Universes as Big Data

Yang-Hui He***<EMAIL_ADDRESS>
Invited review for IJMPA, based on various colloquia, seminars and conference
talks in the 2019-2020 academic year.

1 | Merton College, University of Oxford, OX14JD, UK
---|---
2 | Department of Mathematics, City, University of London, EC1V 0HB, UK
3 | School of Physics, NanKai University, Tianjin, 300071, P.R. China

We briefly overview how, historically, string theory led theoretical physics
first to precise problems in algebraic and differential geometry, and thence
to computational geometry in the last decade or so, and now, in the last few
years, to data science. Using the Calabi-Yau landscape - accumulated by the
collaboration of physicists, mathematicians and computer scientists over the
last 4 decades - as a starting-point and concrete playground, we review some
recent progress in machine-learning applied to the sifting through of possible
universes from compactification, as well as wider problems in geometrical
engineering of quantum field theories. In parallel, we discuss the programme
in machine-learning mathematical structures and address the tantalizing
question of how it helps doing mathematics, ranging from mathematical physics,
to geometry, to representation theory, to combinatorics, and to number theory.

###### Contents

  1. 1 Introduction
  2. 2 Trichotomy and Triadophilia
    1. 2.1 Complex, Kähler, Ricci-flat
    2. 2.2 Low-Energy Physics
    3. 2.3 Early Constructions
  3. 3 Data Explosion
    1. 3.1 The Good, The Bad, and The ?
  4. 4 Deep-Learning the Landscape
    1. 4.1 An Invitation to Machine-Learning
    2. 4.2 Initial Experiments
    3. 4.3 More Success Stories in String/Geometry
  5. 5 Outlook: ML Mathematical Structures

##  1 Introduction

The twentieth century has firmly established that the correct language of
fundamental theoretical physics is that of algebraic/differential
geometry/topology, in all four combinations of these pairs of adjectives and
nouns. Gravity and space-time, should be thought of as the metric and
curvature of Riemannian manifolds; elementary particles, irreducible
representations of the Lorentz group and gauge connections of appropriate
principal Lie-group fibrations, etc. (q.v. an attempted modern summary in
[1]). In some sense, string theory is a brain-child of this tradition. Whether
she will stand as the ultimate theory of everything remains to be seen, but
her rôle in both bearing the torch and ploughing the field of the
conversations between mathematics and physics is unquestionable.

The twenty-first century, with the dramatic progress in computing power and
techniques, is bringing a new interlocutor to this dialogue. Already, in the
first decade, software such as Macaulay2 [2], Singular [3], GAP [4], MAGMA
[5], and the umbrella project of SageMath [6] (launched in 2005), and the
increasing online mathematical databases – [7, 8, 9] to name but a few – are
aiding pure mathematical research in an increasingly prominent way (q.v..
launch of ICMS in 2006 [10]). In parallel – and this is of course in tandem
with experimental physics whose reliance on and interaction with computers has
a rich history of its own – theoretical physics, and string theory in
particular, has benefited from an algorithmic outlook [13, 14, 15, 16].

In our present era of Big Data and AI, it is inevitable that machine-learning
should have an ever-increasing presence in the second decade [17, 18, 19, 20,
21]. The purpose of this talk [11], aimed at a general physics audience, is to
give an overview of some of this activity in the last few years, especially in
the context of machine-learning applied to the string theoretic and geometric
landscape, as well as to other mathematical structures within and beyond
geometry (cf. an attempted pedagogical introduction in [22]). I will take a
somewhat historical approach and start with Calabi-Yau data as a concrete
playground; this is mainly due to the vastness of the subject in physics and
in mathematics, which has consequently led to an abundance of data. The
methodology should be applicable to much more general situations, under the
rubric of machine-learning mathematical structures [17, 23, 12].

The organization of this talk is as follows. We begin by reviewing how 2
parallel traditions, one in theoretical physics and one in pure mathematics,
converging around 1980s and both leading to the study of complex manifolds and
Kähler geometry. This confluence initiated a concerted effort to construct
Ricci-flat such spaces, viz., Calabi-Yau manifolds, in the last decade of the
20th century, continuing into the 21st, developing into an explosion of data.
In §2, we take an overview of this mathematical data, being encouraged by its
availability and plenitude, somehow daunted by the complexity of the
algorithms needed to process them, and compelled by a thirst for techniques
from the “Big Data” revolution and AI research. We then offer the audience an
invitation, in §3, to some modern data science, focusing on machine-learning
and how it may be applied to problems in string phenomenology as well as
algebraic geometry. We conclude with an outlook and report on some recent
results in machine-learning fundamental structures in various branches of
mathematics and relations to physics.

##  2 Trichotomy and Triadophilia

It is well known that string theory is a unified theory of gravity and
elementary particles in high dimensions. Shortly after the First Revolution in
1984 with anomaly cancellation [25] and the discovery of the heterotic string
[26], the subject of “string phenomenology” was born [27]. The reason for this
excitement was that all at once, there was an anomaly-free quantum field
theory which naturally contained the graviton as well as the $E_{8}$ gauge
group. In other words, it presented a unified theory of quantum gravity -
albeit in 10 space-time dimensions - which also, via the embedding
$SU(3)\times SU(2)\times U(1)\subset SU(5)\subset SO(10)\subset E_{6}\subset
E_{8}$, could give rise to the standard model.

###  2.1 Complex, Kähler, Ricci-flat

The solution of [27] was to take inspiration from Kaluza-Klein [28] and treat
the extra $10-4=6$ dimensions as small and space-like, in a compactification
scenario with a 6-manifold $M_{6}$ on top of each point in our 4-dimensional
space-time. Further conditions of supersymmetry ††† Whilst it remains to be
seen whether there is supersymmetry in Nature, it is undisputed from a
theoretical perspective that quantum field theory with supersymmetry (SUSY)
has much richer and tamable structure. A good (and in some sense rigourous)
analogy would be that doing mathematics over $\mathbb{R}$ is difficult, and
this is ameliorated by working over $\mathbb{C}$, the unique (commutative)
algebraic closure. So too, do the theorems of Coleman-Mandula [29] and Haag-
Łopuszański-Sohnius [30], guarantee SUSY as the unique extension to Poincaré
symmetry in a field theory.  and vacuum Einstein solutions constrained the
6-manifold to be (1) complex, (2) Kähler and (3) Ricci-flat, i.e.,
respectively (1) a complex 3-fold, (2) the metric comes from a scalar
potential
$g_{\mu\bar{\nu}}(z,\bar{z})=\partial_{\mu}\overline{\partial}_{\overline{\nu}}K(z,\overline{z})$
and (3) the Ricci curvature for $g_{\mu\bar{\nu}}$ vanishes. We emphasize that
this is the simplest solution. In general, one has to solve the so-called
Hull-Strominger system [31], which would lead to a much wider variety of
possible $M_{6}$. As we shall emphasize later, the Calabi-Yau landscape is
only a corner of possible compactification scenarios.

