The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems

We prove that an $\omega$-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations $\alpha$, $\beta$, $s$ satisfying the identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This establishes an algebraic criterion equivalent to the conjectured borderline between P and NP-complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page

Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials (Representation Theory and Combinatorics)

This article is based on my joint work with Junzo Watanabe [8]. The Lef-schetz property is a ring-theoretic abstraction of the Hard Lefschetz Theorem for compact K\"ahler manifolds. The following are fundamental problems on the study of the Lefschetz property for Artinian graded algebras: Problem 0.1. For a given graded Artinian algebra $A$ , decide whether or not $A$ has the strong (or weak) Lefschetz property. Problem 0.2. When a graded Artinian algebra $A$ has the strong Lefschetz property, determine the set of Lefschetz elements in the part $A_{1}$ of degree one. In this work, we give a characterization of the Lefschetz elements in Ar-tinian Gorenstein rings over a field $k$ of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Goren-stein rings which do not have the strong Lefschetz property

Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.Comment: 50 pages, revised version, accepted for publication on Mathematical Structures in Computer Scienc