9 research outputs found

    The combinatorics of the Jack parameter and the genus series for topological maps

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    Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically

    On discrete surfaces : Enumerative geometry, matrix models and universality classes via topological recursion

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    The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. We approach our study of these objects from different perspectives, namely bijective combinatorics, matrix models and analysis of critical behaviors. Our problems have a powerful relatively recent tool in common, which is the so-called topological recursion introduced by Chekhov, Eynard and Orantin around 2007. Further understanding general properties of this procedure also constitutes a motivation for us. We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorial relation between fully simple and ordinary maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between (first and second order) correlation moments and free cumulants established by Collins--Mingo--'Sniady--Speicher in the setting of free probability, and implement the exchange transformation xleftrightarrowyx leftrightarrow y on the spectral curve in the context of topological recursion. These interesting features motivated us to investigate fully simple maps, which turned out to be interesting combinatorial objects by themselves. We then propose a combinatorial interpretation of the still not well understood exchange symplectic transformation of the topological recursion. We provide a matrix model interpretation for fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters (lambda1,ldots,lambdan)(lambda_1,ldots,lambda_n) are strictly monotone double Hurwitz numbers with ramifications lambdalambda above inftyinfty and (2,ldots,2)(2,ldots,2) above 00. Combining with a recent result of Dubrovin--Liu--Yang--Zhang, this implies an ELSV-like formula for these Hurwitz numbers. Later, we consider ordinary maps endowed with a so-called O(mathsfn)O(mathsf{n}) loop model, which is a classical model in statistical physics. We consider a probability measure on these objects, thus providing a notion of randomness, and our goal is to determine which shapes are more likely to occur regarding the nesting properties of the loops decorating the maps. In this context, we call volume the number of vertices of the map and we want to study the limiting objects when the volume becomes arbitrarily large, which can be done by studying the generating series at dominant singularities. An important motivation comes from the conjecture that the geometry of large random maps is universal. We pursue the analysis of nesting statistics in the O(mathsfn)O(mathsf{n}) loop model on random maps of arbitrary topologies in the presence of large and small boundaries, which was initiated for maps with the topology of disks and cylinders by Borot--Bouttier--Duplantier. For this purpose we rely on topological recursion results for the enumeration of maps in the O(mathsfn)O(mathsf{n}) model. We characterize the generating series of maps of genus gg with kk boundaries and~k′k' marked points which realize a fixed nesting graph, which is associated to every map endowed with loops and encodes the information regarding non-separating loops, which are the non-contractible ones on the complement of the marked elements. These generating series are amenable to explicit computations in the so-called loop model with bending energy on triangulations, and we characterize their behavior at criticality in the dense and in the dilute phases, which are the two universality classes characteristic of the O(mathsfn)O(mathsf{n}) loop model. We extract interesting qualitative conclusions, e.g., which nesting graphs are more probable to occur. We also argue how this analysis can be generalized to other problems in enumerative geometry satisfying the topological recursion, and apply our method to study the fully simple maps introduced in the first part of the thesis

    Ahlfors circle maps and total reality: from Riemann to Rohlin

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    This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

    Subject Index Volumes 1–200

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    Regular pseudo-oriented maps and hypermaps of low genus

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    Pseudo-orientable maps were introduced by Wilson in 1976 to describe non-orientable regular maps for which it is possible to assign an orientation to each vertex in such a way that adjacent vertices have opposite orientations. This property extends naturally to non- orientable and orientable hypermaps. In this paper we classify the regular pseudo-oriented maps and hypermaps of characteristic χ > −3. With the help of GAP (The GAP group, 2014) and its library of small groups, we extend the classification down to characteristic χ = −16 (Tables 7–19 in the Appendix)
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