Research Statement Angel´ Gonz´alezPrieto

My research lies in the interface between complex geometry, algebraic geometry and theoretical physics. I am especially focused on Topological Quantum Field Theories, Geometric Invariant Theory, representation theory and Hodge theory. Moreover, I am interested in algebraic topology, especially in higher category theory and functor calculus. As a byproduct, I am interested in moduli spaces, mainly moduli spaces of parabolic Higgs bundles, and their relation with character varieties, gauge theory and theoretical physics. In addition, I also research in theoretical aspects of machine learning and big data. Particularly, I am interested in collaborative filtering based recommender systems as well as applications of highly parallelizable deep learning methods and generative adversarial networks (GANs) to real-time problems. Finally, I study the transference of geometric techniques to dimensionality reduction problems and manifold learning. 1. Character varieties and Topological Quantum Field Theories Let W be a compact differentiable manifold and let G be an over an algebraically closed field k. The representation variety, XG(W ) is an algebraic variety parametrizing the collection of representations ρ : π1(W ) → G. Moreover, if we want to focus on the of isomorphism classes of representations, we should take the Geometric Invariant Theory (GIT) quotient of XG(W ) under the action of G by conjugation, giving rise to the so-called character variety RG(W ) = XG(W ) G. Furthermore, we can consider a parabolic structure Q on W , determined by a finite collection of co-oriented co-dimension 2 submanifolds, S1,...,Sr ⊆ W with attached conjugacy classes λ1, . . . , λr ⊆ G, called . In this setting, we can consider the the parabolic representation variety, XG(W, Q), parametrizing representations ρ : π1(W − {S1,...,Ss}) → G such that the image of the positive oriented loop around Si lies in λi, as well as the associated parabolic character variety RG(W, Q) = XG,Q(W ) G.

One of the main reasons for studying character varieties is their prominent role in the non-abelian Hodge theory. This beautiful correspondence states that character varieties are just one of the three faces of the same object. Through the Riemann-Hilbert correspondence [36, 37] we get that, for W = Σ a compact Riemann surface and Q a parabolic structure with a single puncture p ∈ Σ with λ = e2πid/n Id (the so-called twisted case), there is a correspondence between R (Σ,Q) and the moduli space of rank n logarithmic flat GLn(C) d bundles with a pole at p with residue − n Id. Furthermore, via the Hitchin-Kobayashi correspondence [35, 9], R (Σ,Q) is shown to be in correspondence with the moduli space of rank n and degree d Higgs bundles. GLn(C) However, even in the simplest cases, the topology and algebraic structure of these character varieties is extremely rich. In particular, their virtual class in the Grothendieck ring of algebraic varieties, KVark, is in general unknown. Many works have focused, in the complex case k = C, on the computation of a coarser invariant called E-polynomial (or Deligne-Hodge polynomial) that collects the compactly supported Hodge number of the character variety. A first approach, based on the ideas of the Weil conjectures, was initiated by Hausel and Rodr´ıguez-Villegasand has led to the computation of the E-polynomial in several cases, like for G = GLn(C) (twisted case) [24], G = SLn(C) [33] or G = GLn(C) and generic semi-simple parabolic structure [23]. A different approach of geometric flavor was initiated by Logares, Mu˜nozand Newstead, based on the stratification of the representation variety. Following this idea, explicit expressions of the E-polynomials for genus g = 1, 2 and G = SL2(C) were computed in [30] and, later, extended for arbitrary genus in [32]. The parabolic case was addressed in [29] for at most two puctures on a torus. The results of [32] by means of the geometric method are extremely suggestive beyond the calculation itself.

