Related papers: First-order model theory and Kaplansky's stable fi…

Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles $P$ over $K$. It is…

The Grothendieck-Serre conjecture predicts that on a regular local ring, no nontrivial reductive torsor becomes trivial over the fraction field. While this conjecture has been proven in the equicharacteristic case, it remains open in the…

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces…

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral…

We define and study a class of subshifts of finite type (SFTs) defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. The main…

This paper investigates situations where a property of a ring can be tested on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff…

Let $p$ be a fixed prime number, and $q$ a power of $p$. For any curve over $\mathbb{F}_q$ and any local system on it, we have a number field generated by the traces of Frobenii at closed points, known as the trace field. We show that as we…

Our result contains as special cases the Frobenius theorem (1895) on the~number of solutions to the equation $x^n=1$ in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer…

Let $\Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $g\geq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of…

I study definable sets in affine continuous logic. Let $T$ be an affine theory. After giving some general results, it is proved that if $T$ has a first order model, its extremal theory is a complete first order theory and first order…

In 1994, the second author and W. van der Kallen showed that the injective stabilization bound for K_1 of general linear group is d+1 over a regular affine algebra over a perfect C_1-field, where d is the krull dimension of the base ring…

We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a…

Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or…

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…

We prove that the first-order theory of any function field K of characteristic p>2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field,…

We show that every separable simple tracially approximately divisible $C^*$-algebra has strict comparison, is either purely infinite, or has stable rank one. As a consequence, we show that every (non-unital) finite simple ${\cal Z}$-stable…

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e.,…

We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois…

For a finite group $G$ acting faithfully on a finite dimensional $F$-vector space $V$, we show that in the modular case, the top degree of the vector coinvariants grows unboundedly: $\lim_{m\to\infty} \topdeg F[V^{m}]_{G}=\infty$. In…

This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. The Lichtenbaum-Quillen Conjecture (now implied by the Voevodsky-Rost Theorem) attempts to describe the algebraic K-theory of rings of integers…