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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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 - page 1/102
  Today I'll talk about my work as a Ph.D. student at the University of Torino.
  I will deal with aspects related to phenomenology from the point of view of strings for which I worked both on theoretical and computational levels.
 - page 2/102
  The plan is to first introduce a semi-phenomenological description of physics from a theory of strings.
  These tools will be used throughout the talk as basis for the rest of the topics.
  I will then mainly deal with open strings and analyse correlators in the presence of twist and spin fields seen as singular points in the string propagation.
  From particle physics, I will then move to a string theory description of cosmology in the presence of time dependent singularities in time dependent orbifods.
  I will try to give a reason for the presence of divergences in amplitudes due to particular interactions of string modes.
  Finally I will focus on computational aspects of string compactifications.
  In particular I will deal with recent advancements in deep learning and artificial intelligence applied to the computation of Hodge numbers of Calab-Yau manifolds.
 - page 3/102
  I will therefore start with the most geometrical aspects of the discussion, reviewing some basics and introducing a framework to deal with open string amplitudes in the presence of twist and spin fields.
 - page 4/102
  As usual in string theory the starting point is Polyakov's action.
  We start directly from the superstring extension where gamma is the worldsheet metric (for the moment generic) and rhos are the 2-dimensional "gamma matrices" forming a representation of the Clifford algebra.
 - page 5/102
  This description presents lots of symmetries which, at the end of the day, provide the key to its success.
  In fact the invariance under Poincaré transformations in the target space, and diffeomorphisms and Weyl transformations on the worldsheet are such that the theory is conformal with a vanishing traceless stress tensor.
  Using all symmetries the worldsheet metric can be brought to diagonal form.
 - page 6/102
  Using standard field theory methods the stress energy tensor of the superstring in D dimensions generates the known Virasoro algebra, extending the classical de Witt's algebra.
  At quantum level it presents a central charge whose value depends on the dimension of the target space.
 - page 9/102
  In order for the stress energy tensor to be a conformal primary field, we need to introduce sets of fields known as conformal ghosts.
  These are conformal fields specified by their weight lambda.
  They are introduced as a first order Lagrangian theory.
 - page 8/102
  The central charge arising from the algebra of the modes of the ghost sector can then be used to compensate the superstring counterpart.
  Consistency of the theory thus fixes the dimension of the target space to be a 10-dimensional spacetime.
 - page 9/102
  The fact that "everyday" physics in accelerators is 4-dimensional is recovered through compactification.
  In simple words we recover 4-dimensional physics at low energy by considering the 10-dimensional Minkowski space as a product of the usual 4-dimensional spacetime and a 6-(real)-dimensional space.
  The internal space has to obey stringent restrictions: in particular it has to be a compact manifold (to "hide" the extra dimensions), it has to break most of the supersymmetry present at high energy, and its arising gauge algebra has to contain the standard model algebra.
 - page 10/102
  These manifolds have been firstly postulated by Eugenio Calabi and later proved to exist by Shing Tung Yau, hence the name Calabi-Yau manifolds.
  In the case at hand, they are a specific class of 3-dimensional complex manifolds with SU(3) holonomy.
  They must be Ricci-flat or equivalently have a vanishing first Chern class.
 - page 11/102
  In general it is not easy to classify these manifolds (as well as computing for instance their metric, which is generally not known).
  However, for instance, we will see that the dimension of the complex cohomology groups, known as Hodge numbers, will play a strategic role.
 - page 12/102
  Going back to Polyakov's action, we solve the equations of motion and the boundary conditions for the string propagation (we focus on the bosonic part for the moment).
  The action in fact introduces naturally the Neumann boundary conditions for the strings.
  Moreover the solution factorises into its holomorphic components (where z is the usual coordinate on the complex plane) due to the equations of motion.
 - page 13/102
  Now consider for a second the simplest toroidal compactification of closed strings, that is suppose we can "hide" the last direction of the string on a circle.
  As in usual quantum mechanics this leads to momentum quantization (defined as an integer n in units of the compactification radius).
  A closed string can moreover wind an integer number of times introducing a "winding number" m stating the number of times which the closed string goes around the compact dimension.
