We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the τ -function of an open KdV hierarchy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the theory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations.Our open KdV and Virasoro constraints uniquely specify a theory of higher genus open descendent integrals. As a result, we obtain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves. Date: Aug. 2015. 1 2.2. Stable disks 2.3. Stable graphs 2.4. Smoothing and boundary 2.5. Moduli and orientations 2.6. Edge labels 2.7. Forgetful maps 3. Line bundles and relative Euler classes 3.1. Cotangent lines and canonical boundary conditions 3.2. Definition of open descendent integrals 3.3. The base 3.4. Abstract vertices 3.5. Special canonical boundary conditions 3.6. Forgetful maps, cotangent lines and base 3.7. Construction of multisections and homotopies 4. Geometric recursions 4.1. Proof of string equation 4.2. Proof of dilaton equation 4.3. Proofs of TRR I and II 5.

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.

Abstract. In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers satisfies the open KdV equations. In this paper we prove this conjecture. Our proof goes through a matrix model and is based on a Kontsevich type combinatorial formula for the intersection numbers that was found by the second author.

Although there is some evidence from twin studies that individual differences in sexual behavior are heritable, little is known about the specific molecular genetic design of human sexuality. Recently, a specific dopamine D4 receptor (DRD4) agonist was shown in rats to induce penile erection through a central mechanism. These findings prompted us to examine possible association between the well-characterized DRD4 gene and core phenotypes of human sexual behavior that included desire, arousal and function in a group of 148 nonclinical university students. We observed association between the exon 3 repeat region, and the C-521T and C-616G promoter region SNPs, with scores on scales that measure human sexual behavior. The single most common DRD4 5-locus haplotype (19%) was significantly associated with Desire, Function and Arousal scores. The current results are consistent with animal studies that show a role for dopamine and specifically the DRD4 receptor in sexual behavior and suggest that one pathway by which individual variation in human desire, arousal and function are mediated is based on allelic variants coding for differences in DRD4 receptor gene expression and protein concentrations in key brain areas.

In recent works, [20,21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we prove a formula of these integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5]. Contents 2.2.4. A comment about the alternative definition in the stable case 2.2.5. Spin graphs 2.2.6. M g,k,l 2.3. The line bundles L i 2.4. Boundary conditions and intersection numbers 2.5. The orientation of M g,k,l 3. Sphere bundles and relative Euler class 4. Symmetric Jenkins-Strebel stratification 4.1. JS stratification for the closed moduli 4.1.1. JS differential and the induced graph 4.1.2. Combinatorial moduli 4.1.3. Tautological line bundles and associated forms 4.2.