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In the Hurwitz space of rational functions on ${{\mathbb{C}}}\textrm{P}^1$ with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

A meromorphic differential on a Riemann surface is said to be {\it real-normalized} if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. Our goal is to develop tools to study this topology. We propose a combinatorial model for the real normalized differentials with a single order $2$ pole and use it to analyze the corresponding absolute period foliation., 25 pages

We prove that the generating function for the symmetric chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev--Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants giving the same solution of the KP. In particular, we introduce the Abel polynomial for graphs and show this for its generating function. The key point here is a Hopf algebra structure on the space spanned by graphs and the behavior of the invariants on its primitive space., Minor corrections

Hurwitz spaces are spaces of meromorphic functions on algebraic curves. B. Dubrovin introduced Frobenius structures on Hurwitz spaces, which serve as one of the most spectacular examples of such structures. On the other hand, Hurwitz spaces admit natural compactification by stable maps. The main goal of the present paper is to formulate a question concerning the behavior of Dubrovin's Frobenius structures on Hurwitz spaces on the boundary of the compactification. Sample computations justifying the question are given.

In the first section of this chapter, we give a proof of the Riemann–Roch formula l(D) − l(K − D) = d − g + 1. In the second section, we present a geometric interpretation of the quantities occurring in the Riemann–Roch formula in terms of canonical curves.

In this chapter, we show how moduli spaces of maps can be applied to compute topological characteristics of various varieties. The notion of a stable map was introduced by Kontsevich. He applied it to solving the classical problem of enumerating rational curves of a given degree in the plane passing through a given collection of points. The methods suggested by Kontsevich turned out to be applicable to a wide circle of problems of enumerative geometry, being now the main tool for computing Gromov–Witten invariants.

In Sect. 6.3, we have already discussed what does the moduli space of elliptic (i.e., genus 1) curves look like. This is a rather typical example, which allows one to observe many features common for all moduli spaces. In this chapter, we will study it in more detail and consider examples of other moduli spaces. From the very beginning, we deal with moduli spaces of curves with marked points: properties of such spaces can be described “in layman’s terms” already for curves of small genus, and these properties are highly nontrivial. As the genus grows, the geometry of the moduli space becomes complicated, and a complete description of this space is usually beyond reach. Nevertheless, one can compute many important geometric characteristics of moduli spaces of curves of high genus.

Every complex algebraic curve is a two-dimensional oriented surface. As we already know, the topology of such surfaces is very simple: for a compact surface, the topology is uniquely determined by its genus (or, equivalently, its Euler characteristic). However, along with a topological structure, a curve has a complex structure. It singles out analytic functions among all the functions on the curve.

We have already come across line bundles over curves: trivial, tangent, and cotangent bundles and their tensor powers. Thus, a natural question arises: given a curve, are there other line bundles over it? For instance, in the case of elliptic curves, all line bundles mentioned above are trivial, so this question means, in particular, whether there are nontrivial line bundles over a given elliptic curve.

The introduction of characteristic classes allows one to see many calculations carried out above in a new light and simplify them. Characteristic classes are a universal tool for computing topological characteristics of algebraic varieties, both smooth and singular. We begin with discussing definitions and general properties of Chern classes of vector bundles, and then show how one can use them to obtain some results we already know and their generalizations. In the next chapter, we will speak about characteristic classes that arise naturally in the study of the topology of moduli spaces of curves.

The line bundles over a given complex curve are in a one-to-one correspondence with the linear equivalence classes of divisors on this curve. Such a class has an integer-valued characteristic, the degree. Since divisors consist of points of the curve, it is natural to expect that the set of classes of divisors of the same degree is endowed with additional structures. It must be a topological space and, moreover, a complex variety. Abel’s theorem identifies the space of classes of divisors of zero degree on a curve of genus g with a g-dimensional complex torus, the Jacobian of the curve.

Constructing moduli spaces is a technically complicated task, involving the analysis of many subtleties. In this chapter, we will discuss, without going into details, one of the possible methods of carrying out such a construction. We will describe the general sequence of steps; numerous results justifying it are either stated without proofs or left as exercises.

A meromorphic function on an algebraic curve is a mapping from this curve into the projective line. However, it is natural to consider also mappings into other complex curves, first of all, one-to-one mappings from a complex curve to itself, i.e., automorphisms of a curve. All automorphisms of a given curve form a group. For a curve of genus 0 (projective line), this group is three-dimensional. For any curve of genus 1 (elliptic curve), it is one-dimensional. For curves of higher genus it is finite, and for curves of genus g > 2 it usually consists only of the identity mapping. Curves with a large symmetry group are of special interest: like any symmetric object, they can be very beautiful.

As we will see below, the projective plane (like any other two-dimensional surface) is too narrow to accommodate every smooth curve. In the three-dimensional space there is much more freedom. However, to define curves in \({\mathbb {C}\mathbb {P}}^3\) and higher dimensional projective spaces is more difficult than in the plane. In this chapter, we discuss methods of defining such curves.