To the theoretical physicist in the mid-1980s, perhaps only the word “Ricci-
flat” was, because of general relativity, familiar. Meanwhile, for the
mathematical community, this was also rather avant-garde. The story goes back
to classical results of Euler, Gauß, and Riemann. Consider a surface $\Sigma$
\- we usually think of a sphere $S^{2}$ or the surface of a doughnut $T^{2}$
\- and its possible topological types, i.e., equivalences up to topology.
Restricting to the cases of smooth, compact (no punctures or boundaries) and
orientable (nothing like Klein bottles or Möbius strips) surfaces, the
familiar shapes, $S^{2}$, $T^{2}$, and those with increasing number of “holes”
(genus) are all there is: any smooth, compact, orientable surface can be
deformed continuously (topologically homeomorphic) to one of these. This
single non-negative integer, the genus $g(\Sigma)$, classifies the topology of
$\Sigma$. Closely related is the quantity $\chi(\Sigma)=2-2g(\Sigma)$, called
the Euler characteristic or Euler number. A high-light of the geometry of the
18th-19th centuries is the chain of equalities

$2-2g(\Sigma)=\chi(\Sigma)=\sum\limits_{i=0}^{\dim_{\mathbb{R}}\Sigma=2}(-1)^{i}b^{i}(\Sigma)=\frac{1}{2\pi}\int_{\Sigma}R\
,$ (2.1)

where one proceeds, from left to right, from topology, to combinatorics, to
Gauß’ Theorema Egregium for differential geometry. Here, $b^{i}$ are the Betti
numbers, counting the number of cycles in dimension $i$ and $R$ is the (Ricci)
curvature. This whole setup above can be complexified where our 2-manifolds –
named Riemann surfaces – become complex 1-folds, named complex curves.
Furthermore, $\Sigma$ are not just complex, but are also Kähler – one can
check that the complex (Hermitian) metric on all such surfaces as complex
1-folds comes from a single potential.

To the equalities (2.1), early 20th century added another, viz.,
$\chi(\Sigma)=[c_{1}(T_{\Sigma})]\cdot[\Sigma]$, where the last integral is
re-interpreted as intersection theory between cohomology (here the Chern class
$c_{1}$) and homology (here the class of the manifold). All of these are
special cases of the index theorem of Atiyah-Singer and Grothendieck-Riemann-
Roch (q.v. [32]) which are applicable to spaces of arbitrary dimension.

Having curvature controlling topology also gives us a natural trichotomy,
which for $\Sigma$, is part of the Riemann Uniformization Theorem.
Specifically, in complex dimension 1, we have

$R\quad\left\\{\begin{array}[]{ll}>0:&g=0,\mbox{ Spherical Geometry }\\\
=0:&g=1,\mbox{ Flat Torus }\\\ <0:&g>1,\mbox{ Hyperbolic Geometry }\\\
\end{array}\right.$ (2.2)

Note that the $R\geq=0$ cases are finite in topological type and the $R<0$
cases are infinite.

Much of modern geometry is concerned with generalizing this beautiful story of
complex dimension 1 to higher dimensions. Expectedly, the situation is much
more involved and many questions still remain open conjectures. Nevertheless,
for Kähler manifolds a conjecture of Calabi [34] dating to the 1950s does give
the analogue of (2.2): essentially, it states that $c_{1}$, the first Chern
class, uniquely controls the Ricci curvature for the Kähler metric. It was not
until the Fields-Medal-deserving work in 1978 by Yau [35] that this was
settled.

Fortuitously, Strominger, one of authors of [27] was visiting Yau at the IAS
in 1985 and were neighbours. Thus, the object onto which physicists stumbled –
Ricci-flat, Kähler manifolds – through string compactification, had the world-
expert literally next door. In fact, such spaces were named Calabi-Yau
manifolds by the physicists.

###  2.2 Low-Energy Physics

Not only did [27] constrain the compactification manifold, they also
established a dictionary between

“Geometry of $X_{6}$ $\longleftrightarrow$ physics of $\mathbb{R}^{1,3}$.”

Purely working from the group theory, the tangent bundle with its $SU(3)$
structure breaks the $E_{8}$ to an $E_{6}$ (SUSY) GUT theory. We will skip the
details, but basically for the fundamental fermions (the choice of which is
anti-generation and generation is by convention):

$\begin{array}[]{rl}\mbox{generations of particles}&\sim h^{2,1}(X)\ ,\\\
\mbox{anti-generations of particles}&\sim h^{1,1}(X)\ ,\end{array}$ (2.3)

where $h^{p,q}$ are the Hodge numbers of the Calabi-Yau manifold $M_{6}$, the
complexified version (hence the double index, for complex and conjugate) of
the Betti numbers mentioned earlier. The alternating sum of Betti numbers to
the Euler number generalizes to a double sum, and in particular
$\chi(M_{6})=2(h^{1,1}(M_{6})-h^{2,1}(M_{6}))$. Since there are 3 generations
of fermions, one of the original constraints of [27] is that

$\left|h^{2,1}(X)-h^{1,1}(X)\right|=3\Rightarrow\chi(X)=\pm 6\ .$ (2.4)

Finding compact, smooth Calabi-Yau 3-folds with Euler number $\pm 6$ was
perhaps historically the first concrete challenge physicists gave to the
algebraic geometry community.

#### Disclaimer:

It must be emphasized that (2.3) is only for the so-called standard embedding
for the heterotic string to get to $E_{6}$-GUT theories, the field has since
evolved to far beyond merely computing Hodge numbers, but to computing
equivariant cohomology of stable bundles (cf. [37, 38, 39, 40]). In addition,
there is a myriad of phenomenological approaches from other
string/M-/F-theoretic constructions, which constitute the vastness of the
“string landscape”, the review of which is not our present intent. The reader
is referred to the wonderful textbooks [36] in general, and to the classic
[24] for an introduction to complex geometry for physicists, and, in the
context of Calabi-Yau spaces, to [33] for a brief invitation and [22] for a
pedagogical textbook.

Although the physics community is no longer searching for manifolds with
property (2.4), there is an entire programme, especially led by Candelas, to
look for Calabi-Yau manifolds of small Hodge numbers [42] which have
interesting mathematics of its own. In any event, the search for a geometric
interpretation, or origin, of 3 generations of particles has been dubbed
“Triadophilia” [41]. In a way, the dictionary started by (2.3), where
properties of our universe are purely phrased in the geometry of some
manifold, is an elegant modern realization of Kepler’s famous adage: “Ubi
materia, ibi geometria” ‡‡‡ “Where there is matter, there is geometry,” from
Johannes Kepler’s Thesis XX from De fundamentis astrologiae certioribus 1602\.
. Perhaps for this reason by itself, it is worth studying string theory as a
theory of physics, let alone its cross-fertilizations to mathematics and - as
we will see - data science.

