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Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications.…

Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in…

We prove a version of Ihara's Lemma for degree q=1,2 cuspidal cohomology of the symmetric space attached to automorphic forms of arbitrary weight (k\geq 2) over an imaginary quadratic field with torsion (prime power) coefficients. This…

For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an…

We construct two-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent…

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C…

Let $p$ denote the characteristic of ${\mathbb F}_q$, the finite field with $q$ elements. We prove that if $q$ is odd then an arc of size $q+2-t$ in the projective plane over ${\mathbb F}_q$, which is not contained in a conic, is contained…

We resolve an algebraic version of Schoenberg's celebrated theorem [Duke Math.J., 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider…

The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a…

Given a field $F$ of characteristic 2, we prove that if every three quadratic $n$-fold Pfister forms have a common quadratic $(n-1)$-fold Pfister factor then $I_q^{n+1} F=0$. As a result, we obtain that if every three quaternion algebras…

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which…

We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a…

Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements, which satisfies $|E| > q$. There exist $x,y \in E$ such that $|E \cdot (y-x)| > q/2.$ In particular, $(E+E) \cdot (E-E) =…

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…

We prove that if $\mathbb{F}$ is an algebraically closed field of zero characteristic which has infinite transcendence degree over $\mathbb{Q}$, then there exists a field automorphism $\varphi$ of ${\rm SL}_n(\mathbb{F})$ and ${\rm…

We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $\Delta(F_i)\leq \Delta$ hold…

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve,…

We consider point sets in the $m$-dimensional affine space $\mathbb{F}_q^m$ where each squared Euclidean distance of two points is a square in $\mathbb{F}_q$. It turns out that the situation in $\mathbb{F}_q^m$ is rather similar to the one…

Let $F$ be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer…

Let k be a field of characteristic not 2, let q be a quadratic space over k and let f be an irreducible polynomial with coefficients in k. In 1969, Milnor raised the following question : how can we decide whether q has an isometry with…