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For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the $n$-point functions produced by the topological recursion on these curves via the $n$-point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions., 16 pages

We prove the 2006 Zvonkine conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called $r$-ELSV formula, as well as its orbifold generalization, the $qr$-ELSV formula, proposed recently in [KLPS17]., 22 pages. Version 4: improved exposition of the proof

We study the $n$-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their $\hbar^2$-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov-Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families we prove in addition the so-called projection property and thus the full statement of the Chekhov-Eynard-Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of $\hbar^2$-deformations of the Orlov-Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck-Riemann-Roch formula this, in turn, gives new and uniform proofs of almost all ELSV-type formulas discussed in the literature., 48 pages; several minor corrections and clarifications

We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopical (necessary to structure chain-level Gromov--Witten invariants) and quantum (with examples found in the works of Buryak--Rossi on integrable systems). We introduce a new version of Kontsevich's graph complex, enriched with tautological classes on the moduli spaces of stable curves. We use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs, by methods due to Merkulov--Willwacher. This group is shown to contain both the prounipotent Grothendieck--Teichm\"uller group and the Givental group., Comment: submitted version, 57 pages, abstract and introduction rewritten

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are the necessary conditions for the Chekhov-Eynard-Orantin topological recursion., 31 pages. Version 2: simplified the computation in section 4.2, also clarifying the deduction of the diagonal part of the A-operators

To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in [1, 2], to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action., Comment: 36 pages

We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus g curves with n marked points.

We study the genus expansion of Barannikov-Kontsevich solutions of the WDVV equation. In terms of the related graph calculus, we give a definition of descendants at one point and prove that this definition satisfies the topological recursion relations in genera 0, 1, and 2, string and dilaton equations, and the pull-back formula.

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are necessary conditions for the Chekhov-Eynard-Orantin topological recursion., DOCUMENTA MATHEMATICA, Vol 24 (2019), p. 857-898