###  2.3 Early Constructions

One of the first questions which the physicists asked Yau was, indubitably,
“how to construct an explicit Calabi-Yau 3-fold?” It is curious that students
in theoretical physics are taught differential geometry first, before
algebraic geometry, whereas the fundamental ideas of the latter - vanishing
loci of polynomials - are certainly more familiar than that of the former -
local patches and differentiable transition functions. We know how to
construct shapes from Cartesian geometry since our early school days. For
example, a quadratic equation in two real variables $(x,y)$ is a conic
section, such as a circle. Thus, the vanishing locus on a quadratic polynomial
in two real variables gives a $2-1=1$ dimensional real manifold in an ambient
$\mathbb{R}^{2}$. We have just created a simple algebraic variety.

Now, we are looking for complex manifolds, we thus construct them as the zero-
locus of multiple polynomials in multiple complex variables. In this way, a
Calabi-Yau 1-fold, a Riemann surface of zero curvature, viz. the torus
$T^{2}=S^{1}\times S^{1}$, is realized as a cubic in two complex variables
given by the so-called Weierstraß equation
$T^{2}\simeq\\{x,y\in\mathbb{C}|y^{2}=x^{3}-g_{2}x-g_{4}\\}\subset\mathbb{C}^{2}$,
where $g_{2,4}$ are complex constants. One can check by writing out $(x,y)$ in
their real and imaginary parts, and the Weierstraß equation becomes 2 real
constraints in 4 real variables, which we can numerically plot by Monte Carlo
to see a torus emerge. Next, compactness can be ensured by including the point
at infinity, where $(x,y)=(\infty,\infty)$. One can do this by so-called
projectivization where instead of $\mathbb{C}^{2}$, we introduce one more
complex coordinate, $z$ such that any point $(x,y,z)\in\mathbb{C}^{3}$ is
identified with the scaled $\lambda(x,y,z)$ for non-zero
$\lambda\in\mathbb{C}$. This scale-invariance brings the point at infinity to
a finite point, rendering the resulting ambient space and the subsequent torus
compact.

What we have done is to construct, from $\mathbb{C}^{3}$ with coordinates
$(x,y,z)$, the complex projective space $\mathbb{C}\mathbb{P}^{2}$ with
homogeneous coordinates $[x:y:z]$. More formally, we define
$\mathbb{C}\mathbb{P}^{n}$ from $\mathbb{C}^{n+1}$ with coordinates
$(z_{0},z_{1},\ldots,z_{n})$ as the quotient by the equivalence relation
$\sim$

$\mathbb{C}\mathbb{P}^{n}:=\mathbb{C}^{n+1}\backslash\\{\vec{0}\\}\bigg{/}(z_{0},z_{1},\ldots,z_{n})\sim\lambda(z_{0},\ldots,z_{n})\
,\qquad\lambda\in\mathbb{C}\backslash\\{0\\}\ .$ (2.5)

The complex $n$-fold $\mathbb{C}\mathbb{P}^{n}$ is smooth, with $n+1$
homogeneous coordinates.

Thus, the Calabi-Yau 1-fold is realized as a so-called projective algebraic
variety inside $\mathbb{C}\mathbb{P}^{2}$

$\\{[x:y:z]|-y^{2}z+x^{3}-4g_{2}xz^{2}-g_{4}z^{3}=0\\}\subset\mathbb{C}\mathbb{P}^{2}\
,$ (2.6)

a homogeneous cubic in the homogeneous coordinates $[x:y:z]$ of
$\mathbb{C}\mathbb{P}^{2}$. Luckily, complex projective space and the zero
loci of any number homogeneous polynomials therein, are guaranteed to be
Kähler. For $\mathbb{C}\mathbb{P}^{n}$, the Kähler metric explicitly comes
from the famous Fubini-Study potential $\log(1+\sum_{i}|z_{i}|^{2})$. This
construction is valid in general: the hypersurface defined by a homogeneous
polynomial of degree $n+1$ in $\mathbb{C}\mathbb{P}^{n}$ is a Calabi-Yau
$(n-1)$-fold. Thus we arrive at our first, and perhaps most famous, example of
a Calabi-Yau 3-fold: the quintic hypersurface in $\mathbb{C}\mathbb{P}^{4}$.
There are many degree 5 monomials one could compose of 5 coordinates, the most
well-studied is the so-called Fermat quintic:

$Q:=\\{x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}=0\\}\subset\mathbb{C}\mathbb{P}^{4}_{[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]}\
.$ (2.7)

What are the topological numbers of $Q$? The Hodge numbers turn out to be
$h^{2,1}(Q)=101$ and $h^{1,1}(Q)=1$ so that $\chi(Q)=2(1-101)=-200$.

Immediately, we also obtain 4 close relatives. Consider the intersection of 2
cubics in $\mathbb{C}\mathbb{P}^{5}$; this is a complete intersection in that
the number of defining polynomials - here 2 - is equal to the codimension –
i.e., the dimension of the ambient $\mathbb{C}\mathbb{P}^{5}$ minus the
dimension of the required manifold, $5-3=2$. We denote this as $[5|3,3]$, much
as we could denote the quintic as $[4|5]$. Note that the number of to the left
of the bar is 1 less than the row-sum (Calabi-Yau condition) and also 3 more
than the number of columns (complete intersection condition). A simple integer
partition shows that there are 5 possibilities in total, including the
quintic, viz.,

$[4|5],\ [5|3,3],\ [5|2,4],\ [6|2,2,3],\ [7|2,2,2,2]\ .$ (2.8)

These are called cyclic Calabi-Yau 3-folds, and are the only ones as complete
intersections in a single projective space.

We need to emphasize that complete intersections are rare and most algebraic
varieties are not so. In fact, there is a general result that (see [44])

######  THEOREM 1.

All Kähler 3-folds can be realized as vanishing loci of systems of polynomials
in $\mathbb{C}\mathbb{P}^{7}$.

Therefore, one could in principle write all sorts of (non-complete-
intersection) polynomials in 8 homogeneous variables and sift out the Calabi-
Yau ones; but this is highly impractical.