The idea is that this paper proves that, if Σg is the compact orientable genus g surface, there exists a recursive algorithm that computes the E-polynomial of X (Σ ) in terms of monodromy data of X (Σ ). This SL2(C) g SL2(C) g−1 recursive pattern arises naturally in Topological Quantum Field Theories (aka. TQFTs), a powerful categorical tool incepted by Witten in his work about the Jones polynomial and Chern-Simons theory [40]. Aware of the importance of this discovery, Atiyah in [2] gave a precise definition of a TQFT as a monoidal functor

Z : Bdn → R-Mod out of the category of n-dimensional bordisms to the category of R-modules. For our purposes, the most important application of TQFTs is that they provide effective methods of com- putation of algebraic invariants. Suppose that we are interested in some algebraic invariant that, for any closed n-dimensional manifold W , assigns a value χ(W ) ∈ R for a fixed ring R. Suppose we can ‘quantize’ χ, that is, we construct a TQFT, Z : Bdn → R-Mod, such that, seeing W as a bordism W : ∅ → ∅, we have that χ(W ) = Z(W )(1) ∈ R. This ‘quantization’ allows us to compute χ in a recursive way. For instance, suppose † g 1 1 1 that W = Σg and decompose Σg = D ◦ L ◦ D, where D : ∅ → S is a disc, L : S → S is a torus with two dics removed and D† : S1 → ∅ is a disc in the other way around, as depicted in the following figure. 1 2

Figure 1. Decomposition of the genus g surface into simple bordisms

† g In that case, the TQFT gives us a decomposition χ(Σg) = Z(Σg)(1) = Z(D ) ◦ Z(L) ◦ Z(D)(1) where Z(D),Z(D†) and Z(L) are R-module homomorphisms. Hence, The desired invariant can be computed auto- matically for all the surfaces Σg from the knowledge of only three linear homomorphisms. With this idea in mind, in [12], together with M. Logares and V. Mu˜noz,we constructed a lax monoidal TQFT computing the E-polynomial of the G-representation varieties, for any complex group G. For that purpose, we used as key ingredients a ‘field theory’ capturing the properties of representation varieties and Saito’s mixed Hodge modules [34] playing the role of a ‘quantisation’. This work was widely generalized by myself in [13]. There, I extended the TQFT to compute virtual classes of representation varieties through a lax monoidal TQFT, ZG : Bdpn → KVark-Mod, for any group G and ground field k, where Bdpn is the category of pairs of n-bordisms. Using this construction, I showed that the geometric method developed in [30] and [32] can be actually understood as implicit calculations of the TQFT

ZG. Moreover, using the TQFT, I provided new calculations of the virtual classes of SL2(k)-representations varieties with no parabolic structure (for which only the E-polynomial was known) and with punctures of Jordan type (that were almost completely unknown). In addition, in [14] I provided a method for computing the virtual classes of character varieties from the one of the representation variety, filling the gap between representation and character varieties. For that purpose, I developed a novel theory based on focusing on the topological properties of Geometric Invariant Theory and how they are preserved under stratifications. In [11], I addressed the parabolic case in full generality, providing a closed formula for the virtual classes of SL2(k)-character varieties with arbitrary semi-simple parabolic structures. This result has attracted the interest of the community since it is known that the techniques of the arithmetic approach cannot address this case, even for E-polynomials. This shows that the TQFT approach is able to capture subtler information of the representation variety, as the change in the geometry when crossing non-generic parabolic structures, in the spirit of a walls-and-chambers problem. A promising new research line in this direction was opened in the recent joint work with M. Logares [17]. In this paper, we gave the first steps towards a full generalization of the TQFT to more general topological spaces with good gluing properties. In particular, we showed that the TQFT can be extended to work with singular manifolds with conic singularities. In this settings, Minor’s results about the local structure of singularities can be understood as induced 2-morphisms that control how degenerations of the manifold affect the geometry of the representation variety. This evidences that, to deal with general topological spaces, we must consider extra 2-category structures preserved by the TQFT. As byproduct of this work, we extended the previous results to obtain virtual classes of character varieties of nodal complex curves, with arbitrary parabolic structure (including the non-generic semi-simple case). Research line 1.1: Higher rank character varietes As we mentioned above, the constructed TQFT is valid for any algebraic group G. For this reason, it can be considered different groups that SL2(C). As a first step, it would be interesting to compute the virtual classes of SL3(C)-parabolic character varieties by means of TQFTs.

