Inspired by the homological mirror symmetry conjecture of Kontsevich [30], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi-projective variety.

We give an interpretation of the path model of a representation (see [18]) of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by Lakshmibai-Seshadri (LS) galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure-Hansen-Bott-Samelson desingularization $\hat\Sigma(\lambda)$ of the closure of an orbit $G(\mathbb{C}[[t]])\cdot\lambda$ in the affine Grassmannian. The homology of $\hat\Sigma(\lambda)$ has a basis given by Białynicki-Birula (BB) cells, which are indexed by the T-fixed points in $\hat\Sigma(\lambda)$. Now the points of $\hat\Sigma(\lambda)$ can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T-fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a nonempty intersection with $G(\mathbb{C}[[t]])\cdot\lambda$ (identified with an open subset of $\hat\Sigma(\lambda)$), and we show that the closures of the strata associated to LS galleries are exactly the Mirković-Vilonen (MV) cycles (see [25]) which form a basis of the representation V(λ) for the Langlands dual group $G^\vee$.

Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin⁻ structure. We show that every irreducible real rational curve with nonempty real part has a canonical spinor state belonging to {± 1}. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real configuration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independently of the choice of the real configuration of points.

Let G be a unitary, symplectic, or orthogonal group over a non-Archimedean local field of residual characteristic different from 2, considered as the fixed-point subgroup in a general linear group $\widetilde{G}$ of an involution. Following [8] and [12], we generalize the notion of a semisimple character for $\widetilde{G}$ and for G. In particular, following the formalism of [4], we show that these semisimple characters have certain functorial properties. Finally, we show that any positive level supercuspidal representation of G contains a semisimple character.

We consider positive solutions of the Dirichlet problem for nonlinear elliptic equations, where the nonlinearity is assumed to satisfy the polynomial upper growth condition f(x,u) ≤ C(1 + u^p). It is known from a classical work of Brezis and Turner that the condition p < p_BT := (N + 1)/(N − 1) implies a uniform a~priori bound for all solutions. Yet this exponent appeared to be technical since Gidas and Spruck later showed that, for nonlinearities with precise power behavior like f(x,u) ~ u^p, the critical exponent is given by the Sobolev number p_S := (N + 2)/(N − 2).

Surprisingly, we show that the exponent p_BT is, however, sharp: whenever p > p_BT, for a suitable nonlinearity f(x,u) = a(x)u^p, with a(x) ≥ 0 and a ∈ L^∞, we prove the existence of an unbounded weak solution.

We next consider the case of systems and show that the polynomial growth conditions for a priori estimates recently obtained by Quittner and the author are also optimal.

Our results are strongly connected with the regularity theory of the Laplace operator in the spaces $L^p_\delta(\Omega)$, the Lebesgue spaces weighted by the distance to the boundary. As a by-product, we in turn establish the optimality of the known linear $L^p_\delta$-regularity estimates. Our proofs are based on the construction of a solution of the Laplace equation with a suitable boundary singularity, with conical support, and we use recent results on the boundary behavior of heat kernels.