  In turn this is reflected on the spectrum of the theory.
  Differently from field theory, shrinking the radius does not decouple the modes, but the compact dimension remains.
  In fact the spectrum of theories compactified on a small radius or a large radius does not change.
  In other words exchanging R and 1/R does not modify the theory.
 - page 14/102
  This so called T-duality can also be applied to open strings with a different outcome.
  At the level of the fields, T-duality switches the sign of the right modes.
  Since open strings cannot wind around the compact dimension, the behaviour of the spectrum is as in field theory: the compact dimension decouples and the open string is constrained on a lower dimensional surface.
  This is the result of introducing Dirichlet boundary conditions on the T-dual coordinate, meaning that the endpoints of the string have to reside on the same surface.
  The procedure can be applied to several dimensions thus introducing surfaces on which the endpoints of the open string live, called D-branes.
 - page 15/102
  D-branes naturally introduce preferred directions of motion thus breaking the D-dimensional Poincaré symmetry.
  This is also seen at the level of the content of the theory.
  It is in fact possible to show that in D-dimensions, the open string sector at massless level contains an Abelian field.
  D-branes split the components into a lower dimensional U(1) field on the D-brane and a vector of spacetime scalars.
 - page 16/102
  In general we have strings whose endpoints are on the same D-brane leading to U(1) gauge theories as well as stretched strings across different branes.
  The Chan-Paton factors can be used to label the endpoints of the strings by the position on the D-branes.
  It is finally possible to show that when the D-branes are coincident, states can be rearranged into enhanced gauge grooups (for instance, unitary), thus creating non Abelian gauge theories.
 - page 17/102
  However, physical constraints on the possible constructions pose serious questions on the D-brane dispositions.
  A quark can be modelled with a string stretched across two distant branes, but such quark would have a mass proportional to the distance to the branes: chirality in particular would not be possible to define, while being one of the defining features of the Standard Model.
  This is where the possibility to put D-branes at an angle with respect to each other becomes crucial.
  While most of the modes will indeed gain a mass, the massless spectrum can support chiral states localised at the intersections.
 - page 18/102
  In this framework we consider models built with D6-branes in such a way that four dimensional Minkowski is filled, thus preserving the "physical" Minkowski invariance.
  We then embed them in 6-dimensional space (we do not worry about compactification for now).
  In this scenario we study correlators of fields in the presence of twist fields, which are a set of conformal fields arising at the intersections.
  These are particularly interesting for phenomenological computations: for instance Yukawa couplings from the string theory definition.
 - page 19/102
  Using the path integral approach, correlators involving twist fields are dominated by the instanton contribution of the classical Euclidean action.
  We focus specifically on its computation in the case of three D-branes necessary for the Yukawa couplings.
 - page 20/102
  The literature already takes into consideration D-branes embedded as lines in a factorised version of the internal space, where the possible relative rotations are Abelian in each plane of the space.
  The contribution of the string in this case is proportional to the area of the triangle formed on the plane by the three D-branes because in fact the string is completely constrained to move on the plane and its worldsheet is in fact the planar area.
 - page 21/102
  In what follows we consider a non completely factorised space and we focus on the 4-dimensional part of it.
  After filling the physical 4-dimensional space, we study the remaining directions of the D-brane in 4-dimensional internal space (that is planes in four dimensions which intersect in a point) using a well-adapted frame of reference which is in general rotated with respect to global coordinates.
  The rotation is not directly an SO(4) element, but it is an element of a particular Grassmannian: in fact separately rotating the Dirichlet and Neumann components does not change anything (we just need to relabel the coordinates), as long as no reflections are involved.
  The rotation involved is therefore a representative of a left equivalence class of possible rotations.
 - page 22/102
  Once the geometry is specified, we then consider the open strings.
  Introducing the usual conformal transformations mapping the strip to the complex plane, the intersecting branes are mapped to the real axis.
  D-branes are therefore real intervals on the space, between their intersection points, mapped to the real axis as well.
