Related papers: $m$-Rigidity and Finite-One Degrees Inside Typical…

Fields with only finitely many maximal subrings are completely determined. We show that such fields are certain absolutely algebraic fields and give some characterization of them. In particular, we show that the following conditions are…

Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\pd M < \infty$, we study the asymptotic intersection multiplicity $\chi_\infty(M, A/\underline{x})$, where $\underline{x} = (x_1, \ldots, x_r)$ is a…

In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete $n$-manifold $M$ of Ricci curvature, $\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $\pi$, then $M$ is isometric to the unit sphere…

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has…

In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is even, $m(s,t)\in\{2\}\cup 4\N$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd,…

Andrews, Brietzke, R\o dseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$…

L-Infinity structures have been a subject of recent interest in physics, where they occur in closed string theory and in gauge theory. This paper provides a class of easily constructible examples of $L_n$ and $L_{\infty}$ structures on…

Let $\mathcal{A}$ be a mathematical structure with an additional relation $R$. We are interested in the degree spectrum of $R$, either among computable copies of $\mathcal{A}$ when $(\mathcal{A},R)$ is a "natural" structure, or (to make…

Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef…

Lance Bryant noticed in his thesis that there was a flaw in our paper "Associated graded rings of one-dimensional analytically irreducible rings", J. Algebra 304 (2006), 349-358. It can be fixed by adding a condition, called the BF…

We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the…

Separations among the first order logic ${\cal R}ing(0,+,*)$ of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory.…

We prove that $n$-dimensional smooth hypersurfaces of degree $n+1$ are superrigid. Starting with the work of Fano in 1915, the proof of this Theorem took 100 years and a dozen researchers to construct. Here I give complete proofs, aiming to…

We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq…

Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square Sq_{B/A} M, which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction…

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible…

Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely\u \i\ curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from…

We introduce a general bipartite-like representation and Schmidt decomposition of an arbitrary pure state of $N$ indistinguishable fermions, based on states of $M<N$ and $(N-M)$ fermions. It is directly connected with the reduced $M$- and…

We settle a conjecture by Bik and Marigliano stating that the degree of a one-dimensional discrete model with rational maximum likelihood estimator is bounded above by a linear function in the size of its support, therefore showing that…

We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 -…