.
Any such function, as $\theta$ ranges over
(a suitable open subset of) $\R^d$ is called an {\em inference
function} for the model $f$. The number $|{\rm Exp}|^N$ of
all conceivable functions is astronomical. The result by
Elizalde and Woods says
that only a tiny, tiny fraction of all these functions are actual
inference functions. The polynomial growth rate in Theorem 4
makes it feasible, at least in principle, to pre-compute all such
inference functions ahead of time, once per graphical model.
This is important for {\em parametric inference}.
Two recent examples of concrete bio-medical applications of parametric inference
can be found in \cite{BDW} and \cite{DHSWP}.
One way you can tell a biology paper from
a mathematics paper is that the order of the authors' names
has a meaning and is thus rarely alphabetic.
This concludes my discussion of four recent
theorems that were inspired by biology. All four stem from
my own limited field of expertise, and hence the selection
has been very biased. A feature that Theorems 1, 2, 3 and 4
have in common is
that they are meaningful as statements of pure mathematics.
I must sincerely apologize to my colleagues in mathematical
biology for having failed to give proper credit to their
many many important research contributions. My only
excuse is the hope that they will agree with my view
that the answer to the question in the title is affirmative.
\vskip0.15in
\begin{thebibliography}{10}
\bibitem{AR} E.~Allman and J.~Rhodes:
{\em Phylogenetic ideals and varieties for the general Markov model},
{\tt math.AG/0410604}.
\bibitem{BD} H-J~Bandelt and A.~Dress: {\em A canonical decomposition
theory for metrics on a finite set\/},
Advances in Mathematics {\bf 92} (1992) 47--105.
\bibitem{BDW} N.~Beerenwinkel, C.~Dewey and K.~Woods:
{\em Parametric inference of recombination in HIV genomes},
{\tt q-bio.GN/0512019}.
\bibitem{BHV} L.~Billera, S.~Holmes and K.~Vogtman:
{\em Geometry of the space of phylogenetic trees},
Advances in Applied Mathematics {\bf 27} (2001) 733-767.
\bibitem{BW} W.~Buczynska and J.~Wisniewski:
{\em On phylogenetic trees - a geometer's view},
{\tt math.AG/0601357}.
\bibitem{Coh} J.E.~Cohen: {\em
Mathematics is biology's next microscope, only better; biology
is mathematics' next physics, only better},
PLOS Biology {\bf 2} (2004) No.12.
\bibitem{DHSWP}
C.~Dewey, P.~Huggins, K.~Woods, B.~Sturmfels and L.~Pachter:
{\em Parametric alignment of Drosophila genomes},
PLOS Comput.~Biology {\bf 2} (2006) No. 6.
\bibitem{DL} M.~D\'eza and M.~Laurent:
{\em Geometry of Cuts and Metrics},
Springer, New York, 1997.
\bibitem{DHM} A.~Dress, K.~Huber and V.~Moulton:
{\em Metric spaces in pure and applied mathematics},
Documenta Mathematica, Quadratic Forms LSU (2001) 121-139.
\bibitem{DT} A.~Dress and W.~Terhalle:
{\em The tree of life and other affine buildings},
Documenta Mathematica,
Extra Volume ICM III (1998) 565-574
\bibitem{Eli} S.~Elizalde:
{\em Inference functions},
Chapter 9 in \cite{ASCB}, pp.~215--225.
\bibitem{EW} S.~Elizalde and K.~Woods:
{\em Bounds on the number of inference functions of a
graphical model},
Formal Power Series and Algebraic Combinatorics (FPSAC 18),
San Diego, June 2006.
\bibitem{ERSS}
N. Eriksson, K. Ranestad, B. Sturmfels and S. Sullivant:
{\em Phylogenetic algebraic geometry}, in Projective Varieties with Unexpected Properties,
(editors C.~Ciliberto, A.~Geramita, B.~Harbourne, R-M.~Roig and K.~Ranestad), De Gruyter, Berlin, 2005, pp. 237-255.
\bibitem{J} M.~Joswig:
{\em Tight spans},
Introduction with link to the software {\tt POLYMAKE} and
an example of six bees,
{\tt www.mathematik.tu-darmstadt.de/$\sim$joswig/tightspans/index.html}.
\bibitem{HB} D. H.~Huson and D. Bryant:
{\em Application of phylogenetic networks in evolutionary studies}
{\tt Molecular Biology and Evolution} {\bf 23} (2006) 254-267.
(Software at {\tt www.splitstree.org})
\bibitem{Rota} M.~Kac, G-C.~Rota and J.~T.~Schwartz: {\em Discrete
Thoughts}, Birkh\"{a}user, Boston,1986.
\bibitem{KT} S.~Keel and J.~Tevelev: {\em Equations for $\bar M_{0,n}$}, {\tt math.AG/0507093}.
\bibitem{LM} JM~Landsberg and L.~Manivel:
{\em On the ideals of secant varieties of Segre varieties},
Found Comput. Math. {\bf 4} (2004) 397-422
\bibitem{LYP}
D. Levy, R. Yoshida and L. Pachter:
{\em Beyond pairwise distances: neighbor joining with phylogenetic diversity estimates},
Molecular Biology and Evolution {\bf 23} (2006) 491--498.
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