Of the 5 immediate ones, none has the property (2.4), so the community turned
to more general constructions. Again, as mentioned earlier, today physicists
are no longer limited to (2.4) and (2.8) have all been met with renew zest. In
the late 1980s, however, a different path was undertaken, and an industry of
subsequent generalizations to (2.8) was initiated:

CICYs

The first generalization is to take, instead of a single
$\mathbb{C}\mathbb{P}^{n}$, a product $A$ of projective spaces. That is, let
$A=\mathbb{C}\mathbb{P}^{n_{1}}\times\ldots\times\mathbb{C}\mathbb{P}^{n_{m}}$,
of dimension $n=n_{1}+n_{2}+\ldots+n_{m}$ and each having homogeneous
coordinates $[x_{1}^{(r)}:x_{2}^{(r)}:\ldots:x_{n_{r}}^{(r)}]$ with the
superscript $(r)=n_{1},n_{2},\ldots,n_{m}$ indexing the projective space
factors. The Calabi-Yau 3-fold is then defined as the complete intersection of
$K=n-3$ homogeneous polynomials in the coordinates $x_{j}^{(r)}$. Succinctly
§§§ Importantly, the Chern classes and the Euler number can be read off the
matrix configuration explicitly. The individual terms $(h^{1,1},h^{2,1})$,
however, cannot be deduced from the configuration matrix directly. This is one
of the short-comings of the index theorem: the integral of curvature and the
intersection of the Chern classes give only the alternating sum (Euler number)
in (co-)homology, but not the individual terms. , this information can be
written into an $m\times K$ configuration matrix which generalize (2.8):

$X=\left[\begin{array}[]{c|cccc}\mathbb{C}\mathbb{P}^{n_{1}}&q_{1}^{1}&q_{2}^{1}&\ldots&q_{K}^{1}\\\
\mathbb{C}\mathbb{P}^{n_{2}}&q_{1}^{2}&q_{2}^{2}&\ldots&q_{K}^{2}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\mathbb{C}\mathbb{P}^{n_{m}}&q_{1}^{m}&q_{2}^{m}&\ldots&q_{K}^{m}\\\
\end{array}\right]_{m\times K\
,}\quad\begin{array}[]{l}K=\sum\limits_{r=1}^{m}n_{r}-3\ ,\\\
\sum\limits_{j=1}^{K}q^{r}_{j}=n_{r}+1\ ,\ \forall\;r=1,\ldots,m\
.\end{array}$ (2.9)

These manifolds defined by (2.9) were called CICYs (complete intersection
Calabi-Yau manifolds) and were explicitly constructed by Candelas et al. [45]
(q.v. Hübsch’s classic book [46]) in the early 1990s. The combinatorial
problem for these integer matrices turned out to be rather non-trivial and one
of the most powerful super-computers then available, the one at CERN, was
recruited. To our knowledge, this might have been the first “data-base” in
algebraic geometry.

Up to trivial equivalence such as row/column permutations as well as non-
trivial ones such as so-called splitting, CICYs were shown to be finite in
number, a total of 7890 configurations, with a maximum of 12 rows, a maximum
of 15 columns, and all having entries $q_{j}^{r}\in[0,5]$. There are 266
distinct Hodge pairs $(h^{1,1},h^{2,1})=(1,65),\ldots,(19,19)$, giving 70
distinct Euler numbers $\chi\in[-200,0]$.

WP4s

Noticing that the CICY data is rather skewed in that all Euler numbers were
non-positive, the constructions went on. The reason for this is that
physicists knew about mirror symmetry by then, one of whose most salient
features is the exchange of $h^{1,1}\leftrightarrow h^{2,1}$, which would flip
the sign of $\chi$. Another way to generalize projective space is to introduce
weights ¶¶¶ In fact, products of projective spaces can also be thought of as a
weighted projective space with vector-valued grading. . That is, one takes
weighted projective space $\mathbb{C}\mathbb{P}^{4}_{[d_{0}:\ldots:d_{4}]}$ as
the ambient space $A$, which generalizes (2.5) by having integer “weights”
$(d_{0},d_{1},d_{2},d_{3},d_{4})\in\mathbb{Z}_{+}$ as

$\mathbb{C}\mathbb{P}^{4}_{[d_{0}:\ldots:d_{4}]}:=\mathbb{C}^{5}\backslash\\{\vec{0}\\}\bigg{/}\left((z_{0},z_{1},\ldots,z_{4})\sim(\lambda^{d_{0}}z_{0},\ldots,\lambda^{d_{4}}z_{4})\right)\
,\qquad\lambda\in\mathbb{C}\backslash\\{0\\}\ .$ (2.10)

Taking all weights $d_{i}=1$ is the ordinary $\mathbb{C}\mathbb{P}^{4}$. As
with $Q$, if we embed a hypersurface of degree $d_{0}+d_{1}+\ldots+d_{4}$ into
$\mathbb{C}\mathbb{P}^{4}_{[d_{0}:\ldots:d_{4}]}$, it defines a CY3. The
classification of such manifolds was performed in [47] and a total of 7555 is
found, with 2780 distinct Hodge pairs and a more balanced $\chi\in[-960,960]$.

Reflexive Polytopes

The next systematic generalization of weighted projective space is a toric
variety, which, instead of having a single list of weights as in (2.10), has a
list of $m$ weights (giving a so-called charge-matrix) acting on
$\mathbb{C}^{n+m}$ to give an $n$-fold. Based on the theorem of Batyrev-
Borisov [48], Kreuzer and Skarke spent almost a decade explicitly constructing
such Calabi-Yau manifolds, culminating in the early 2000s with the
construction of the most extensive database of CY3 so far, the Toric
Hypersurfaces [13].

In brief, the ambient space is a toric 4-fold $A$, constructed from an integer
polytope $\Delta\subset\mathbb{R}^{4}$ which is reflexive, meaning that
$\Delta$ has a single interior point (which can be taken to be the origin) and
all bounding hyperplanes are distance 1 from this point. Furthermore, a
particular hypersurface in the toric variety $A$ is a CY with defining
equation of the CY3 is given by

$X=\\{\sum\limits_{\vec{m}\in\Delta}c_{\vec{m}}\prod\limits_{j=1}^{k}x_{j}^{\vec{m}\cdot\vec{v}_{j}+1}=0\\}\subset
A\ ,$ (2.11)

with $x_{j}$ coordinates of the ambient toric 4-fold, $c_{\vec{m}}$ complex
coefficients, and $\vec{v}_{j}$ the (integer) vertices of $\Delta^{\circ}$.
The weighted $\mathbb{C}\mathbb{P}^{4}$ hypersurfaces are special cases of
(2.11).

