Some families of polynomial automorphisms

We give a family of polynomial automorphisms of the complex affine plane whose generic length is 3 and degenerating in an automorphism of length 1 with surprisingly high degree.     

Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms

We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.     

The automorphism group of a rigid affine variety

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of  is pointed, the group  is a finite extension of . As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.     

Multiplication operators on the Bergman space via analytic continuation

In this paper, using the group-like property of local inverses of a finite Blaschke product ?, we will show that the largest C^?-algebra in the commutant of the multiplication operator M? by ? on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ?−1°? over the unit disk. If the order of the Blaschke product ? is less than or equal to eight, then every C^?-algebra contained in the commutant of M? is abelian and hence the number of minimal reducing subspaces of M? equals the number of connected components of the Riemann surface of ?−1°? over the unit disk.     

Cyclic homology of affine hypersurfaces with isolated Singularities

We consider reduced, affine hypersurfaces with only isolated singularities. We give an explicit computation of the Hodge-components of their cyclic homology in terms of de Rham cohomology and torsion modules of differentials for large n. It turns out that the vector spaces HC_n(A) are finite dimensional for n ≥ N − 1.     

Optimal inverse Littlewood–Offord theorems

Let ηi, i=1,…,n, be iid Bernoulli random variables, taking values ±1 with probability . Given a multiset V of n integers v₁,…,vn, we define the concentration probability as A classical result of Littlewood–Offord and Erd?s from the 1940s asserts that, if the vi are non-zero, then ρ(V) is O(n−1/2). Since then, many researchers have obtained improved bounds by assuming various extra restrictions on V.About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the inverse Littlewood–Offord problem. In the inverse problem, one would like to characterize the set V, given that ρ(V) is relatively large.In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen the previous result of Tao and Vu, obtaining an optimal characterization for V. This immediately implies several classical theorems, such as those of Sárközy and Szemerédi and Halász.The method also applies to the continuous setting and leads to a simple proof for the β-net theorem of Tao and Vu, which plays a key role in their recent studies of random matrices.All results extend to the general case when V is a subset of an abelian torsion-free group, and ηi are independent variables satisfying some weak conditions.     

A Reilly inequality for the first Steklov eigenvalue

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p₁ of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.     

An algebraic slice in the coadjoint space of the Borel and the Coxeter element

Let g be a complex simple Lie algebra and b a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual b^? of b is a polynomial algebra on rank g generators (Grothendieck and Dieudonné (1965–1967)) [16]. The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate y+V of a vector subspace of b^? such that the restriction map induces an isomorphism of Y onto the algebra R[y+V] of regular functions on y+V. This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) [20]; but the construction fails in general even with respect to the Borel. Moreover already in type C(2) no algebraic slice exists.Very surprisingly the exception of type C(2) is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types B(2m), C(n) and F(4).Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction.     

On the automorphism groups of algebraic bounded domains

This paper is a part of author's doctoral thesis. It was partially supported by Sonderforschungsbereich 237 Unordnung und große Fluktuatuionen of the Deutsche Forschungsgemeinschaft     

The existence and asymptotics of solutions for the Abelian Chern–Simons system with two Higgs fields and two gauge fields

In this paper, we study a system of elliptic equations in R² which arises from the self-dual equations for the Abelian Chern–Simons system with two Higgs fields and two gauge fields. We provide a new proof for the existence of topological solutions by constructing explicit supersolutions and subsolutions. We also study the asymptotic behavior of condensate solutions on the torus. It is shown that the maximal solutions converge uniformly to zero away from the vortex points, and the convergence rate is computed.     

Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure.     

LG/CY correspondence: The state space isomorphism

We prove the mirror duality conjecture for the mirror pairs constructed by Berglund, Hübsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi–Yau hypersurfaces inside a weighted projective space and the Fan–Jarvis–Ruan–Witten state space of the associated Landau–Ginzburg singularity theory.     

A geometric degree formula for A-discriminants and Euler obstructions of toric varieties

We give explicit formulas for the dimensions and the degrees of A-discriminant varieties introduced by Gelfand, Kapranov and Zelevinsky. Our formulas can be applied also to the case where the A-discriminant varieties are higher-codimensional and their degrees are described by the geometry of the configurations A. Moreover combinatorial formulas for the Euler obstructions of general (not necessarily normal) toric varieties will be also given.     

Cancellation Problems and Dimension Theory

In this note, we study the global dimension of coalgebras and discuss the class of coalgebras of global dimension less or equal to 1. The coalgebras in this class, which contains all the cosemisimple coalgebras, are called hereditary coalgebras. If C is a finite dimensional coalgebra, then C is hereditary if and only if C (the convolution algebra of C) is a hereditary algebra. Any direct sum of hereditary coalgebras is hereditary too. This gives us many examples of infinite dimensional hereditary coalgebras. A coalgebra is left hereditary if and only if it is right hereditary. Moreover, there do not exist hereditary Hopf algebras of finite dimension which are not cosemisimple.     

On topological Radon transformations

We construct a functor, which we call the topological Radon transform, from a category of complex algebraic varieties with morphisms given by divergent diagrams, to constructible functions. The topological Radon transform is thus the composition of a pull-back and a push-forward of constructible functions. We show that the Chern-Schwartz-MacPherson transformation makes the topological Radon transform of constructible functions compatible with a certain homological Verdier-Radon transform. We use this set-up to prove, given a projective variety X, a formula for the Chern-Mather class of the dual variety in terms of that of X.     

The stable equivalence and cancellation problems

Let K be an arbitrary fieldof characteristic 0, and $\mathbf{A}^n$ then-dimensional affine spaceover K.A well-known cancellation problem asks, given twoalgebraic varieties $V_1, V_2 \subseteq \mathbf{A}^n$ with isomorphiccylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, whether$V_1$ and $V_2$ themselves are isomorphic.In this paper, we focus on a related problem: given twovarieties with equivalent (under an automorphism of $\mathbf{A}^{n+1}$)cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, are$V_1$ and $V_2$ equivalent under an automorphism of $\mathbf{A}^n$?We call this stable equivalence problem.We show that the answer is positivefor any two curves $V_1, V_2 \subseteq \mathbf{A}^2$.For an arbitrary $n \ge 2$, we consider a special, arguably themost important, case of both problems, where one of the varieties isa hyperplane. We show that a positive solution of the stableequivalence problem in this case implies a positive solution ofthe cancellation problem.     

Superharmonic functions are locally renormalized solutions

We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.