In this paper we give a refinement of the Burgess bound for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to Friedlander, Iwaniec, and Kowalski. In particular, for any nontrivial multiplicative character $\chi$ modulo a prime $q$ and any integer $r⩾2$, we show that $${\sum }_{M<n⩽M+N}\chi \left(n\right)=O\left(N^{1-1/r}{q}^{(r+1)/4r^2}\right(logq{)}^{1/4r}),$$ which sharpens the previous results by a factor $(logq{)}^{1/4r}$. Our improvement comes from averaging over numbers with no small prime factors rather than over an interval as in the previous approaches.

Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize $2$-dimensional $CAT\left(0\right)$ cube complexes and are a square analog of systolic complexes. We introduce and study the basic properties of these complexes. Using a form of dismantlability for the $1$-skeleta of finite quadric complexes, we show that every finite group acting on a quadric complex stabilizes a complete bipartite subgraph of its $1$-skeleton. Finally, we prove that $\mathrm{C}\left(4\right)\text{-}\mathrm{T}\left(4\right)$ small cancelation groups act on quadric complexes.

We give bounds for various homological invariants (including Castelnuovo–Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the nondescribing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

We prove that the $\mathrm{Aut}$-invariant word norm on right-angled Artin and right-angled Coxeter groups is unbounded (except in few special cases). To prove unboundedness, we exhibit certain characteristic subgroups. This allows us to find unbounded quasi-morphisms which are Lipschitz with respect to the $\mathrm{Aut}$-invariant word norm.

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph $\mathcal{F}$, whose vertices are reduced Gaussian rationals in ${\mathbb{Q}}_{\infty }(i)$ and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in ${\mathbb{H}}^3$ where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of ${\mathbb{H}}^3$ by Farey octahedrons. A natural distance $\varrho$ on ${\mathbb{Q}}_{\infty }(i)$ is also recalled, where $\varrho (\infty ,w)=n$ is the least number of edges in $\mathcal{F}$ from $\infty$ to $w\in \mathbb{Q}(i)$, where $n$ is called the generation of w and a relevant path in $\mathcal{F}$ is called a geodesic expansion for $w$. The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation $n-1$, $n$, and $n+1$. Subsequently, it is seen that there can be at most four Farey neighbors of generation $n-1$ in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any $w$ is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

We prove Fujita’s freeness conjecture for Gorenstein complexity-one $T$-varieties with rational singularities.

In this article, we determine all equivariant compactifications of the three-dimensional vector group ${\mathbf{G}}_a^3$ that are smooth Fano threefolds with Picard number greater than or equal to two.

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the $4$-genus having their origins in the Murasugi–Tristram inequality, and at the same time extends previously known results about concordance invariance of the signature to a bigger set of allowed variables. Finally, we show that the multivariable signature and nullity are also invariant under $0.5$-solvable cobordism.

Lenstra introduced the notion of Euclidean ideal classes for number fields to study cyclicity of their class groups. In particular, he showed that the class group of a number field with unit rank greater than or equal to one is cyclic if and only if it has a Euclidean ideal class. The only if part in the above result is conditional on the extended Riemann hypothesis. Graves and Murty showed that one does not require the extended Riemann hypothesis if the unit rank of the number field is greater than or equal to four and its Hilbert class field is abelian over rationals. In this article, we study real cubic and quadratic fields with cyclic class groups and show that they have a Euclidean ideal class under certain conditions.