 - page 23/102
  In this definition, the boundary conditions of the strings become a set of discontinuities on the real axis, one for each D-brane, and the embedding equation specifying the intersections.
 - page 24/102
  Instead of dealing directly with the discontinuities, it is more suitable to introduce an auxiliary function defined over the entire complex plane by gluing functions defined on the upper plane and on the bottom plane on any arbitrary interval.
  This "doubling trick" transforms the discontinuity into a monodromy factor, when looping any base point through the glued interval.
 - page 25/102
  The fact that rotations are non Abelian leads to two different monodromies for a base points starting in the upper plane or in the bottom plane.
  This is a general feature, but the 3 D-branes case simplifies this enough.
 - page 26/102
  Dealing with these 4 x 4 matrices is delicate.
  Using a known isomorphism between SO(4) matrices and SU(2) x SU(2), the monodromy matrix can be cast into a tensor product of two 2 x 2 matrices.
  The matrices completely encode the rotations of the D-branes: the solution to the string propagation is therefore given by a holomorphic function (due to the equations of motion), satisfying these boundary conditions for each of the three D-branes we consider.
  In this matrix form we therefore look for a tensor product of 2-dimensional basis of holomorphic functions with three distinct regular (Fuchsian) monodromies.
 - page 27/102
  The usual SL(2,R) invariance allows us to fix the monodromies in 0, 1 and infinity.
  The overall solution will then be a superposition of all possible basis.
  Given the previous properties we therefore look for a basis of Gauss hypergeometric functions.
 - page 28/102
  Since we deal with rotations, the parameters of the hypergeometric functions involved are indeed connected to the parameters of the rotation (that is the rotation vectors).
  However the choice is not unique and labelled by the periodicity of the rotations.
  The superposition of the solutions is however still not the final result.
 -------------- TODO
 - page 36/102
  The reason is a huge redundancy in the description: using the free parameters of the rotations we should in fact fix all degrees of freedom in the solution, which at the moment is an infinite sum involving an infinite amount of free parameters.
  For the moment we only showed that the rotation matrix is equivalent to a monodromy matrix from which can build an initial solution.
 - page 37/102
  Using contiguity relations we can then restrict the sum over independent functions (that is functions which cannot be written as rational functions of contiguous hypergeometrics).
 - page 38/102
  Finally requiring the Euclidean action to be finite restricts the sum to only two terms.
 - page 39/102
  Imposing the boundary conditions (that is fixing the intersection points) fixes the free constants in the solution.
 - page 40/102
  The physical interpretation of the solution is finally straightforward in the Abelian case, where the action can be reduced to a sum of the area of the triangles.
 - page 41/102
  In the non Abelian case we considered there is no simple way to write the action using global data.
  However the contribution to the Euclidean action is larger than the Abelian case: the strings are in fact no longer constrained on a plane and, in order to stretch across the boundaries, they have to form a small bump while detaching from the D-brane.
  The Yukawa coupling in this case is therefore suppressed with respect to the Abelian case.
  Phenomenologically speaking, since the couplings are proportional to the mass of the scalar involved, the non Abelian case describes the coupling of lighter states. 
 - page 42/102
  We then turn the attention to fermions and the computation of correlators involving spin fields.
  Though ideally extending the framework introduced before, we abandon the intersecting D-brane scenario, and we introduce point-like defects on one boundary of the superstring worldsheeet in its time direction in such a way that the superstring undergoes a change of its boundary conditions when meeting a defect.
 - page 43/102
  It is possible to show that in this case the Hamiltonian of the theory develops a time dependence: it is in fact conserved only between consecutive defects.
 - page 44/102
  Suppose now that we could expand the field in a basis of solutions to the boundary conditions and work, as before, on the entire complex plane.
 - page 45/102
  Ideally we would be interested in extracting the modes in order to perform any computation of amplitudes.
  The definition of the operation is connected to a dual basis whose form is completely fixed by the original field (which we know) and the request of time independence.
 - page 46/102
  The resulting algebra of the operators is in fact defined through such operation and is therefore time independent (as it should be for consistency).
 - page 47/102
  Differently from what done in the bosonic case, we focus on U(1) boundary change operators.