Thus the question of finding toric hypersurface CY3 is the classification of
reflexive integer 4-polytopes (up to $SL(4;\mathbb{Z})$, under which the toric
4-folds are equivalent). In $\mathbb{R}^{1}$, there is trivially 1 reflexive
polytope (the pair of points $\pm 1$). In $\mathbb{R}^{2}$, it is known at
least to 19-th century mathematics, that there are 16 reflexive polygons up to
$SL(2;\mathbb{Z})$. Unfortunately (and perhaps shockingly), the next number is
already unknown until the work of Kreuzer-Skarke. They found 4319 reflexive
polyhedra in $\mathbb{R}^{3}$. For $\mathbb{R}^{4}$, 6 months of computation
on the best computer available to the late 1990s gave an astounding
473,800,776. Each of these gives ∥∥∥ It should be emphasized that most of
these toric ambient spaces $A$ (as with weighted $\mathbb{C}\mathbb{P}^{4}$)
are not smooth, and requires smoothing or resolution of singularities:
different resolutions give rise to potentially different CY3s. Thus, the
actual number of Calabi-Yau 3-folds from this construction is estimated to be
many orders of magnitude larger. For a given $\Delta$, the Hodge pair will be
the same, different resolutions will give different intersection numbers and
Chern classes. Up to $h^{1,1}=7$, this was done exhaustively in [49], while
for the highest $h^{1,1}\sim 490$, this was done in [50]. The full list of
CY3s, after all the resolutions, has been recently estimated to be as large as
$10^{10^{5}}$ [51].  a hypersurface Calabi-Yau 3-fold. Thus, our zoo of
manifolds increased from 5, to some 10 thousand, and to some half-billion.
Interestingly, the next number, that of reflexive polytopes in
$\mathbb{R}^{5}$ up to $SL(5;\mathbb{Z})$, is unknown. It would be great to
have a generating function for the sequence $1;16;4319;473,800,776;\ldots$.

The KS dataset produced 30,108 distinct Hodge pairs and $\chi\in[-960,960]$,
with the extremal values of $\pm 960$ being the weighted
$\mathbb{C}\mathbb{P}^{4}$ cases. No CY construction so far has ever produced
an Euler number whose magnitude exceeds 960. A conjecture of Yau states that
the topological type of (connected, smooth, compact) Calabi-Yau manifolds is
finite in every dimension (we already see this in complex dimensions 1 and 2)
and it could well be that 960 is the upper bound in dimension 3. There has
been nice parallel directions of work in infinite families of Calabi-Yaus [52,
53] beyond topological type such as Gromow-Witten invariants, as well as in
zooming in on special corners of small Hodge numbers [41, 42, 43].

Thus, by the turn of the century, there is an data-base of Calabi-Yau
manifolds whose size is “big” even by today’s standards. There is an internet
meme, that “technically, Moses was the first person to download data from the
cloud using a tablet” [54]. This amusing anachronism is a fitting analogy to
how the age of “big data” in theoretical physics and algebraic geometry really
goes back to the 1980s.

##  3 Data Explosion

Meanwhile, by the mid to late 1990s, in parallel to the heterotic programme
outlined above, the discovery of D-branes [55], M-theory and
$G_{2}$-compactification [56, 57], F-theory [58, 64], AdS/CFT [59], etc., as
well as the wealth of dualities linking them begat the Second String
Revolution. As with the First, this gave rise, and is still continuing to
engender, a plethora of mathematical data, leading to various estimates of the
“string landscape” [60, 15], which was already anticipated in [61]. Numbers
such as $10^{500}$ and, as aforementioned, today’s $10^{10^{5}}$ began to
enter the string and popular psyche.

In some sense, string theory has traded one difficult problem – the
quantization of gravity – with another: the selection of the right vacuum. The
latter is perhaps of more and certainly increasing interest to pure
mathematicians, because the largess of data provides an inspiring playground
for generating, testing and proving new conjectures.

Ultimately, whichever scenario one prefers to geometrically engineer (to use
the phrase of [62]) one’s preferred quantum field theory, including the
standard model, the procedure can be algorithmized. Indeed, any problem in
algebraic geometry (over $\mathbb{C}$) reduces to finding an appropriate
Gröbner basis and then to finding (co-)kernels of integer matrices (in a
corresponding monomial basis) [2].

Take, as an example, AdS/CFT from the point of view of computational geometry:
this is a correspondence between a SUSY conformal QFT and a (non-compact)
Calabi-Yau cone $M$ over a Sasaki-Einstein manifold $X$. Moreover
specifically, this is a mapping between a quiver representation and the
geometric data of $X$ (see e.g., [63] for a quick review). When $M$ is toric,
for instance, the graph data of the quiver and the combinatorial data of $M$
are both amenable to an algorithmic treatment.

###  3.1 The Good, The Bad, and The ?

Taking stock of the progress up to the second decade of this century, we hope
to have given the reader a glimpse of how computational and algorithmic
geometry has enriched the classical dialogue between physics and mathematics.
The ever increasing number of (freely available) mathematical databases online
(typically of size $\sim 1-10$ Gb) is augmented by ever-more efficient
software developed to address them (especially the umbrella project of
SageMath [6]) as well as by the growing power of the personal laptop. This,
certainly can be considered “the Good.”

Unfortunately, most algorithms needed to compute anything, whether it be
finding Gröbner bases, obtaining triangulations of polytopes, or extracting
dual cones, are exponential in complexity. Thus, if one aims to sift through
vacua to find the standard model or to understand the minimal model approach
to algebraic varieties, case-by-case checks is impossible, even with the best
HPC available. This, certainly needs to be rendered as “the Bad”.

While the statistics of the vacuum degeneracy in string theory had been
considered in the last decade [15], it is only expedient, given the breath-
taking speed with which the Big Data Revolution is taking over every aspect of
civilization, especially in this decade of the new millenium, that one should
apply the most recent techniques to address the landscape of mathematical
data.

In many ways, the search of the standard model within the string landscape
reminds us of the hunt for exo-planets. The latter scans the heavens for
habitable earths and the former, for universes akin to ours. The latter
accumulates more and more real data with the betterment of technology and the
former, theoretical data with furtherance of methodology. Whether one believes
our universe is “special” by some anthropic argument, or by a selection
principle, or is a mere point in the multiverse, is currently still a matter
of debate, but the big data of mathematical universes beckon exploration.

##  4 Deep-Learning the Landscape

A question which instinctively occurred whilst contemplating the big data of
universes [17, 18], was that the typical problem in string theory, or, in
algebraic geometry for that matter, is of the form

$\stackrel{{\scriptstyle INPUT}}{{\framebox{\mbox{integer
tensor}}}}\longrightarrow\stackrel{{\scriptstyle
OUTPUT}}{{\framebox{\mbox{integer}}}}\ .$

That one has integer output, especially in string theory, is because presently
much of the field is still at the stage of finding quantities such as number
generations, or the charges of particles. The fact that there is no known non-
trivial (compact) Calabi-Yau metric analytically (Yau’s proof of the Calabi
Conjecture is famously non-constructive and relies on subtle existence
statements of Monge-Ampère PDEs) hinders questions such as finding masses
****** We will later address some of the recent advances in numerical metrics
and connections. . In geometry, much of the field is concerned with finding
topological invariants such as indices or Betti numbers (as mentioned in
introduction in Gauß’ theorema egregium) because when complicated integrals
become integers there is usually some deep mathematics going on. On the other
hand, one has integer tensor input is seen in a multitude of examples above:
whether we are dealing with polytopes or CICY configurations or quiver
adjacency matrices. Of course, in general cases (such as numerical metrics),
the integer condition case be relaxed to numerical tensor input going to some
numerical output.