A key space in the calculation of SL2(C)-representation varieties was the orbit space SL2(C)/SL2(C) under the action by inner automorphisms. In this case, the quotient map is given by the trace, so SL2(C)/SL2(C) is the affine line with two doubled points, corresponding to the Jordan forms. It is precisely around these two points where the monodromy information captured by the TQFT concentrates. I expect to find a similar behavior in the rank 3 case with the particularity that, now, SL3(C)/SL3(C) is the affine plane with some doubled curves corresponding to the collapsing Jordan type orbits. Thus, we expect that the computations of the TQFT will be similar to the rank 2 case but with monodromy information supported on curves instead of points. Research line 1.2: TQFTs across the non-abelian Hodge correspondence So far, I have focused my study on the character varieties part of the non-abelian Hodge theory. However, it would be very interesting to extend this construction to the other sides of the non-abelian Hodge correspondence and to construct TQFTs computing virtual classes of moduli spaces of Higgs bundles and flat connections. Indeed, thanks to the hyperk¨ahlerstructure induced by this triad, we expect to find a sort of emerging quaternionic structure on the TQFTs. 3

In principle, this task can be addressed with the developed techniques just by changing the field theory associated to the TQFT. However, the spaces parametrizing Higgs bundles or flat connections before quotienting by the action of the gauge group are no longer algebraic varieties on the nose, in sharp contrast with character varieties, where the representation variety is a genuine algebraic variety. In order to tackle this problem, as a first step we propose to extend the TQFT to compute virtual classes of the moduli stack of representations. Roughly speaking, the notion of stacks as sheaves fits perfectly in the spirit of TQFTs: it is precisely this kind of sheaf-theoretic property on the studied spaces what allows us to glue the information along boundaries, as dictated by a TQFT. Indeed, quotient stacks are implicitly presented in the known construction, for instance when studying the orbit space of the underlying group. In this way, we expect that from this stacky version of the TQFT, it will be much more natural to extend it across the non-abelian Hodge correspondence to moduli stacks of Higgs bundles and flat connections. Research line 1.3: Mirror symmetry Another framework in which character varieties are central is the geometric Langlands program. In [25], Hausel and Thaddeus proved that the moduli space of SL2(C)-Higgs bundles is the first non-trivial example of the Strominger-Yau-Zaslow conditions [39] for mirror symmetry for Calabi-Yau manifolds. As shown in [22] and [24], this conjecturally would imply that a collection of conjectures concerning E-polynomials of G-character varieties and their Langlands dual LG-character varieties must hold. The validity of these conjectures has been L discussed in some cases as in [30] and [31] for G = SL2(C)( G = PGL2(C)). Despite of that, the general case remains unsolved. I expect that the ideas introduced in my work by means of the constructed TQFT for representation varieties will be useful to shed light on these issues. In particular, it would be interesting to study whether the TQFTs of G and LG are somehow related and what kind of implications this has for mirror symmetry.