  The resulting monodromy on the complex plane is therefore a phase factor.
 - page 48/102
  As in the previous case we can write a basis of solutions which incorporates the behaviour looping the point-like defects.
  Consequently we can also define a dual basis.
  Notice that both fields are defined up to integer factors, since we are dealing with rotations (not unlike the previous bosonic case).
 - page 49/102
  In order to compute amplitudes we then need to define the space on which the representation of the algebra acts.
  We define an excited vacuum, annihilated by positive frequency modes, and the lowest energy vacuum (from the strip definition).
  None of these vacua is actually the usual invariant vacuum, as the short distance behaviour with the stress energy tensor shows the presence of operators responsible of the change in the boundary conditions.
 - page 50/102
  The vacua need to be consistent, leading to conditions labelled by an integer factor L relating the basis of solutions with its dual (and ultimately the algebra of operators).
  In fact the vacuum should always be correctly normalised and the description of physics using any two of the vacuum definitions should be consistently equivalent.
 - page 51/102
  To avoid having overlapping in- and out-annihilators, the label L must vanish.
 - page 52/102
  With this frameword, the stress energy tensor displays as expected a time dependence due to the presence of the point-like defects.
  Specifically it shows that in each defect we have a primary boundary changing operators (whose weight depends on the monodromy epsilon) and which creates the excited vacuum from the invariant vacuum.
  This is by all means an excited spin field.
  Moreover the first order singularities display the interaction between pairs of excited spin fields.
  Finally (and definitely fascinating), the stress energy tensor obeys the canonical OPE, that is the theory is still conformal (even though there is a time dependence). 
 - page 53/102
  In formulae, the excited vacuum used in computations is created by a radially ordered product of excited spin fields.
 - page 54/102
  We are therefore in a position to compute the correlators involving such spin fields (however since we cannot compute the normalisation, we can compute only quantities not involving it).
  For instance we reproduce the known result of bosonization where the boundary changing operator is realised through the exponential of a different operator.
  Moreover, since we have complete control over the algebra of the fermionic fields, we can also compute any correlator involving both spin and matter fields.
 - page 55/102
  We therefore showed that semi-phenomenological models need the ability to compute correlators involving twist and spin fields.
 - page 56/102
  We then introduced a framework to compute the instanton contribution to the correlators using intersecting D-branes.
 - page 57/102
  We then showed how to compute correlators in the fermionic case involving spin fields as point-like defects on the string worldsheet.
 - page 58/102
  The question would now be how to extend this to non Abelian spin fields and, most importantly, to twist fields, where there is no framework such as bosonization.
 - page 59/102
  After considering defects and singular points in particle physics, we analyse time dependent singularities.
  These are defining properties of many cosmological models (such as the Big Bang singularity), and can be studied in different ways.
 - page 60/102
  As string theory is considered a theory of everything, its phenomenological description should in fact include both strong and electroweak forces as well as gravity.
 - page 61/102
  In particular from the gravity and cosmology side, we would like to have a better view of the cosmological implications in string theory.
 - page 62/102
  For instance we could try to study Big Bang models to gain an improved insight with respect to field theory.
 - page 63/102
  For this, one way would be to build toy models of singularities in time, in which the singular point exists in one specific moment, rather than place.
 - page 64/102
  A simple way to make it so is to build toy models from time-dependent orbifolds.
 - page 65/102
  Orbifolds were introduced in physics before their formalisation as rigorous mathematical entities.
  The mathematical idea of orbifold is similar in construction to the construction of differential manifolds, where the topological space inherits the topology of the left coset of M and a Lie Group G and where the charts encode the orbital partitions through a projection map.
  All in all, orbifolds are manifolds locally isomorphic to a quotient.
  From the physical point of view, the requirements are relaxed.
  In fact we usually focus on global quotients where the orbifold group is the isometry group.
  This leads to the presence of fixed points on the orbifold where additional states are located, the so called twisted sectors of the theory.
  They usually appear as singular limits of Calabi-Yau manifolds.
 - page 66/102
  We shall use these to introduce singularities in time.