As discussed repeatedly, the machinery of computational geometry has developed
sophisticated algorithms to obtain the output from the given input, even
though the generic such algorithm is expensive. Nevertheless, physicists and
mathematicians have bitten the bullet over the last 20 years or so and
computed extensive examples. For instance, all Hodge numbers for the
1/2-billion Kreuzer-Skarke Calabi-Yau manifolds have been calculated using
combinatorics of polytopes [65], likewise, all those for CICYs have been
obtained by chasing exact sequences [46].

The situation is rather reminiscent of hand-writing recognition, the
archetypal problem in machine-learning. For example, I write 0 to 9 as follows

(4.12)

and we wish to let the computer recognize them. The input is an image, which
is an $m\times n$ matrix (indexing the pixels in a 2-dimensional grid) each
entry of which is a 3-vector of a real value between 0 and 1, denoting the
percentage of RGB values. If we only wish to keep gray-scale information, each
entry is then a real number between 0 and 1. Or, if we only want black-white,
the input is just a binary matrix. The output is an integer from 0 to 9,
called a 10-channel output.

As mathematicians or theoretical physicists, we might solve this problem by
exploiting the geometry and find, say, a clever Morse function as we scan the
input matrix row-wise and column-wise and detect the critical points. This is,
of course, very expensive. What Google or your smart-phone does, is to turn to
labeled data. Such data has been painfully collected over the years by NIST
(National institute of Standards), and look like the following (each is given
as, for example, by a $28\times 28$ pixelated image):

$\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,clip,width=180.67499pt]{./PICTS/digitSample.jpg}\ldots\quad\framebox{\includegraphics[trim=0.0pt
0.0pt 0.0pt 0.0pt,clip,width=36.135pt]{./PICTS/pixel3.pdf}}\quad 28\times
28\times(RGB)$ (4.13)

The difficult part of labeling each image with the correct channel has been
done, and still adjusting with new usage by new users.

In summary, what happens is the following:

Data Acquisition:

the collection of known cases (input $\to$ output), such as (4.13), gives us
training data;

Machine-Learning:

setting up some algorithm to optimize parameters which does the classification
best;

Data Validation:

once the machine has “learnt” the training data, we can take a set of
validation data, which, importantly, the machine has not seen before. This is
in the same format as the training data, with given input and output and we
check the actual with the predicted outputs.

How different, really, is a problem in algebraic geometry? For example,
computing the Hodge number of a given CICY, after all the work in long exact
sequences in cohomology, gives the association rule

$X={\tiny\left(\begin{array}[]{cccccccc}1&1&0&0&0&0&0&0\\\ 1&0&1&0&0&0&0&0\\\
0&0&0&1&0&1&0&0\\\ 0&0&0&0&1&0&1&0\\\ 0&0&0&0&0&0&2&0\\\ 0&1&1&0&0&0&0&1\\\
1&0&0&0&0&1&1&0\\\ 0&0&0&1&1&0&0&1\\\ \end{array}\right)}\ ,\quad
h^{2,1}(X)=22;\quad\begin{tabular}[]{c}\includegraphics[trim=199.16928pt
142.26378pt 284.52756pt
142.26378pt,clip,width=144.54pt]{./PICTS/cicy2000}\end{tabular}\rightarrow 22\
.$ (4.14)

To the right, we have purposefully represented the CICY configuration as a
pixelated image, since all CICYs can be embedded, after right-bottom zero-
padding, into a $12\times 15$ integer matrix with entries $\in[0,5]$. We have
7890 labeled data-points, from which we can take, say, 80% for training, to be
validated on the remaining 20%. The programme of machine-learning algebraic
geometry was thus initiated [17, 18].

It is timely, that in 2017 (the same year that Sophia, the AI robot, became
the first non-human citizen of a country), 4 independent groups were thinking
about various aspects of machine-learning the string landscape [17, 19, 20,
21]. Thinking back, let us see the sequence of the starting year of annual
series of conferences in the string community: “Strings” (1986-),
‘StringPheno” (2002-), “NSF String Vacuum Project” (2006 - 2010), “String-
Math” (2011-), session of stringy mathematics and physics at “SIAM” (2014-),
and now, “String-Data” (2017-).

###  4.1 An Invitation to Machine-Learning

We refer the reader to the now classic introduction to machine-learning in
[66] as well as a wonderful new monograph for physicists in [67]. Here, it is
expedient to give a rapid taster of this vast subject.

Contrary to expectations, the field of machine-learning and neural networks
goes as far back as cybernetics in the 1940s. In 1957, the first perceptron
was set up by MIT-Cornell, where a wall of CdS photo-receptors was set up to
emulate neurons firing. In the 1980 - 90s, artificial neural networks went
under the philosophy of connectivism where computational power emerged from
inter-connectivity. Slowly the word “artificial” disappeared and such
algorithms were simply called neural networks (NNs). By 2006, the phrase
“Deep” NN came into being, a term which we will explain shortly.

In general, sorting data into discrete categories is done by classifiers and
predicting continuous values, regressors. Given data, machine-learning (ML)
roughly fall under the headings of unsupervised, where patterns are to be
extracted, and supervised where labeled data, such as the ones in (4.13) and
(4.14), where the ML algorithms are trained to associate input to output.
Examples of unsupervised ML include clustering analysis, auto-encoders,
principle component analyses (PCA), etc., and those of supervised ML include
support vector machines (SVM), neural regressors and neural classifiers, etc.
In this talk, I will concentrate, because of the nature of the problem, on
supervised ML.

Let us start with a single neuron (the perceptron), which consists of a
(ususally analytic) function $f(z_{i})$ called the activation function, for
some input tensor $z_{i}$ with multi-index $i$. We then consider
$f(w_{i}z_{i}+b)$ with weights $w_{i}$ and bias $b$. Typical activation
functions include: (1) Logistic Sigmoid: $\left(1+e^{-x}\right)^{-1}$; (2)
Hyperbolic tangent: $\tanh(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$; (3)
Softplus: $\log\left(1+e^{x}\right)$, a “softened” version of ReLu (Rectified
Linear Unit): $\max(0,x)$; (4) Softmax:
$x_{i}\rightarrow\frac{e^{x_{i}}}{\sum_{i}e^{x_{i}}}$; (5) Identity: $x_{i}\to
x_{i}$ (which, with weights and biases, becomes the general affine
transformation).

Given Training data: ${\cal D}=\\{(x_{i}^{(j)},d^{(j)}\\}$ with input $x_{i}$
and known output $d^{(j)}$, we minimize some appropriate cost/loss function to
find optimal $w_{i}$ and $b$ (this is the “learning”). Then, with parameters
fixed, we can check against Validation Data. Common cost functions include SEL
(squared-error-loss)

$SEL:=\sum\limits_{j}\left[f\left(\sum_{i}w_{i}x_{i}^{(j)}+b\right)-d^{(j)}\right]^{2}$
(4.15)

for continuous output, and XC (cross-entropy)

$XE:=-\frac{1}{n}\sum\limits_{j}\left[d^{(j)}\log
f(x^{(j)})+(1-d^{(j)})\log(1-f(x^{(j)}))\right]$ (4.16)

for discrete (categoritcal) data.

