We first describe how one associates a cubic curve to a given ternary trilinear form. We explore relations between the rank and border rank of the form and the geometry of the corresponding cubic curve. When the curve is smooth, we show there is no relation. When the curve is singular, normal forms are available, and we review the explicit correspondence between the normal forms, rank and border rank.

CAT(0) spaces with boundary the join of two cantor sets, Khek Lun Harold Chao (Indiana University)We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete, then X is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees.

Existence of almost contact structures on manifolds with G_2-structures, Hyunjoo Cho, Sema Salur, and Firat Arikan (University of Rochester)We show the existence (co-oriented) contact structures on certain classes of G2-manifolds, and that these two structures are compatible in certain ways. We also prove that any seven-manifold with a spin structure has an almost contact structure, and construct explicit almost contact structures on manifolds with G2-structures. Moreover, we also show that one can extend any almost contact structure on an associative submanifold to whole G2-manifold.

An Algebraic Topology Model for Research and Development, Thanos Gentimis and Maria Bampasidou (University Of Florida)In this paper we describe a model based on basic algebraic topology that describes interactions between companies in terms of research and development. We then apply our model to collaborations between mathematicians in the creation of mathematical papers.

Essential manifolds with extra structures, Sergii Kutsak (University of Florida)I consider classes of algebraic manifold , of symplectic manifolds , of symplectic manifolds with the hard Lefschetz property and the class of cohomologically symplectic manifolds . For every class of manifolds , I denote by a subclass of -dimensional rationally essential manifolds with fundamental group . I will prove that for all the above classes with symplectically aspherical form the condition implies that for every . Also I will show that all the inclusions are proper.

Partial compactification of the zero section of the universal abelian variety, Dmitry Zakharov and Samuel Grushevsky (Stony Brook University)

The moduli space of principally polarized abelian varieties is one of the central objects of study in algebraic geometry. The moduli space is not compact, and admits several natural compactifications. All of these compactifications are extensions of Mumford's partial compactification by semi-abelic varieties of torus rank one. The partial compactification is the base for a universal family that admits a zero-section. In our joint work with Samuel Grushevsky, we calculate the class of the zero section in the Chow ring of the partial compactification of the universal abelian variety.

Representation Theorey and Hilbert Schemes of Points on K3 Surfaces, Letao Zhang (Rice University)Let X denote a general deformation of the Hilbert schemes of n points on K3 surfaces, and Gx be the associated group acting on the cohomology ring. We computed the graded character formula associated to the Gx action on the cohomology ring of X. Also, we could use the formula to deduct the generating series of the number of canonical Hodge classes for each n.