 - page 67/102
  In the literature we can already find efforts in the computation of amplitudes (mainly closed strings, since we are dealing with gravitational interactions).
  The presence of divergences in N-point correlators is however usually associated to a gravitational backreaction due to the exchange of gravitons.
 - page 68/102
  However the 4-tachyon amplitude in string theory is divergent already in the open string sector (at tree level, thus we are sure no gravitational interaction is present).
  The effective field theory interpretation would be a 4-point interaction of scalar fields (higher spins would only spoil the behaviour).
 - page 69/102
  To investigate further, consider the null boost orbifold from D-dimensional Minkowski space through a change of coordinates (as such it shares most of the properties with the usual spacetime).
 - page 70/102
  The orbifold is built through the periodic identification of one coordinate along the direction of its Killing vector.
  Notice that at this time the momentum in this direction will have to be quantized to be consistent.
 - page 71/102
  From these identifications, we can build the usual scalar wave function obeying the standard equations of motion.
  Notice the behaviour in the time direction u which already takes a peculiar form and the presence of the quantized momentum in a strategic place.
 - page 72/102
  In order to introduce the problem we first consider a theory of scalar QED.
 - page 73/102
  When computing the interactions between the fields, the terms involved are entirely defined by two main integrals.
  It might not be immediately visible, but given the behaviour of the scalar functions, any vertex interaction with more than 3 fields diverges.
 - page 74/102
  The reason for the divergence is connected to the "strategically placed" quantized momentum.
  When when all quantized momenta vanish, in the limit of small u (that is near the singularity) the integrands develop isolated zeros which prevent the convergence.
  In fact, in this case, even a distributional interpretation (not unlike the derivative of a delta function) fails.
 - page 75/102
  So far the situation is therefore somewhat troublesome.
  In fact even the simplest theory presents divergences.
 - page 76/102
  Moreover, obvious ways to regularise the theory do not work: for instance adding a Wilson line does not cure the problem as divergences also involve neutral strings which would not feel the regularisation.
 - page 77/102
  The nature of the divergence is therefore not just gravitational, but there must be something hidden.
 - page 78/102
  In fact the problems seem to arise from the vanishing volume in phase space along the compact direction: the issue looks like geometrical, rather than strictly gravitational.
 - page 79/102
  Since the field theory fails to give a reasonable value for amplitudes involving time-dependent singularities, we could therefore ask whether string theory can shed some light.
 - page 80/102
  The relevant divergent integrals are in fact present also in string theory.
  They arise from interactions of massive vertices (the first massive vertex is shown here).
  These vertices are usually overlooked as they do not play in general a relevant role.
  However it is possible that near the singularity they might actually come into game and give a contribution.
  These vertices are involved at low energy in the definition of contact terms (that is terms which do not involve exchange of vector bosons) in the effective field theory, which therefore is lacking their definition.
 - page 81/102
  In this sense even string theory cannot give a solution to the problem.
  In other words since the effective theory does not even exist, its high energy completion cannot be capable of providing a better description.
 - page 82/102
  There is however one geometric way to escape this.
  Since the issues are related to a vanishing phase space volume, it is sufficient to add a non compact direction to the orbifold in which the particle is "free to escape".
 - page 83/102
  While the generalised null boost orbifold has basically the same definition through its Killing vector, the presence of the additional direction acts in a different way on the definition of the scalar functions.
  As you can see the new time behaviour ensures better convergence properties, and the presence of the continuous momentum ensures that no isolated zeros are present at any time.
  In fact even in the worst case scenario, the arising amplitudes would still have a distributional interpretation.
 - page 84/102
  We therefore showed that divergences in the simplest theories are present both in field theory and string theory.
 - page 85/102
  And that in the presence of singularities, the string massive states start to play a role.
 - page 86/102
  The nature of the divergences is entirely due to vanishing volumes in phase space.
 - page 87/102
  But the introduction of "escape routes" for fields establishes a distributional interpretation of the amplitudes.
 - page 88/102
  It is also possible to show that this is not restricted to "null boost" types of orbifolds, but even other kinds of orbifolds present the same issues.
