A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold $M^n$ in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of $M^n$ has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable $k$-currents, we eliminate $H_k (M^n, {\bb Z})$ for all $1 < k < n - 1$. We then observe that the fundamental group of $M^n$ is trivial. It should be emphasized that our result is optimal.

We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason’s insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan–Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan–Lusztig cell modules for Hecke algebras of type A from [V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (2008), 1363–1426].

We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(x) defined by the q-expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.

We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV [Compositio Math. 143 (2007), 112–164] for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one- and two-level densities to study the distribution of low-lying zeros for one-parameter rational families of elliptic curves over $\mathbb{Q}(t)$. Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the two-level densities of some constant sign families, we find the first examples of families of elliptic curves where we can distinguish SO(even) from SO(odd) symmetry.

In this article we introduce a natural ‘elliptic’ generalization of the classical polylogarithms, study the properties of these functions and their relations with Eisenstein series.

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric ω∈c1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that on P2 blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

I give the necessary and sufficient conditions for the existence of Unitary local systems with prescribed local monodromies on $\mathbb P$1 − S where S is a finite set. This is used to give an algorithm to decide if a rigid local system on $\mathbb P$1 − S has finite global monodromy, thereby answering a question of N. Katz. The methods of this article (use of Harder–Narasimhan filtrations) are used to strengthen Klyachko's theorem on sums of Hermitian matrices. In the Appendix, I give a reformulation of Mehta–Seshadri theorem in the SU(n) setting.

In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda–Tate-like formula for elliptic threefolds, and a relation between the ‘average’ number of rational points on singular fibers and the Galois action on those fibers.

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example $F=\mathbb{Q}_p$), and let GF be the absolute Galois group of F. The main result of this article is the following: a p-adic representation of GF, which is a limit of subquotients of crystalline representations with Hodge–Tate weights in an interval [a; b], is itself crystalline with Hodge–Tate weights in [a; b]. In order to show this, we study the $(\phi,\Gamma)$-modules attached to crystalline representations, which allows us to improve some results of Fontaine, Wach and Colmez.

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field $\Bbbk$ of characteristic zero.

For a fixed parabolic subalgebra 𝔭 of we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪0𝔭. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s [les ] 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2] manifolds with polarisation of degree 2d and split type is of general type if d≥12.

Let A be a finite-dimensional k-algebra over algebraically closed field k and $m{\rm mod}\, A$ be the category of finite-dimensional left A-modules. We show that a module M in ${\rm mod}\, A$ degenerates to another module N in ${\rm mod}\, A$ if and only if there is an exact sequence $0\to N\to M\oplus Z\to Z\to 0$ in ${\rm mod}\, A$ for some A-module Z. Moreover, we give a description of minimal degenerations of finite-dimensional A-modules.

We give a natural construction of framed mixed Tate motives unramified over $\mathbb{Z}$ whose periods are the multiple $\zeta$-values. Namely, for each convergent multiple $\zeta$-value we define two boundary divisors A and B in the moduli space $\overline{\mathcal{M}}_{0,n+3}$ of stable curves of genus zero. The corresponding multiple zeta-motive is the nth cohomology of the pair $(\overline{\mathcal{M}}_{0,n+3}-A,B)$.