By Mark Gross, Daniel Huybrechts, Dominic Joyce, Geir Ellingsrud, Loren Olson, Kristian Ranestad, Stein A. Stromme

This booklet is an extended model of lectures given at a summer season university on symplectic geometry in Nordfjordeid, Norway, in June 2001. The unifying function of the publication is an emphasis on Calabi-Yau manifolds. the 1st half discusses holonomy teams and calibrated submanifolds, targeting certain Lagrangian submanifolds and the SYZ conjecture. the second one reports Calabi-Yau manifolds and replicate symmetry, utilizing algebraic geometry. the ultimate half describes compact hyperkahler manifolds, that have a geometrical constitution very heavily concerning Calabi-Yau manifolds.

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**Example text**

N = 1. At this step we involve the following observation made by D. Cordero-Erausquin, M. Fradelizi and B. Maurey in their study and proof of the so-called B-conjecture, cf. [CE-F-M], Theorem 1. It is stated below as a lemma, where D(λ) is treated as a linear map. 2 (D. Cordero-Erausquin, M. Fradelizi and B. Maurey [CE-F-M]). For any symmetric convex body K in Rn , the function (t1 , . . , tn ) −→ γ(D(et1 , . . , etn )(K)) is log-concave on Rn . 1, introduce the function on Rn−1 v(t1 , . .

1), and the (Shannon) entropy power of X is N (X) = e2h(X)/n . We limit ourselves to random vectors X with h(X) < +∞; in this case, N (X) is a non-negative real number. Building on work of [32] and resolving a conjecture they made, [33] recently showed the following result. 1. Let X1 , . . , XM be independent Rn -valued random vectors, such that the entropy of each exists and is ﬁnite. Let β be a fractional partition using a collection G of subsets of [M ]. Then N (X1 + . . + XM ) ≥ βs N s∈G Xj .

Some applications of duality relations. , 1469, Springer, Berlin, 1991. [M-P] Milman, V. , and Pajor, A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. , 1376, Springer, Berlin, 1989. [P] Pisier, G. The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989, xvi+250 pp. C. The diﬀerence body of a convex body. Arch. Math. (Basel) 8 (1957), 220–233. C. Convex bodies associated with a given convex body.