good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations. The proofs of some theorems are continued after the statements and proofs of needed lemmas with no helpful demarcations (“Continuation of proof of Theorem . . . ”). Finally, at least one lemma (3.34) has a corollary with no theorem or proposition in sight; the status of Lemma 3.36 is also unclear. Lemma 1.16 on page 12 giving a closed formula for Stirling numbers of the second kind is important enough to figure in the proof of capstone Theorem 3.71 (Schlömilch’s formula) on page 114. This nonstandard approach to chapter organization may impede some readers’ comprehending the structure of the author’s highly purposeful discussion. Review of 3 Enumerative Combinatorics Author of Book: Charalambos A. Charalambides Publisher: Chapman & Hall/CRC ISBN L-58488-290-5, Hard Cover, 609 pages

Enumerative Combinatorics

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Advances in mathematics

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see Toen and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toen in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (n−d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E,T∗X), for E an elliptic curve and T∗X is the total space of the cotangent bundle of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).

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Shifted Symplectic Structures

We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.

/pdf/homological-projective-duality-2z95vh9cwk.pdf

Homological projective duality

We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane, and singular cubics, and discuss its relation to the rationality of these cubics.

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Derived Categories of Cubic Fourfolds

Derived categories of quadric fibrations and intersections of quadrics

Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative ℙ2, certain complexes of sheaves on a noncommutative ℙ3, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative ℙ2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative ℙ2. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative ℝ4 than the one considered by Nekrasov and Schwarz (a q-deformed ℝ4).

/pdf/noncommutative-instantons-and-twistor-transform-3lcz15hl7d.pdf

Noncommutative Instantons and Twistor Transform

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.

Alexander Kuznetsov

Papers

Homological projective duality

Derived Categories of Cubic Fourfolds

Derived categories of quadric fibrations and intersections of quadrics

Noncommutative Instantons and Twistor Transform

Semiorthogonal decompositions in algebraic geometry