In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection numbers of psi classes.Second, we examine the Dubrovin-Zhang hierarchy, an integrable system constructed from a Frobenius manifold by deforming its associated pencil of Poisson structures of hydrodynamic type. This integrable hierarchy was proved to be Hamiltonian and tau-symmetric, and conjectured to be bi-Hamiltonian. We prove a vanishing theorem for the negative degree terms of the second Poisson bracket, thus providing strong evidence to support this conjecture. The proof of this theorem demonstrates the implications the bi-Hamiltonian recursion relation and tautological relations in the cohomology rings of the moduli spaces of stable curves have on the bi-Hamiltonian structure of the Dubrovin-Zhang hierarchies.Third, we propose a conjectural formula for the simplest non-trivial product of doubleramification cycles DR_g(1,1)lambda_g in terms of cohomology classes represented by standard strata. Although there are known formulas relating double ramification cycles to other, more natural tautological classes, they are much more complicated than the one conjectured here. This conjecture refines the one point case of the Buryak-Guéré-Rossi conjectural tautological relations, which are equivalent to the existence of a Miura transformation relating Buryak's double ramification hierarchies and the Dubrovin-Zhang ones.Finally, we analyze the differential geometry of (2 + 1) integrable systems through infinitedimensional Frobenius manifolds. More concretely, we study, both formally and analytically, the Dubrovin equation of the 2D Toda Frobenius manifold at its irregular singularity. The fact that it is infinite-dimensional implies a qualitatively different behavior than its finite-dimensional analogue, the Frobenius manifold underlying the extended Toda hierarchy. The two most remarkable differences are non-uniqueness of formal solutions to the Dubrovin equation and non-completeness of the analytic ones. These features together greatly complicate the analysis of Stokes phenomenon, which we perform by splitting the space of solutions into infinitely many two-dimensional subspaces., Dans cette thèse, nous étudions la géométrie sous-jacente des systèmes intégrables. Nous nous intéressons particulièrement aux hiérarchies d'EDPs d'évolution, tau-symétriques et bi-Hamiltoniennes.D'abord, nous explorons la relation étroite entre les champs des systèmes intégrables et la géométrie algébrique en donnant une nouvelle démonstration de la conjecture de Witten, qui construit la string tau-fonction de la hiérarchie de Korteweg-de Vries par théorie d'intersection des espaces de modules des courbes stables avec des points marqués. Cette nouvelle démonstration se base sur la géométrie des cycles de ramification double, des classes tautologiques dont le comportement sous des pullbacks des applications forgetful et gluing facilitent le calcul des nombres d'intersection des psi classes.Dans un deuxième temps, nous examinons la hiérarchie de Dubrovin et Zhang, un système intégrable construit en déformant la structure bi-Hamiltonienne de type hydrodynamique associée à une variété de Frobenius. Cette hiérarchie intégrable est Hamiltonienne et tau-symétrique, et est conjecturée bi-Hamiltonienne. Nous démontrons un théorème d'annulation des termes de degrés négatifs du deuxième crochet de Poisson qui fournit des preuves fortes pour soutenir cette conjecture. La démonstration de ce théorème illustre les implications que la récursivité bi-Hamiltonienne et les relations tautologiques en cohomologie des espaces de modules des courbes stables ont sur la structure bi-Hamiltonienne des hiérarchies de Dubrovin et Zhang.Dans un troisième temps, nous conjecturons une formule pour le plus simple des produits non triviaux des cycles de ramification double DR_g(1,1)lambda_g en termes des classes de cohomologie réprésentées par les strates standards. Malgré l'existence de formules qui mettent en relation des cycles de ramification double avec autres classes tautologiques plus naturelles, elles sont beaucoup plus compliquées que celle proposée ici. Cette conjecture précise dans le cas d'un point les relations tautologiques conjecturales de Buryak, Guéré et Rossi, qui sont équivalentes à l'existence d'une transformation de Miura qui relie la hiérarchie de ramification double de Buryak et celle de Dubrovin et Zhang.Finalement, nous analysons la géométrie différentielle des systèmes intégrables en (2 + 1) dimensions par variétés de Frobenius de dimension infinie. Plus concrètement, nous étudions, formèlement et analytiquement, l'équation de Dubrovin de la variété de Frobenius de la hiérarchie de Toda bidimensionnelle à sa singularité irrégulière. Le fait qu'elle est de dimension infinie implique un comportement qualitativement différent de celui de son analogue en dimension finie, la variété de Frobenius sous-jacente à la hiérarchie de Toda élargie. Les deux différences les plus rémarcables sont que les solutions formèles de l'équation de Dubrovin ne sont pas uniques et que les solutions analytiques ne forment pas un système complet. Conjointement ces deux caractéristiques compliquent l'analyse du phénomène de Stokes, que nous réalisons en divisant l'espace des solutions en une infinité des sous-espaces de dimension deux.

The first topic of this dissertation is the moduli space of curves. I define half-spin relations, specialising Pandharipande-Pixton-Zvonkine’s spin relations, to reprove Buryak-Shadrin-Zvonkine’s result on the dimension of the top Chow group for the open moduli space, and give new bounds for the lower groups. I also use these relations to reduce Faber’s intersection number conjecture to a combinatorial identity. The second is Hurwitz numbers. I prove quasi-polynomiality for orbifold simple, weakly and strictly monotone, and spin Hurwitz numbers. In the first case, this was already known, via the Johnson-Pandharipande-Tseng formula, and in the other cases this is new. I also prove that triply mixed double Hurwitz numbers satisfy piecewise polynomiality. For monotone and for orbifold spin Hurwitz numbers, I derive a cut-and-join-equation and prove topological recursion (for r = 2 and for genus zero, general r in the latter case). The former was known by Goulden-Guay-Paquet-Novak. The latter proves Zvonkine’s r-ELSV formula for these cases. The third is integrable hierarchies. I reprove a result by Alexandrov, showing that triple Hodge integrals on the moduli space of curves satisfy the Kadomtsev-Petviashvili hierarchy, generalising Kazarian’s single Hodge integral proof. I give a new and purely cohomological proof of Dubrovin-Liu-Zhang’s theorem, that integrable hierarchies described by dispersive semi-simple Poisson pencils are classified by central invariants, and I streamline Carlet-Posthuma-Shadrin’s proof that this is an equivalence.