2. Spaces of knots Other of my research interests is the study of the higher homotopy groups of the spaces of knots in Rn for n ≥ 4. It is very well-known that, for n = 3, the space of embeddings of S1 → R3 is not connected (in fact, this is the germ of the classical knot theory). However, for ambient dimension n ≥ 4, this space of embedding Emb(S1, Rn) can be shown to be connected through basic transversality arguments, so it makes sense to ask 1 n for the higher homotopy groups πk(Emb(S , R )) for k ≥ 1. Surprisingly, this computation is an open problem that has been stuck for almost 20 years and is strongly 1 4 1 4 related to Gromov’s h-principle [21]. For instance, for n = 4, only π1(Emb(S , R )) = 0 and π2(Emb(S , R )) = Z ⊕ Z are known. These calculations were accomplished by Sinha in [38] by means of Goodwillie’s functor calculus. For that purpose, Sinha gave simpler models for the Taylor approximations of the embeddings functor in terms of spaces of configurations. However, it is well-known that, in order to compute higher homotopy groups, the models of the Taylor approximation become too involved and no homotopic information can be extracted. Research line 2.1: Computation of higher homotopy groups of smooth knots Using the expertise obtained after dealing with TQFTs, together with F. Preasas’ group, we decided to import cobordism-theoretic tools to the problem of computing homotopy groups of spaces of knots. In this way, we propose a completely new approach for computing these higher homotopy groups based on assigning a new invariant, a kind of generalized linking number, to the moduli space of strict immersions with values in the ring of framed bordisms. This gives you a between the homotopy groups of immersions relative to embeddings and the ring of framed bordisms (which is isomorphic to stable homotopy groups of spheres). In this way, the computation reduces to understanding the kernel of a group homomorphism, a problem that can be addressed by studying the flexibility-rigidity duality in lower homotopy groups of knots in the complement of 1-skeleta through classical obstruction theory. In this way, using this approach we are able to easily recover all the known results about the homotopy of spaces of knots. Moreover, in principle, the techniques can be pushed beyond the frontier of knowledge to compute unknown homotopy groups of spaces of knots. Nevertheless, in this high rank, new singularities appear in the crossing points of immersions. It is precisely the arising of this singularities what prevents the functor calculus approach to extend its results to high degree. However, using our obstruction-theoretic approach, these singularity models can be studied through Arnold’s singularity classification [1], giving rise to new obstructions depending on the local model of the crossing. This work is gathered in the paper [7], very close to completion. 4

Research line 2.2: Application of the results to Engel knots Complementing the previous objective, we plan to translate the results for smooth embedding to the case of Engel knots. Recall that Engel structures are an extension of contact structures (which are forced to lie in odd dimensional manifolds) to 4-manifolds. Precisely, an Engel structure on a 4-manifold M is a completely non-integrable 2-dimensional distribution D in TM. In particular, there is a standard Engel structure, D, R4, with coordinates (x, y, z, w), given as the intersection of the kernel of the 1-forms dy − zdx and dz − wdx. In this way, instead of arbitrary embeddings into R4, it can be considered horizontal embeddings with respect 1 4 0 1 to the standard Engel structure i.e. embeddings γ : S → R such that γ (t) ∈ Dγ(t) for all t ∈ S . In principle, it seems like this space is much more restrictive than the usual space of smooth knots. However, recently Casals and Pino proved that there exists complete h-principle for Engels knots [8]. This has the striking byproduct that the higher homotopy groups of the spaces of smooth knots can be computed from the corresponding ones of horizontal knots up to known topological data. On the other hand, the aforementioned approach of using obstruction theory for understanding smooth knots can be easily translated to the horizontal framework. In this way, this kind of techniques can be applied, verbatim, to understand the homotopy groups of spaces of horizontal knots. However, the horizontal setting has an important advantage: singularities appearing at the crossings are no longer singularities in R4 and can be studied as singularities of real-valued functions. This fact, together with Igusa’s theorem [26] shows that higher singularies are removable, simplifying the study of the strict immersions appearing in the moduli space of immersions. We expect that his ideas of using horizontal knots will allow us to push even further the rank of known homotopy groups of spaces of knots in R4. Research line 2.3: Character varieties of knots This excursion into knot theory from TQFTs is actually a round trip. Nowadays, I am working to translate these methods to understand the virtual classes of G-character varieties over complements of knots. In this line, recently with V. Mu˜noz,we have completed the paper [19] in which we study the virtual class of the character variety over complements of torus knots for G = SL4(C). For this purpose, we propose a novel stratification of the character variety based on its canonical filtration with semi-simple quotients, which reduces the problem of computing the virtual class to a combinatorial problem. In low rank, this combinatorial problem can be addressed by hand, but in high rank the combinatorial explosion forces that the computation must be computer-aided. The developed method is, in principle, general and provides a tool for dealing with arbitrary rank. However, in practice, in order to extend the method to rank 5 it is necessary a subtle analysis of the configuration space of two 2-dimensional irreducible representations, that can be bypassed in the rank 4 case. In this direction, we expect to extend the methods in the future to the high rank setting, reducing the computation of virtual classes of torus knots to a fully combinatorial problem that can be addressed with a computational approach.