 - page 89/102
  In summary we showed that the divergences cannot be regarded as simply gravitational, but even gauge theories present issues.
  Their nature is however subtle: string massive states are not usually taken into account when computing amplitudes.
 - page 90/102 
  We finally move to the last chapter involving methods for phenomenology in string theory.
  After the analysis of semi-phenomenological analytical models, we now consider a computational task related to compactifications of extra-dimensions.
 - page 91/102
  We focus on Calabi-Yau manifolds in three complex dimensions.
  Due to their properties and their symmetries, the relevant topological invariants are two Hodge numbers: they are integer numbers and in general can be difficult to compute.
  As the number of possible Calabi-Yau 3-folds is an astonishingly huge number, we focus on a subset.
 - page 92/102
  Specifically we focus on manifolds built as intersections of projective spaces, that is intersections of several homogeneous equations in the complex coordinates of the manifold.
  As we are interested in studying these manifolds as topological spaces we do not care about the coefficients, but only the exponents.
  The intersection is complete in the sense that it is non degenerate.
 - page 93/102
  The intersections can be generalised to multiple projective spaces and equations.
  The problem we are interested is therefore to be able to take the so called "configuration matrix" of the manifolds and predict the value of the Hodge numbers.
  Formally this is a map from a matrix to a natural number.
 - page 94/102
  The real issue is now how to treat the configuration matrix and how to build such map.
 - page 95/102
  We use a machine learning approach.
  In very simple words it means that we want to find a new representation of the input (possibly parametrized by some weights which we can tune and control) such that the predicted Hodge numbers are as close as possible to the correct result.
  In this sense the machine has to learn some way to transform the input to get a result close to what in the computer science literature is called the "ground truth".
  The measure of proximity is called "loss function" or "Lagrangian function" (with a slight abuse of naming conventions).
  The machine then learns some way to minimise this function (for instance using gradient descent methods and updating the previously mentioned weights).
 - page 96/102
  We thus exchange the difficult problem of finding an analytical solution with an optimisation problem (it does not imply "easy", but it is at least doable).
 - page 97/102
  In order to learn the best way to change the input representation, we can rely on a vast computer science literature and use large physics datasets containing lots of samples from which to infer a structure.
 - page 98/102
  In this sense the approach can merge techniques from physics, mathematics and computer science benefiting from advancements in all fields.
 - page 99/102
  The approach can furthermore provide a good way to analyse data and infer structure and advance hypothesis, which could end up overlooked using traditional brute force algorithms.
  In this case we focus on the prediction of two Hodge numbers with very different distributions and ranges.
  The data we consider were computed using top of the class computing power at CERN in the 80s, with a huge effort of the string theory community.
  In this sense Complete Intersection Calabi-Yau manifolds are a good starting point to investigate the application of machine learning techniques: they are well studied and completely characterised.
 - page 100/102
  The dataset we use contains less than 10000 manifolds.
 - page 101/102
  From these we remove product spaces (recognisable by their block diagonal form of the configuration matrix) and we remove very high values of the Hodge numbers from training.
  Mind that in this sense we are simply not feeding the machine "extremal" configurations in an attempt to push as far as possible the application: should the machine learn a good representation, it should be automatically capable of learning also those configurations without a human manually feeding them.
 - page 102/102
  With respect to previous distributions, we therefore consider a smaller subset of matrices for training.
 - page 103/102
  We then define three separate folds: the largest contains training data used by the machine to adjust the parametrisation, 10% of the data is then used for intermediate evaluation of the process (for instance to avoid the machine to overfit the data in the training set), while the last subset is used to give the final predictions.
  Differently from the validation set, the test set has not been seen by the machine and therefore can reliably test the generalisation ability of the algorithm.
 - page 104/102
  Differently from previous approaches we consider this as a regression task in the attempt to let the machine learn a true map between the configuration matrix and the Hodge numbers, rather than a classification algorithm (in case we can also discuss the classification approach as it has some interesting applications itself).
 - page 105/102
  The distributions of the Hodge numbers therefore present less outliers than the initial dataset, but as you can see we expect the result to be similar even without the procedure, since the number of outliers removed is small.
