Physique mathematique des nombres Hurwitz pour debutants
6
2014
170
10 Stunden 13 Minuten
6 Ergebnisse
1:41:20
16Kazarian, MaximHurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasoro-type constraints, Chekhov-Eynard-Orantin-type recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, Gromov-Witten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'.
When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.
2014Institut des Hautes Études Scientifiques (IHÉS)
2:03:38
14Kazarian, MaximHurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasoro-type constraints, Chekhov-Eynard-Orantin-type recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, Gromov-Witten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'.
When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.
2014Institut des Hautes Études Scientifiques (IHÉS)
1:56:04
65Zvonkine, DimitriWe will use the example of Hurwitz numbers to make an introduction into the intersection theory of moduli spaces of curves and into the subject of topological recursion.
2014Institut des Hautes Études Scientifiques (IHÉS)
2:06:58
20Kazarian, MaximHurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasoro-type constraints, Chekhov-Eynard-Orantin-type recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, Gromov-Witten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'.
When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.
2014Institut des Hautes Études Scientifiques (IHÉS)
1:21:20
26Eynard, BertrandThe "topological recursion" defines a double family of "invariants" $W_{g,n}$ associated to a "spectral curve" (which we shall define). The invariants $W_{g,n}$ are meromorphic $n$-forms defined by a universal recursion relation on $|\chi|=2g-2+n$, the initial terms $W_{0,1}$ and $W_{0,2}$ being the canonical 1-form and 2-form on the spectral curve. Those invariants have fascinating mathematical properties, they are "symplectic invariants" (invariants under some symplectic transformations of the spectral curve), they are almost modular forms, they satisfy Hirota-like equations, they satisfy some form-cycle duality deformation relations (generalization of Seiberg-Witten), they are stable under many singular limits, and enjoy many other fascinating properties... Moreover, specializations of those invariants recover many known invariants, including Hurwitz numbers to which this conference is dedicated (see M. Kazarian' lecture), intersection numbers, Gromov-Witten invariants, numbers of maps (Tutte's enumeration of maps), or asymptotics of random matrices expectation values. And since very recently, it is conjectured that they also include knot polynomials (Jones, HOMFLY, super polynomials...), which provides an extension of the volume conjecture. We shall present a few examples and mention how these invariants were first discovered in random matrix theory, and then observed or conjectured in many other areas of maths and physics.
2014Institut des Hautes Études Scientifiques (IHÉS)
1:04:06
29Kontsevich, Maxim2014Institut des Hautes Études Scientifiques (IHÉS)