This thesis studies proper Lie groupoids on three levels: the groupoids themselves, their induced foliations, and their orbit spaces. Proper Lie groupoids are shown to admit desingularisations via a successive blow-up procedure, whereby orbits are systematically added to achieve regularity. Regarding their foliations, a thorough treatment of the known integration results for singular foliations is included. Moreover, the underlying orbit spaces of proper Lie groupoids are studied. This is done by first providing an intrinsic definition of a so-called orbispace using atlases, both in the language of Morita bibundles, and in the language of Morita equivalences through fractions. An orbispace is said to be proper if it admits a proper defining atlas. It is then shown that proper Lie groupoids, up to a precise notion of Morita equivalence, correspond exactly to such proper orbispaces. This can be interpreted as the statement that proper orbispaces form a subcategory of all differentiable stacks. All of these developments mirror the well-known correspondences between regular proper Lie groupoids, regular foliations, and orbifolds. The above results are further shown to hold in the setting of proper Riemannian groupoids. In particular the desingularisation procedure can be performed in such a way that the regularised groupoid has arbitrarily small Gromov—Hausdorff distance from the original groupoid. Moreover, proper Riemannian orbispaces are defined and shown to correspond precisely to appropriate equivalence classes of proper Riemannian groupoids. This thesis contains various other results, including those on holonomy groupoids of orbit-like foliations, and a de Rham theorem for orbispaces.

In my thesis I consider interplay between several different structures in mathematical physics. These structures are used to solve a large class of problems in enumerative algebraic geometry and combinatorics in a universal way. The problems can range from counting certain one-dimensional drawings on two-dimensional surfaces to counting maps of certain type from a two-dimensional surface to some higher-dimensional space. The structures that we study in this thesis allow to encode the solutions to this type of enumerative and combinatorial problems in some general compact form. In one approach the solutions to the enumerative problems are encoded in a complex algebraic curve with certain functions on it. From this initial small set of data one can reconstruct the full solution with the help of a recursive procedure that is absolutely universal and does not depend on a particular problem. In another approach the solutions to the enumerative problems are encoded as certain integrals over some complicated spaces that parametrize different complex structures on two-dimensional surfaces. This reveals that the solutions to the enumerative problems reflect the geometric properties of the space of complex structures, also in a universal way. These two approaches turn out to be related in many different ways. In this thesis their relation is studied in the framework of an advanced differential geometric structure called Frobenius manifold.

The thesis considers several enumerative geometric problems concerning the topology of the moduli space of curves and their combinatorics. These enumerative geometric problems are analysed from different intertwined points of view and using different mathematical tools, including Hurwitz theory, Givental theory, cohomological field theories, integrable hierarchies, Fock spaces, quantum curves, and a relatively new powerful technique introduced by Chekhov, Eynard and Orantin known as topological recursion. These subjects lie in the interplay between enumerative algebraic geometry, differential geometry and mathematical physics.

Gromov-Witten theory and spectral curve topological recursion are important parts of modern algebraic geometry and mathematical physics. In my thesis I study relations between these theories and some important new aspects and applications of them. In particular, a construction for a local spectral curve which produces the same invariants as a given Gromov-Witten theory is presented in the thesis, as well as constructions for quantum spectral curves for several important theories, and a new proof of the so-called ELSV formula.

In the present thesis we consider polynomial knot invariants and their properties. We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion (also known as the large N expansion) and discuss its properties. Then we also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Next we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero-Moser-Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.

The study of curves is an important area of research in algebraic geometry and mathematical physics. In my thesis I study so-called moduli spaces of curves; these are spaces that parametrize all curves with some specified properties. In particular, I study maps from curves to other spaces, recursive relations between numbers of curves with specified properties and equations for generating functions of these numbers. Krommen vormen een belangrijk onderzoeksgebied binnen de algebraïsche meetkunde. In mijn proefschrift bestudeer ik zogenaamde moduliruimten van krommen; dit zijn ruimten die alle krommen met gegeven eigenschappen parametriseren. Verschillende aspecten die ik onderzoek zijn afbeeldingen van krommen naar andere ruimten, recursieve relaties tussen aantallen krommen met gegeven eigenschappen en vergelijkingen voor genererende functies van dit soort aantallen.