 - page 106/102
  The pipeline we adopt is the same used at industrial level by companies and data scientists.
  We in fact heavily rely on data analysis to improve as much as possible the output.
 - page 107/102
  This for instance can be done by including redundant information, that is by feeding the machine variables which can be manually derived: by definition they are redundant but can be used to easily learn a pattern.
  In fact as we can see most of the features such as the number of projective spaces or the number of equations in the matrix are heavily correlated with the Hodge numbers.
  Moreover even using algorithms to produce a ranking of the variables show that such "engineered features" are much more important than the configuration matrix itself.
  Here we can see some of the scalar variables ranked across each other.
 - page 108/102
  Using the "engineered data", we now get to the choice of the algorithm.
  There is no general rule in this, even though there might be good guidelines to follow.
 - page 109/102
  Though the approach is clearly "supervised" in the sense that the machine learns by approximating a known result, we also tried other approaches in an attempt to generate additional information which the machine could use.
  The first approach is a clustering algorithm, intuitively used to look for a notion of "proximity" between the matrices.
  This however did not play a role in the analysis.
  The other is definitely more interesting and it consists in finding a better representation of the configuration matrix using less components.
  The idea is therefore to "squeeze" or "concentrate" the information in a lower dimension (matrices in our case have 180 components, so we are trying to aim for something less than that).
 - page 110/102
  In general however we first relied on traditional regression algorithms, such as linear models, support vector machines and boosted decision trees.
  I will not enter into the details and differences between the algorithms, but we can indeed discuss them.
 - page 111/102
  Let me however say a few words a dimensionality reduction procedure known as "principal components analysis" (or PCA for short).
  Suppose that we have a rectangular matrix (which could be number of samples in the dataset times the number of components of the matrix once it has been flattened).
  The idea is to project the data onto a lower dimensional surface where the variance if maximised in order to retain as much information as possible.
  This is usually used to isolate a signal from a noisy background.
  Thus by isolating only the meaningful components of the matrix we can hope to help the algorithm.
 - page 112/102
  Visually PCA is used to isolate the eigenvalues of the covariance matrix (or the singular values of the matrix) which do not belong to the background.
  From random matrix theory we know that the eigenvalues of a independently and identically distributed matrix (a Wishart matrix) follow a Marchenko-Pastur distribution.
  Such matrix containing a signal would therefore be recognised by the presence of eigenvalues outside this probability distribution.
  We could therefore simply keep the corresponding eigenvectors.
  In our case this resulted in an improvement of the accuracy, obtained by retaining less than half of the components of the matrix (corresponding to 99% of the variance of the initial set).
 - page 113/102
  As we can see we used several algorithms to evaluate the procedure.
  Previous approaches in the literature mainly relied on the direct application of algorithms to the configuration matrix.
  It seems that the family of support vector algorithms work best with a large number of data used for training.
 - page 114/102
  Techniques such as feature engineering and PCA provide a huge improvement (even with less training data).
  Let me for instance point out the fact the even a simple linear regression represents a large improvement over previous attempts.
 - page 115/102
  However this does not conclude the landscape of algorithms used in machine learning.
  In fact we also used neural networks architectures.
  They are an entire class of function approximators which use (some variants of) gradient descent to optimise the weights.
  Their layered structure is key to learn highly non linear and complicated functions.
  We focused on two distinct architectures.
  The older fully connected network were mostly employed in previous attempts at predicting the Hodge numbers.
  They rely on a series of matrix operations to create new outputs from previous layers.
  In this sense the matrix W and the bias term b are the weights which need to be updated.
  Each node is connected to all the outputs, hence the name fully connected or densely connected.
  To learn non linear functions this is however not sufficient: an iterated application of these linear maps would simply result in a linear function to be learned.
  We "break" linearity by introducing an "activation function" at each layer.
  The second architecture is called convolutional from its iterated application of "sliding window function" (that is convolutions) applied on the layers.
 - page 116/102
  Convolutional networks have several advantages over a fully connected approach.