3. Theoretical machine learning In the last years, I have incorporated to my research interests theoretical aspects of machine learning and applications of statistics to computer sciences. From this point of view, I am interested in non-linear models for recommender systems, deep learning with special attention to Generative Adversarial Networks (GANS), and geometric methods in dimensional reduction. Collaborative work with A. Br´u,J. Bobadilla, S. Gomez-Canaval, R. Lara-Cabrera, A. Mozo, F. Ortega and E. Talavera (Universidad Polit´ecnica de Madrid). Research line 3.1: Improving fairness in recommender systems Researching in recommender systems is, currently, one of the most important topics in artificial intelligence, mostly due to their application to service providing companies like Netflix, Amazon or Facebook. In this direction, with J. Bobadilla, R. Lara-Cabrera and F. Ortega, we have proposed new collaborative filtering techniques improving the known baselines. For instance, in [27] and [4], we designed new techniques of matrix factorization based on deep learning and Dirichlet distributions. Moreover, the proposed model in [4] extends the pre-existing methods in the sense that it not only returns a prediction of most liked items, but also a reliability measure about how accurate the prediction is. This reliability coefficient is particularly relevant in the context of beyond-accuracy quality measures i.e. measures that provide an extra value to the prediction (for instance, a novelty factor that may surprise a user). In this spirit, one of the most challenging problems is to improve the fairness in recommender systems. Due 5 to its statistical nature, recommender systems tend to be biased to demographic groups with most of the users (for instance, young people in social networks) and produce bad prediction to minority groups. In order to address this problem, in [6] and [5], we proposed two models based on deep learning that allow us to unravel the hidden features of the minority groups and to compensate the demographic bias, providing fairer and more accurate recommendations. As future work, we plan to incorporate this fairness technology to the matrix factorization algo- rithm at the heart of the recommender system, using the reliability measure developed in [4] to evaluate and refine the predictions for minority groups, according to the hidden features unhided by [6]. In this line, jointly with A. Br´u,J.C. Nu˜noand J.L. Gonz´alez, we work on the application of machine learning techniques to predict and prevent recidivism in gender violence [15]. Research line 3.2: Stability and convergence of Generative Adversarial Networks Related with my research in deep learning, I am interested in the theoretical aspects of Generative Adversarial Networks (GANs). Since their inception in [20], GANs have revolutionized the research in neural networks thanks to their versatility. Roughtly speaking, a GAN is a two-players game in which one player, the generator (typically a convolutional neural network) tries to generate, from white noise, very realistic data (say, paintings or human faces). On the other hand, the other player, the discriminator (typically a deep feed-forwarded neural network) tries to distinguish the fake data rendered by the generator from real data as best as possible. In this way, both players are in competition where the generator tries to cheat as much as possible the discriminator and the discriminator tries to be very accurate modelling the generation patterns of the discriminator. Eventually, when the system learning is properly trained, the result is a generator that renders incredibly realistic data. Despite their power, the GAN game is a min-max non-convex game for which no general optimization methods for converging to a Nash equilibrium are known. Essentially, the used approach is based in a gradient descent training that gives rise to a dynamical system that may not converge. Only very partial results about the dynamics of GANs are known, being the most important works [3, 28]. In this way, with S. G´omez-Ca˜naval, A. Mozo and E. Talavera, I am currently investigating convergence and optimality issues of GANs, both from a dynamical and a statistical point of view. To address this problem, in the recent paper [18] we propose to analyze the GAN game through the Fourier modes of its associated cost function. In the case of a GAN over a torus, this approach gives rise to interesting dynamical systems that can be studied through techniques of Morse theory and bifurcation theory. Using these results, we propose a novel method to quantify and analyze the convergence issues of GANs. Additionally, in [10], we analyze the convergence in the problem of generating probability distributions using GANs, showing that certain phenomena like discrete data or non-smooth density function are behind the bad convergence. In this direction, I would like to incorporate algebro-geometric tools to this applied problem. For instance, it would be interesting to understand more deeply the topological behavior of the GAN flow on surfaces, say whether it can be given a mechanical interpretations through a symplectic structure, and their bifurcation according to the parametric variation of the underlying distribution of the real data. Research line 3.3: Manifold learning Finally, I am very interested in the incorporation of techniques coming from differential geometry to dimen- sional reduction problems. Despite of the important advances in dimensional reduction, the known algorithms tend to exploit the information in a very linear way (like PCA and variants) or using only metric information (like Isomap or Spectral Embedding). For this reason, typically, the known methods are not able to properly project data distributed according to a complex topology, since they do not take into account any information about the curvature of the underlying manifold. In this line, it would be interesting to study the behavior of the known methods under varying curvature though a benchmark. In a joint work with R. Lara-Cabrera and F. Ortega [16], we propose to study the geometric properties of the diffeomorphism of Rn obtained after composing a known immersion Rn → RN (N > n), simulating a real data distribution, with the projection RN → Rn computed by a dimensionality reduction algorithm. For instance, a very desirable property of this diffeomorphism would be to be locally conformally flat (i.e. locally the pullbacked metric is conformal to the standard flat metric). This property can be measured through the Cotton tensor (in dimension 3) and through the Weyl tensor (in dimension ≥ 4), so we can use these tensors as a quality measure for the known reduction algorithms. A step further in this project, with the knowledge extracted from this analysis, would be to design new dimensional reduction methods based on modifying the curvature through a geometric flow, like the Ricci flow with surgery to get constant sectional curvature or the Yamabe flow to converge to constant scalar curvature. 6

