Related papers: A note on the integrality of mirror maps

By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in…

It is shown that any finite-dimensional homomorphic image of an inverse limit of nilpotent not-necessarily-associative algebras over a field is nilpotent. More generally, this is true of algebras over a general commutative ring k, with…

Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $\infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms…

We present an elementary and self-contained construction of $A_\infty$-algebras, $A_\infty$-bimodules and their Hochschild homology and cohomology groups. In addition, we discuss the cup product in Hochschild cohomology and the spectral…

We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma'$ immering into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the…

We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…

Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $\theta : E\to F$ is a linear map preserving orthogonality, i.e., $<\theta(x), \theta(y) > = 0$ whenever $<x, y > = 0$. We show in this…

We describe for any Riemannian manifold a certain infinitesimal neighbourhood of the diagonal. Semi-conformal maps are analyzed as those that preserve such neighbourhoods; harmonic maps are analyzed as those that preserve mirror image…

We will show in this paper that if $\lambda$ is very close to 1, then $$I(M,\lambda,m)= \sup_{u\in H^{1,n}_0(M) ,\int_M|\nabla u|^ndV=1}\int_\Omega (e^{\alpha_n |u|^\frac{n}{n-1}}-\lambda\sum\limits_{k=1}^m\frac{|\alpha_nu^\frac{n}{n-1}|^k}…

We construct countable groups $G$ with the following new degree of W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$, then the groups…

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental…

We introduce the notion of G-hypergeometric function, where G is a complex Lie group. In the case when G is a complex torus, this notion amounts to the notion of Gelfand's A-hypergeometric function. We show that the integral $\int…

Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on various hypersurfaces which satisfy…

We present natural conjectural generalizations of the `positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano…

Let $\Sigma$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within…

The hypergeometric function method naturally provides the analytic expressions of scalar integrals from concerned Feynman diagrams in some connected regions of independent kinematic variables, also presents the systems of homogeneous linear…

Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched…

We prove several superrigidity results for isometric actions on metric spaces satisfying some convexity properties. First, we extend some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of…

We prove one direction of homological mirror symmetry for complete intersections in algebraic tori, in all dimensions. The mirror geometry is not a space but a LG model, i.e. a pair given by a space and a regular function. We show that the…

We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for affine-K\"{a}hler manifolds and for elliptic curves. Their geometric properties, especially the link with special lagrangian fibrations and the…