The astute reader would recognize that we have done is precisely (non-linear)
regression. With a single neuron, supervised ML is exactly that. When we link
up a multitude of neurons into a directed graph, complexity emerges through
connectivity in a gestalt-philosophical way; this is the NN. A common type of
NN is when the graph organizes into “layers” as in

$\begin{array}[]{cc}\rotatebox[origin={c}]{90.0}{$\longleftarrow$ width
$\longrightarrow$}&\begin{array}[]{c}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,clip,width=325.215pt]{./PICTS/mlp.jpg}\end{array}\\\
&\longleftarrow\mbox{depth}\longrightarrow\end{array}$ (4.17)

which is called a forward-feeding NN (or, a more dated acronym, MLP, for
multi-layer perceptron). The MLP is composed of an input layer, an output
layer, and a number of hidden layers. The total number of layers is called the
depth and the rough number of neurons per layer, the width. ML with large
depth NNs is, for obvious reasons, called deep learning.

The precise choice of activation functions and inter-connectivity of the NN is
called the architecture. The various parameters - not the variables like
weights and biases to be optimized during training - such as depth, width,
learning-rate (this the step-size for any gradient descent method using for
finding minima), batch-size (the training data is usually passed in batches at
a time), etc., are called hyper-parameters. As one can imagine, there is a
variety of universal approximation theorems which essentially state that for
sufficiently large width, or depth, any output can be approximated to
arbitrary precision. In fact, a forward-feeding fully-connected NN with only
ReLU activation is good enough to approximate any integrable function.

As with all models of statistical prediction, it is good to have a measure of
“goodness of fit”. Some standard ones are as follows

Naïve Precision:

This is particularly useful when the output is discrete (and belonging to a
relatively small number of categories), and we simply compute the percentage
of agreed cases between the predicted and actual.

R-squared:

For continuous output, suppose on validation dataset ${\cal
V}=\\{x_{i}^{(j)}\longrightarrow{d^{(j)}}\\}_{j=1,2,\ldots,m}$, the predicted
values are $\\{x_{i}^{(j)}\longrightarrow{\hat{d}^{(j)}}\\}_{j}$. Then the
Coefficient of Determination, or simply R-squared, is defined to be
$R^{2}:=1-\frac{SS_{\mbox{res}}}{SS_{\mbox{tot}}}$, where data variance is
$SS_{\mbox{tot}}:=\sum_{j}(d^{(j)}-\overline{d^{(j)}})^{2}$ for mean
$\overline{d^{(j)}}$, and residual sum of squares is
$SS_{\mbox{res}}:=\sum_{j}(d^{(j)}-\hat{d}^{(j)})^{2}$. A bad fit is when
$R^{2}$ is close to 0, and a perfect fit, when $R^{2}=1$.

Confusion Matrix:

For discrete output (say $n$ categories), we can establish an $n\times n$
matrix with the $(i,j)$-th entry being the number of cases predicted to be $j$
while the actual value is $i$. Ideally, we wish this to be a diagonal matrix.
A measure of how close to the diagonal is the Matthews’ $\phi$-coefficient
defined to be $\sqrt{\chi^{2}/n}$ where $\chi^{2}$ is the Chi-square of the
matrix treated as a contingency table. A value of $\phi=0$ means the
correlation is random and $\phi=1$ is a perfect fit (incidentally, $\phi=-1$
mean complete anti-correlation); thus we can use $\phi$ as a measure of
confidence (avoiding false positives and false negatives) in addition to the
naïve precision.

###  4.2 Initial Experiments

Armed with the appropriate mathematical data and the technique of ML,
implemented via either Python’s Keras/TensorFlow [68] or Wolfram’s Mathematica
version $>11.0$[69]. One can take a simple MLP of the form (the hyper-
parameters and architecture differ for specific cases and the following is
only an illustration of a typical case)

$\begin{array}[]{c}\includegraphics[trim=28.45274pt 0.0pt 0.0pt
0.0pt,clip,width=361.34999pt]{./PICTS/NNclass}\end{array}$ (4.18)

where the hidden layers are (1) a fully-connected linear layer of 500 nodes;
(2) element-wise sigmoid activation $\sigma(z):=(1+e^{-z})^{-1}$; (3) dropout
layer (we switch off neurons with some probability in order not to over-fit);
(3) linear layer of 100 nodes; (4) sigmoid; (5) linear layer of 2 nodes; (6)
Softmax to output. Such an NN was found to estimate the size of Hodge numbers
for CICYs and WP4 hypersurfaces very well in a matter of seconds on an
ordinary laptop [17]. One could think of this setup, a fully connected neural
networks of depth $d$, as the following composition of maps:

$\mathbb{R}^{n_{0}}\stackrel{{\scriptstyle
L_{n_{1}}}}{{\longrightarrow}}\mathbb{R}^{n_{1}}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}\mathbb{R}^{n_{1}}\stackrel{{\scriptstyle
L_{n_{2}}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
L_{n_{d}}}}{{\longrightarrow}}\mathbb{R}^{n_{d}}\rightarrow\mathbb{R}\;,$
(4.19)

where $L_{n}$ are activation functions (such as sigmoids) with trainable
weights and biases and co-domain dimension $n$ and the last layer outputs some
real value or discrete value. The power of the MLP is the harnessing of the
ultimately complicated (not even necessarily analytic) structure of the
composite map.

A detailed analysis was carried out in [70] where the 19-way
classifier/regressor in an architecture similar to (4.18) as well as an SVM,
performed to about 90% accuracy in an 80-20% training-validation split ††††††
We remark that for CICY 4-folds, where the data is about a million, accuracies
to around 96% was achieved [71]. . It is interesting that in these
experiments, one never exploited the matrix structure of the input: e.g., the
CICY configuration was flattened a long vector of integers. This is quite
contrary to the image processing of (4.13) where a convolutional network (CNN)
would be used which “convolves” with nearest neighbours. Such CNNs were indeed
tried more recently and $>99\%$ accuracies were reached [73].

###  4.3 More Success Stories in String/Geometry

One can imagine that all computational problems in string phenomenology and
more generally in computational algebraic geometry could benefit from the
paradigm of machine-learning. Indeed, the initial explorations of [19] on
Calabi-Yau volumes, of [20] on line-bundle cohomology, and of [21] on F-theory
compactifications, in conjunctions with [17], launched the String Data
conference series from 2017 on-wards.