References

[1] V. I. Arnol’d, V. I. Arnol’d, V. Arnold, et al. Singularity theory, volume 53. Cambridge University Press, 1981. [2] M. Atiyah. Topological quantum field theories. Inst. Hautes Etudes´ Sci. Publ. Math., (68):175–186 (1989), 1988. [3] S. A. Barnett. Convergence problems with generative adversarial networks (gans). arXiv arXiv:1806.11382, 2018. [4] J. Bobadilla, A. Gonz´alez-Prieto,R. Lara-Cabrera, and F. Ortega. Providing reliability in recommender systems through Bernoulli matrix factorization. Information Sciences, 553:110–128, 2021. [5] J. Bobadilla, A.´ Gonz´alez-Prieto,F. Ortega, and R. Lara-Cabrera. Deep learning feature selection to unhide demographic recommender systems factors. Neural Computing and Applications, pages 1–18, 2020. [6] J. Bobadilla, R. Lara-Cabrera, A. Gonz´alez-Prieto,and F. Ortega. Deepfair: Deep learning for improving fairness in recom- mender systems. International Journal of Interactive Multimedia and Artificial Intelligence, 2020. [7] F. Cantero, E. Fern´andez,A. Gonz´alez-Prieto,and F. Presas. Obstruction calculus and homotopy of spaces of knots. In preparation. [8] R. Casals and A.´ del Pino. Classification of engel knots. Mathematische Annalen, 371(1-2):391–404, 2018. [9] K. Corlette. Flat G-bundles with canonical metrics. J. Differential Geom., 28(3):361–382, 1988. [10] S. G´omez-Ca˜naval, A. Gonz´alez-Prieto,A. Mozo, and E. Talavera. Convergence of GAN in statistical problems. In preparation. ´ [11] A. Gonz´alez-Prieto.Virtual classes of parabolic SL2(C)-character varieties. Adv. Math., 368:107148, Jul 2020. [12] A.´ Gonz´alez-Prieto,M. Logares, and V. Mu˜noz.A lax monoidal topological quantum field theory for representation varieties. Bulletin des Sciences Math´ematiques, 161:102871, Jul 2020. [13] A.´ Gonz´alez-Prieto. Motivic theory of representation varieties via Topological Quantum Field Theories. Preprint arXiv:1810.09714v2, 2018. [14] A.´ Gonz´alez-Prieto.Topological geometric invariant theory and character varieties. arXiv:1807.08540, 2018. [15] A. Gonz´alez-Prieto,A. Br´u,J. C. Nu˜no,and J. L. Gonz´alez.Machine learning for prevention of recidivism in gender violence. In preparation. [16] A. Gonz´alez-Prieto,R. Lara-Cabrera, and F. Ortega. Quality measures for dimensional reduction through riemannian curva- ture. In preparation. [17] A. Gonz´alez-Prietoand M. Logares. On character varieties of singular manifolds. arXiv:2011.04603, 2020. [18] A. Gonz´alez-Prieto,A. Mozo, E. Talavera, and S. G´omez-Ca˜naval. Dynamics of Fourier modes in torus Generative Adversarial Networks. Mathematics, 2021. [19] A.