Volume 2017, Issue 727

For any finite subgroup G⊂SL3⁡(ℂ){G\subset\operatorname{SL}_{3}(\mathbb{C})}, work of Bridgeland–King–Reid constructs an equivalence between the G -equivariant derived category of ℂ3{\mathbb{C}^{3}} and the derived category of the crepant resolution Y=G⁢-⁢Hilbℂ3{Y=\mathop{{G}\hbox{-}\mathrm{Hilb}}\mathbb{C}^{3}} of ℂ3/G{\mathbb{C}^{3}/G}. When G is abelian, we show that this equivalence gives a natural correspondence between irreducible representations of G and exceptional subvarieties of Y , thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid’s recipe and extends earlier work from [3] and [14] which dealt only with the case when ℂ3/G{\mathbb{C}^{3}/G} has one isolated singularity.

The notions 𝒲{\mathscr{W}}-entropy and reduced volume for Ricci flow were introduced by Perelman, which had proved their importance in the study of Ricci flow. Lei Ni studied the analogous concepts for heat equation on static manifolds, and proved limt→∞\lim_{t\rightarrow\infty}𝒲\mathscr{W}== ln(limt→∞\lim_{t\rightarrow\infty}V¯\bar{V}) which links the large time behavior of these two. Due to the surprising similarity between those concepts in the Ricci flow and the linear heat equation, a natural question whether such an equation holds for the Ricci flow ancient solution was asked by Lei Ni. In this note, we give an alternative proof to Lei Ni’s equation based on a new method. Following the same philosophy of this method, we answer Lei Ni’s question positively for type I κ-solutions of Ricci flow.

We study commutative rings R for which all projective schemes over R have finite morphisms to a projective space of R . We then use our results to define a new kind of capacity for adelic subsets of projective flat schemes X{{X}} over global fields. This capacity can be used to generalize the converse part of the Fekete–Szegő Theorem.

We prove that a pair of integral quadratic forms in five or more variables will simultaneously represent “almost all” pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

We introduce generalized Monge–Ampère capacities and use these to study complex Monge–Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor. Our results still hold if the divisor is replaced by any closed subset.

We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining speed and curvature pinching estimates for the flows by the approximating speeds, independent of the smoothing parameter, we may pass to the limit to deduce that the flow by the nonsmooth speed converges to a point in finite time that, under a suitable rescaling, is round in the C2{C^{2}} sense, with the convergence being exponential.

In this paper we classify codimension 1 Mukai foliations on complex projective manifolds.

For a discrete group G , we consider the minimal C*C^{*}-subalgebra of ℓ∞\ell^{\infty}( G ) that arises as the image of a unital positive G -equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra C (∂F⁡G\partial_{F}G) of continuous functions on Furstenberg’s universal G -boundary ∂F⁡G{\partial_{F}G}. This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G -action on ∂F⁡G{\partial_{F}G} is amenable, and use this fact to prove Ozawa’s conjecture that if G is exact, then there is an embedding of the reduced C*C^{*}-algebra Cr*⁢(G){\mathrm{C}^{*}_{r}(G)} of G into a nuclear C*C^{*}-algebra which is contained in the injective envelope of Cr*⁢(G){\mathrm{C}^{*}_{r}(G)}. The algebra C (∂F⁡G\partial_{F}G) arises as an injective envelope in the sense of Hamana, which implies rigidity results for certain G -equivariant maps. We prove a generalization of a rigidity result of Ozawa for G -equivariant maps between spaces of functions on the hyperbolic boundary of a hyperbolic group. Our result applies to hyperbolic groups, but also to groups that are not hyperbolic or even relatively hyperbolic, including certain mapping class groups. It is a longstanding open problem to determine which groups are C*C^{*}-simple, in the sense that the algebra Cr*⁢(G){\mathrm{C}^{*}_{r}(G)} is simple. We prove that this problem can be reformulated as a problem about the structure of the G -action on the Furstenberg boundary. Specifically, we prove that a discrete group G is C*C^{*}-simple if and only if the G -action on the Furstenberg boundary is topologically free. We apply this result to prove that Tarski monster groups are C*C^{*}-simple. This provides another solution to a problem of de la Harpe (recently answered by Olshanskii and Osin) about the existence of C*C^{*}-simple groups with no free subgroups.

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in 𝔼n+1{\mathbb{E}^{n+1}}. We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching 0, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov–Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.