Though it is difficult to review all the works since, I will give a
bird’s-eye-view of the the various directions taken, first within
string/geometry, and then more generally to other branches of mathematics. In
the former, some major directions and success stories (with $0.90$ accuracies
with relatively simple architectures) have included

Heterotic:

Selection of MSSM from heterotic orbifold constructions [92, 93],
distinguishing standard models from heterotic line bundles [96, 99]. Machine-
learning of bundle cohomology of surfaces [95] as well as toric hypersurfaces
[97]. One points out [98] where exact formulae were found for line-bundle
cohomology through an MLP exploration of the regions in moduli space.

F/M-Theory:

Finding gauge groups [88] and matter-content [89] within F-theory
compactifications. Distinguishing elliptically fibered manifolds within the
CICYs [91]. Decidability issue of diophantine systems in Kähler stabilization
[90].

Type II:

topological data analysis [84] of, and genetic algorithms for searching within
[85] flux vacua in type II. Reinforcement learning explorations of IIA brane
configurations [86] and IIB landscape [87]. Seiberg duality in type IIB quiver
theories [112].

Physical Symmetries:

symmetries in various physical systems (including representations of CICYs)
[105], and CFT symmetries [106].

Metric:

As mentioned several times, there is no known analytic Calabi-Yau metric on a
non-trivial compact Kähler manifold. Donaldson developed an efficient
numerical algorithm using the method of balanced metrics from a potential
formed by increasing powers of monomial sections [100], which were then nicely
implemented in [101, 102] (q.v.. also the functional method of [103]). It was
shown in [104] that Donaldson’s algorithm can be machine-learnt (and to
10-100-fold increase in efficiency).

Cosmology:

The cosmic landscape [80], especially vacuum selection from cosmology
constraints [81], were studied. Interesting network structures were found in
[81] and [83] studied certain accessibility measures in inflation and [82],
machine learning in inflation.

“Meta” Physics:

A fun experiment was undertaken in [107] where all titles from hep-th and four
related sections of the arXiv: hep-ph, hep-lat, gr-qc, and math-ph were
downloaded since the beginning and fed into the NN Word2Vec (about $10^{6}$
titles). Interesting linear syntactical identities such as “holography +
quantum + string + ads = extremal-black-hole” presented themselves and the
syntactical structure of the different sections were indeed found to be
distinct.

Of particular note are the striking ideas in [74, 75, 76, 77] where the
fundamentals of quantum field theory, holography and renormalization group
flow, are phrased in terms of appropriate neural networks. Indeed, the reader
is also referred to the recent works of [78, 79] on the possible computational
nature of reality itself.

##  5 Outlook: ML Mathematical Structures

Given the efficacy of ML in so many directions in string/geometry, it is
natural to ask whether and how different problems in mathematics respond to
ML. We leave a detailed discussion of this to [23], but for now, it is perhaps
fitting that we conclude this talk with some conducive experiments which have
been performed as well as some speculations for the future. Let us
approximately group the successful experiments by subject:

Algebraic Geometry over $\mathbb{C}$:

Most of the problems mentioned above fall under this heading. We need to
emphasize that we work over $\mathbb{C}$, an algebraically closed number
field. Any problem in computational algebraic geometry essentially boils down
to finding kernels and co-kernels of integer matrices (in appropriate monomial
bases), something quite adaptable to ML. A recent work on using reinforcement
learning to perform the key step of finding S-pairs in constructing Gröbner
bases was done in [118].

Representation Theory:

Preliminary investigation on whether SVMs and MLPs can distinguish finite
groups and finite rings from random matrix structures was initiated in [108];
more surprising was the fact that simple groups seemed to be distinguishable.
For continuous groups, lengths of branching rules and tensor decomposition in
Lie algebras can also be learned by looking at weight vectors [109]; this is
obviously also of importance to particle physics.

Knot Theory:

Jones polynomials and complementary volume of knots are studied from ML in
[110] and letting ML find configurations of knots themselves, in [111].

Graph Theory and Combinatorics:

Cluster mutation on quivers was studied in [112]. On a more basic level,
properties of finite simple graphs, such as whether it possesses Euler or
Hamilton cycles, whether it is flat (there is a notion of Ricci-flatness for
finite graphs), etc. were studied in [113].

Number Theory:

As one might imagine, a direct attack on predicting the next prime number by
ML is most likely unfruitful [17, 22]. Likewise, predicting quantities
relevant to the Birch-Swinnerton-Dyer Conjecture was also difficult [114].
Surprisingly, however, problems in arithmetic geometry, ranging from dessins
d’enfant [115] ($>0.9$ accuracy), to arithmetic properties of hyper-elliptic
curves [116] ($\sim 0.99-1.00$ accuracies) and Galois number field extensions
of the rationals [117] ($>0.9$ accuracy) behaved very well to simple
classifiers such as Naïve Bayes.

Symbolic Manipulation:

Recent advances in generating new identities in calculus [119] and continued
fractions [120] have met with impressive success. So too, have there been
tools to extract fundamental laws [121] and formulae [122] of physics.

With these tantalizing thoughts let us conclude my talk here. We have seen how
into the alembic of mathematics and fundamental physics is now infused, over
the last few years, new techniques of the data revolution, especially the
predictive power of machine-learning and neural networks. We are, of course,
only at the early stage. Having an ML predict a result, even to 100% accuracy,
does not always mean one could obtain analytic information as to why. What we
hope for, is what the physicist Max Tegmark calls “intelligible intelligence”,
where we can formulate new results, or at least conjecture precise statements,
when we are given an ML algorithm which performs superbly well. When I shared
my initial excitement back in 2017, of the prospects to machine-learn problems
ranging from geometry to algebra, to my friend the logician Boris Zilber, he
astutely remarked: “now you have syntax, it would be good to find the
semantics.”

## Acknowledgments

We are grateful for the kind invitations, in person and over Zoom, of the
various institutions over this most extraordinary year of 2020 – the
hospitality and conversations before the lock-down and the opportunity for a
glimpse of the outside world during: Harvard University, Tsinghua
University/BIMSA, Universidad Católica del Norte Chile, London Institute of
Mathematical Sciences, Queen’s Belfast, King’s College London, University of
Connecticut, “Clifford Algebra & Applications 2020” at UST China, “String
Maths 2020” at Capetown, “International Congress Mathematical Software 2020”
at Braunschweig, University of Torino, “SageMath/M2 - an Open Source
Initiative” at the University of Minnesota, “East Asia Strings” at Taipei-
Seoul-Tokyo, Nankai University, Imperial College London, and Nottingham
University. The work, as always, is indebted to STFC UK for grant ST/J00037X/1
and Merton College, Oxford for a quiet corner of paradise.

## References

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http://geometry.ma.ic.ac.uk/3CinG/index.php/data/
http://coates.ma.ic.ac.uk/fanosearch/

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StringMaths 2020, https://www.youtube.com/watch?v=GqoqxFsaogY

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