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Research Statement 1. Character Varieties and Topological Quantum

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Research Statement 1. Character Varieties and Topological Quantum 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 algebraic group 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 moduli space 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 holonomies. In this setting, we can consider the the parabolic representation variety, XG(W; Q), parametrizing representations ρ : π1(W − fS1;:::;Ssg) ! 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 2 Σ with holonomy 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 ) 2 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) 2 R. This `quantization' allows us to compute χ in a recursive way. For instance, suppose y 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 Dy : 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 y 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(Dy) 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.
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