Related papers: Fixed point theorems involving numerical invariant…

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in…

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of…

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some…

We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant…

We prove the basic properties of determinantal semi-invariants for presentation spaces over any finite dimensional hereditary algebra over any field. These include the virtual generic decomposition theorem, stability theorem and the…

Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a neighborhood of…

We construct Steinberg summands of $G$-equivariant spectra with $\mathrm{GL}_n(\mathbb{F}_p)$-action. We prove a lemma about their fixed points when $G$ is a $p$-group, and then use this lemma to compute the fixed points of the Steinberg…

We establish that smooth, geometrically integral projective varieties of small degree are not pointless in suitable solvable extensions of their field of definition, provided that this field is algebraic over $\Bbb Q$.

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of…

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism…

Assume a fixed point v in a G-module V can be separated from zero by a homogeneous invariant of degree dp^r where p>0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by…

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \…

We consider rational representations of a connected linear algebraic group $\mathbb G$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto \Pi(\mathbb G)_M$ to $\mathbb G$-modules of the…

We establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices. The conclusions are stronger if such a mapping also respects matrix similarities, i.e., is a…

We give an axiomatic characterization of the fixed point index of an $n$-valued map. For $n$-valued maps on a polyhedron, the fixed point index is shown to be unique with respect to axioms of homotopy invariance, additivity, and a splitting…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A Boardman-Vogt style homotopy invariance result about algebras over…

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational…

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and…

We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the…