Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

We characterize genus g canonical curves by the vanishing of combinatorial products of g+1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities. A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of Szego kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added. Accepted for publication in Math. An

Generalized boundary strata classes

We describe a generalization of the usual boundary strata classes in the Chow ring of $\overline{\mathcal{M}}_{g,n}$. The generalized boundary strata classes additively span a subring of the tautological ring. We describe a multiplication law satisfied by these classes and check that every double ramification cycle lies in this subring.Comment: For the Proceedings of the 2017 Abel Symposium, 10 page

Computing top intersections in the tautological ring of MgM_g

We derive effective recursion formulae of top intersections in the tautological ring $R^*(M_g)$ of the moduli space of curves of genus $g\geq 2$. As an application, we prove a convolution-type tautological relation in $R^{g-2}(M_g)$.Comment: 18 page

Lectures on the Asymptotic Expansion of a Hermitian Matrix Integral

In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann $\zeta$-function. The third method is derived from a formula for the $\tau$-function solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are \emph{transcendental}, in the sense that they cannot be obtained by the celebrated Krichever construction and its generalizations based on algebraic geometry of vector bundles on Riemann surfaces. In each case a mathematically rigorous way of dealing with asymptotic series in an infinite number of variables is established

A new species of the deep-bodied actinopterygian Dapedium from the Middle Jurassic (Aalenian) o f southwestern Germany

Dapedium is one of the most abundant and diverse genera of ganoid fishes from the Early Jurassic fossil lagerstatte of Europe. In spite of its abundance, however, its timing of extinction is poorly constrained, with the youngest described material being Early Jurassic in age. We describe new diagnostic and relatively complete material of a large species of Dapedium (standard length estimated at 50 cm) from the Middle Jurassic (earliest Aalenian) Opalinuston Formation of Baden-Wurttemberg, Germany. The Aalenian material represents a distinct species, D. ballei sp. nov., differing from Early Jurassic species in a unique combination of characters pertaining to the shape of the dermal skull elements, pectoral fin position, and scale shape and ornamentation. However, although D. ballei sp. nov. exhibits a unique combination of characters, there are no autapomorphies with which to distinguish it from the Toarcian species of Dapedium. Dapedium ballei represents the geologically youngest species of Dapedium, extending the range of this genus into the Middle Jurassic. The Opalinuston Formation fills an important gap in the marine vertebrate fossil record, and finds from this horizon have the potential to greatly improve our understanding of evolutionary dynamics over this period of faunal transition

On third Poisson structure of KdV equation

The third Poisson structure of KdV equation in terms of canonical ``free fields'' and reduced WZNW model is discussed. We prove that it is ``diagonalized'' in the Lagrange variables which were used before in formulation of 2D gravity. We propose a quantum path integral for KdV equation based on this representation.Comment: 6pp, Latex. to appear in ``Proceedings of V conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, June 1994'' Teor.Mat.Fiz. 199

Conformal blocks and generalized theta functions

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks", which is defined in terms of representations of the Lie algebra sl(r, C((z))).Comment: 43 pages, Plain Te

On some differential-geometric aspects of the Torelli map

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.Comment: To appear on Boll. UMI, special volume in memory of Paolo de Bartolomei

Menelaus relation and Fay's trisecant formula are associativity equations

It is shown that the celebrated Menelaus relation and Fay's trisecant formula similar to the WDVV equation are associativity conditions for structure constants of certain three-dimensional algebra.Comment: Talk given at the Conference " Mathematics and Physics of Solitons and Integrable Systems", Dijon, 28.6-2.7, 2009. Minor misprints correcte

Brill–Noether general K3 surfaces with the maximal number of elliptic pencils of minimal degree

We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal c2.publishedVersio