´ Gonzalez-Prieto and V. Mu˜noz. Motive of the SL4-character variety of torus knots. arXiv:2006.01810, 2020. [20] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672–2680, 2014. [21] M. Gromov. Partial differential relations, volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1986. [22] T. Hausel. Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve. In Geometric methods in algebra and number theory, volume 235 of Progr. Math., pages 193–217. Birkh¨auserBoston, Boston, MA, 2005. [23] T. Hausel, E. Letellier, and F. Rodriguez-Villegas. Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J., 160(2):323–400, 2011. [24] T. Hausel and F. Rodriguez-Villegas. Mixed Hodge polynomials of character varieties. Invent. Math., 174(3):555–624, 2008. With an appendix by Nicholas M. Katz. [25] T. Hausel and M. Thaddeus. Mirror symmetry, Langlands duality, and the . Invent. Math., 153(1):197–229, 2003. [26] K. Igusa. Higher singularities of smooth functions are unnecessary. Ann. of Math. (2), 119(1):1–58, 1984. [27] R. Lara-Cabrera, A.´ Gonz´alez-Prieto,and F. Ortega. Deep matrix factorization approach for collaborative filtering recom- mender systems. Applied Sciences, 10(14):4926, 2020. [28] S. Liu, O. Bousquet, and K. Chaudhuri. Approximation and convergence properties of generative adversarial learning. In Advances in Neural Information Processing Systems, pages 5545–5553, 2017. [29] M. Logares and V. Mu˜noz.Hodge polynomials of the SL(2, C)-character variety of an elliptic curve with two marked points. Internat. J. Math., 25(14):1450125, 22, 2014. [30] M. Logares, V. Mu˜noz,and P. E. Newstead. Hodge polynomials of SL(2, C)-character varieties for curves of small genus. Rev. Mat. Complut., 26(2):635–703, 2013. [31] J. Mart´ınez.E-polynomials of P GL(2, C)-character varieties of surface groups. Preprint. arXiv:1705.04649, 2017. [32] J. Mart´ınez and V. Mu˜noz.E-polynomials of the SL(2, C)-character varieties of surface groups. Int. Math. Res. Not. IMRN, (3):926–961, 2016. [33] M. Mereb. On the E-polynomials of a family of SLn-character varieties. Math. Ann., 363(3-4):857–892, 2015. [34] M. Saito. Mixed Hodge modules. Publ. Res. Inst. Math. Sci., 26(2):221–333, 1990. [35] C. T. Simpson. Higgs bundles and local systems. Inst. Hautes Etudes´ Sci. Publ. Math., (75):5–95, 1992. [36] C. T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Etudes´ Sci. Publ. Math., (79):47–129, 1994. [37] C. T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Etudes´ Sci. Publ. Math., (80):5–79 (1995), 1994. [38] D. P. Sinha. The topology of spaces of knots: cosimplicial models. American journal of mathematics, 131(4):945–980, 2009. [39] A. Strominger, S.-T. Yau, and E. Zaslow. Mirror symmetry is T -duality. Nuclear Phys. B, 479(1-2):243–259, 1996. [40] E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989.