Volume 2019, Issue 757

We explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space as in a paper of Cheeger, Colding and Tian. Our results will be applied to study the limit spaces for a sequence of Kähler metrics arising from solutions of certain complex Monge–Ampère equations for the existence problem of Kähler–Ricci solitons on a Fano manifold via the continuity method.

We prove best-possible bounds for bilinear forms in Kloosterman sums for GL⁡(3)\operatorname{GL}(3) associated with the long Weyl element. As an application we derive a best-possible spectral large sieve inequality on GL⁡(3)\operatorname{GL}(3).

Studying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE′⁢(n,0){\mathrm{CDE}^{\prime}(n,0)}, which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying CDE′⁢(n,0){\mathrm{CDE}^{\prime}(n,0)}) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.

We prove the all-time existence of non-compact, complete, strictly convex solutions to the α-Gauss curvature flow for any positive power α.

We study the Littlewood–Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K -theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang–Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood–Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson–Tao and Vakil.

We prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II1{\mathrm{II}_{1}} factor L⁢(Γ){L(\Gamma)} is prime. In particular, we deduce that the II1{\mathrm{II}_{1}} factors associated to the arithmetic groups PSL2⁢(ℤ⁢[d]){\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and PSL2⁢(ℤ⁢[S-1]){\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer d≥2{d\geq 2} with d≢1⁢(mod⁡ 4){d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S . This provides the first examples of prime II1{\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of L⁢(Γ){L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that L⁢(Γ){L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for L⁢(Γ){L(\Gamma)}.

Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called simplicial complexity that allows to obtain a quite satisfactory answer to his question. Using this new complexity, we also derive new results on systolic area for groups that specify its topological behaviour.

A homogeneous polynomial of degree d in n+1{n+1} variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in n+1{n+1} variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided n≥25⁢R{n\geq 25R}, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in 25⁢R{25R} variables, previous work having required that n≫R2{n\gg R^{2}}. One conjectures that n≥6⁢R+1{n\geq 6R+1} should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.