Abstract:
Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer to a question of Ostrover and Tyomkin: the quantum cohomology algebra of a toric Fano manifold contains a field as a direct summand. Moreover, it gives a good evidence that the same statement holds for any Fano manifold.

Abstract:
We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.

Abstract:
We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.

Abstract:
We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case.

Abstract:
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's $\Gamma$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

Abstract:
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52].

Abstract:
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52].

Abstract:
We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class G_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by G_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_coh(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces and Grassmannians.

Abstract:
We show that the big quantum cohomology of the symplectic isotropic Grassmanian $IG(2,6)$ is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation.

Abstract:
data from the hydrothermally influenced guaymas basin of the gulf of california are presented on the concentration and distribution of ag, as, au, ba, cd, co, cr, cu, fe, hg, mn, pb, sb, se, and zn in different tissues of dominant hydrothermal vent animals such as vestimentifera riftia pachyptila and vesicomyid clams archivesica gigas and other organisms, including spongia, bivalve mollusks nuculana grasslei, phelliactis pabista, and crab munidopsis alvisca. chemical element content was measured by atomic absorption spectrometry (flame and graphite furnace methods) and instrumental neutron activation analysis. in the dominant specialized taxa, the main target organs of metals were the trophosome and obturaculae of riftia pachyptila, the gills and mantle of archivesica gigas. the other organisms also demonstrated high bioaccumulation of metals. especially high levels of most of the metals (excluding mn) were detected in the soft body of nuculana grasslei. the highest mn content was found in the whole body of spongia. bioconcentration factor of the trace metals studied varies within three orders of magnitude from 5 (mn) to 3？104 (cd). this testifies apparently a selectivity of trace metal bioaccumulation by the organisms which is determined by metal bioavailability independently of metal concentration in the water column. variability in the molar ratio fe/mn allows us to assume that these metals undergo fractionation during migration from the hydrothermal fluids to the interior organs of animals. insignificant differences between the cd, cu, fe, hg, pb, and zn levels in the guaymas basin vent clams versus that in the bivalve mollusks from polluted areas of the gulf of california might suggest that the metal bioavailability play an important role in the bioaccumulation.