Tur\'an and Ramsey problems for alternating multilinear maps

Guided by the connections between hypergraphs and exterior algebras, we study Tur\'an and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and has origins in the works of Lov\'asz (Proc. Sixth British Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and Feldman and Propp (Adv. Math., 1992). Our main result is a Ramsey theorem for alternating bilinear maps. Given $s, t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$ subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace $W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural group-theoretic (for finite $p$-groups) and geometric (for Grassmannians) implications, and leads to new Ramsey-type questions for varieties of groups and Grassmannians.Comment: 20 pages. v3: rewrite introductio

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.Comment: 46 pages. Final version, to appear in Advances in Mathematic

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe

Lines in Euclidean Ramsey theory

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $K$.Comment: 7 page

The wonderland of reflections

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable \omega-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable $\omega$-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.Comment: 24 page