We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and combinatorics of symmetric groups and their representations is explained. Applications of the results to constructing efficient computations in problems related to models of quantum field theories are given., 32 pages

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations between the homology classes represented by various strata.

In 1891 A. Hurwitz considered the problem of enumerating the ramified coverings of the two-dimensional sphere by two-dimensional surfaces with fixed types of branching over the branch points. In the original setting the problem was reformulated in terms of characters of the symmetric group. Recently it turned out that the problem is also very closely connected with diverse physical theories, with singularity theory, and with the geometry of the moduli spaces of complex curves. The discovery of these relationships has led to an enlargement of the class of cases in which the enumeration yields explicit formulae, and a clarification of the nature of the classical results. This survey is devoted to a description of the contemporary state of this thriving topic and is intended for experts in topology, the theory of Riemann surfaces, combinatorics, singularity theory, and mathematical physics. It can also serve as a guide to the modern literature on coverings of the sphere.

This paper considers combinatorial properties of meanders — nonself-intersecting curves on the euclidean and projective plane. Both theoretical and numerical results are given, and various connections with other domains in mathematics are indicated.

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems., 13 pages, many figures

We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2-sphere., Comment: 12 pages, no figures

The theory of maps (sometimes also called embedded graphs, or ribbon graphs, or fat graphs, or graphs with rotation), or, otherwise, the topological graph theory, is an old and well established branch of combinatorics. It may justifiably be proud of such classical results as the Euler formula (relating the number of vertices, edges, and faces of a map with the genus of the corresponding surface), and of such notably difficult modern achievements as the Four Color Theorem. But the last two decades witnessed a kind of a volcanic activity in this domain which would have been difficult to predict some 30 years ago.

A substantial part of the theory of dessins d’enfants could have been developed in the 19th century. However, the fundamental theorem that lies at the basis of the theory was established by G. V. Belyi1 [21] only in 1979. Below we formulate the theorem, but postpone its proof until the very end of the chapter (Sec. 2.6). This order of exposition permits the reader to begin by acquiring some experience, by seeing examples and following the accompanying discussion.

This chapter is devoted mainly to the description of algebraic and combinatorial constructions related to Vassiliev’s theory of knot invariants [283], [284], [285]. The main combinatorial object of the theory is a chord diagram, or, which is the same, a one-vertex map. We start with a description of this notion and of the famous 4-term relation for for the chord diagrams. Our presentation follows the main lines of Bar-Natan [18], but we also pay a lot of attention to the recent development in the analysis of intersection graphs of chord diagrams. We do not present the proof of Kontsevich’s theorem for Vassiliev invariants since its machinery is too far from the subject of our book. Good explanations are available in [50], [195].

We start with a combinatorial—group theoretic notion of constellation. Then we pass on to topology and study ramified coverings of the sphere. Next we introduce various types of “pictures”, or embedded graphs: maps, hypermaps, trees, cacti and so on, and clarify their relation to the ramified coverings and the constellations. Finally, at the end of the chapter we discuss Riemann surfaces; they are constructed as ramified coverings of the complex Riemann sphere.

This chapter is devoted to a description of the intriguing connection between map enumeration and matrix integrals. This connection was first established in [143] for the purposes of matrix models of quantum gravity. It was later reinvented by mathematicians [138] in the computation of the Euler characteristic of moduli spaces of complex curves. Our presentation follows [25] and we show, along the lines of [138], how it leads to the enumeration of one-face maps. We also relate, following [87], [299], the universal one-matrix model to the Korteweg—de Vries (KdV) hierarchy of partial differential equations. The results of this chapter will be used in the next one in the description of the Harer—Zagier computation of the Euler characteristic of moduli spaces of curves, as well as in the study of Witten’s conjecture, which is now Kontsevich’s theorem.

In this chapter we present an overview of the connection between the geometry of moduli spaces of complex curves with marked points and the topology of embedded graphs. According to Harer [137], the idea of this connection belongs to Mumford. It proved to be extremely fruitful. The most celebrated results here are the calculation of the orbifold Euler characteristic of moduli spaces of smooth curves due to Harer and Zagier [138] (we present also Kontsevich’s calculation based on similar ideas) and Kontsevich’s proof of Witten’s conjecture. We must note that a complete exposition of this proof containing all the details has not yet been published. Our text does not fulfill this mission either. In the next chapter we will show how the geometry of moduli spaces of curves is related to that of Hurwitz spaces, that is, the moduli spaces of meromorphic functions on complex curves. Since a meromorphic function is also associated to an embedded graph, we obtain another facet of the same connection.

The starting point of the research described in this chapter is the question formulated as Problem 1.1.11 in Sec. 1.1. This question consists in enumerating constellations with given passport, and we call it the Hurwitz problem. In this general form, it has no satisfactory answer up to now, and the present chapter describes some partial results reflecting the best of the current knowledge. The known results are, however, rather general.