Related papers: A Bollob\'as-type theorem for affine subspaces

We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations.

The main aim of this paper is to prove a Bishop-Phelps-Bollob\'as type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the…

We prove various results on the size and structure of subsets of vector spaces over finite fields which, in some sense, have too many mutually orthogonal pairs of vectors. In particular, we obtain sharp finite field variants of a theorem of…

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular…

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices…

In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.

We prove finiteness results on integral points on complements of large divisors in projective varieties over finitely generated fields of characteristic zero. To do so, we prove a function field analogue of arithmetic finiteness results of…

In this survey we give an overview of recent developments on the Quantitative Subspace Theorem. In particular, we discuss a new upper bound for the number of subspaces containing the "large" solutions, obtained jointly with Roberto…

We construct explicitly the quantum symplectic affine algebra $U_q(\widehat{sp}_{2n})$ using bosonic fields. The Fock space decomposes into irreducible modules of level -1/2, quantizing the Feingold-Frenkel construction for q=1.

We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of…

We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to…

Let $V$ be an $n$-dimension real vector space with a direct sum decomposition $V = V_1 \oplus \cdots \oplus V_r$. Let $\mathcal{P} = \{(A_i, B_i) : i \in [m]\}$ be a skew Bollob\'as system of subspaces of $V$ such that each $i\in [m]$, $…

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

In this paper, we classify irreducible representations of affine group superschemes over fields $F$ of characteristic not two in terms of those over a separable closure $F^{\mathrm{sep}}$ and their Galois twists. We also compute the…

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues…

A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming…

We generalize to the super context, the known fact that if an affine algebraic group $G$ over a commutative ring $k$ acts freely (in an appropriate sense) on an affine scheme $X$ over $k$, then the dur sheaf $X\tilde{\tilde{/}}G$ of…

We prove a geometric property of the set A^{-1} of inverses of the nonzero elements of an F_q-subspace A of a finite field involving the size of its intersection with two-dimensional F_q-subspaces. We give some applications, including a new…