  Since the input in this case does not need to flattened, convolutions retain the notion of vicinity between cells in a grid (here we have an example of a configuration matrix as seen by a convolutional neural network).
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  Since they do not have one weight for each connection, they have a smaller number of parameters to be updated (in our specific case we cut more more than one order of magnitude the number of parameters used).
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  Moreover weights are shared by adjacent cells, meaning that if there is a structure to be inferred, this is the way to go.
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  In this sense a convolutional architecture can isolate defining features of the output and pass them to the following layer as in the animation.
  Using a computer science analogy, this is used to classify objects given a picture: a convolutional neural network is literally capable of isolating what makes a dog a dog and what distinguishes it from a cat (even more specific it can separate a Labrador from a Golden Retriever).
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  This has in fact been used in computer vision tasks in recent year for pattern recognitions, object detections and even spatial awareness tasks (for instance isolate the foreground from the background).
  In this sense this is the closest approximation of artificial intelligence in supervised tasks.
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  My contribution in this sense is inspired by deep learning research at Google.
  In recent years they were able to devise new architectures using so called "inception modules" in which different convolution operations are used concurrently.
  The architecture holds better generalisation properties since more features can be detected and processed at the same time.
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  In our case we decided to go for two concurrent convolutions one scanning each equation (the vertical kernel) of the configuration matrix, while a second convolutions scans each projective space (in horizontal).
  The layer structure is then concatenated until a single output is produced (the Hodge number that is).
  The idea is to be able to learn a relation between between projective spaces and equations and recombine them to find a new representation.
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  As we can see even the simple introduction of a traditional convolutional kernel (it was a 5x5 kernel in this case) is sufficient to boost the accuracy of the predictions (results by Bull et al. in 2018 reached only 77% of accuracy on h^{1,1}).
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  The introduction of the Inception architecture has however has major advantages: it uses even less parameters than "traditional" convolutional networks, it boosts the performance reaching near perfect accuracy, it needs a lot less data (even with just 30% of the data for training, the accuracy is already near perfect).
  Moreover with this architecture we were able to predict also h^{2,1} with 50% accuracy: even if does not look a reliable method to predict it (I agree, for now), mind that previous attempts have usually avoided computing it, or they reached accuracies as high as 8-9%.
  The network is also solid enough to predict both Hodge numbers at the same time: trading a bit of the accuracy for better generalisation, it is in fact possible to let the machine learn the existing relation between the Hodge numbers without specifically inputing anything (for instance by inserting the fact that the difference of the Hodge numbers is the Euler characteristic).
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  Deep learning can therefore be used conscientiously (and I cannot stress this enough) as a predictive method.
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  Provided that one is able to analyse the data (no black boxes should ever be admitted).
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  It can also be used a source of inspiration for inquiries and investigations.
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  Always provided a good analysis is done beforehand (deep learning is a black box in the sense that once it starts it is difficult to keep track of what is happening under the bonnet, but not because we supposedly do not know what is going on in general).
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  Deep learning can also be used for generalisation of patterns and relations.
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  As always only after careful consideration.
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  Moreover convolutional networks look promising and with a lot of unexplored potential.
  This is in fact the first time in which they have been successfully used in theoretical physics.
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  Finally, this is an interdisciplinary approach in which a lot is yet to be learned from different perspective.
  Just think of the entire domain of geometric deep learning in computer science where the underlying structures of the process are investigated: surely mathematics and theoretical physics could provide a framework for it.
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  More directions to investigate now remain.
  In fact one could in principle exploit freedom in representing the configuration matrices to learn the best possible representation, and use all symmetries to try new strategies to improve the results.
  Otherwise one could start to think about this in a mathematical embedding and study what happens in higher dimensions (where the number of manifolds is larger: almost one million complete intersections).
  Moreover, as I was saying, this could be used as an attempt to study formal aspects of deep learning, or even more to directly dive into the "real artificial intelligence" and start to study the problem in a reinforcement learning environment where the machine automatically learns a task without knowing the final result.
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  I will therefore leave the open question as to whether this is actually going to be the end or just the start of something else.
  In the meantime I thank you for your attention.