Related papers: Stratified Picard--Lefschetz theory
We study a class of $p$-Laplacian Dirichlet problems with weights that are possibly singular on the boundary of the domain, and obtain nontrivial solutions using Morse theory. In the absence of a direct sum decomposition, we use a…
We introduce a class of "Lipschitz horizontal" vector fields in homogeneous groups, for which we show equivalent descriptions, e.g. in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness…
The notion of local equivalence relation on a topological space is generalised to that of local subgroupoid. The main result is the construction of the holonomy and monodromy groupoids of certain Lie local subgroupoids, and the formulation…
We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present a homology theory for such systems which is based on the dimension group in the…
We propose a new stratification of the reduced subschemes of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties that highlights the relation between the geometry of these spaces and the action of the associated automorphism group.…
We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is…
A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real…
We consider a functional calculus for compact operators, acting on the singular values rather than the spectrum, which appears frequently in applied mathematics. Necessary and sufficient conditions for this singular value functional…
We establish curious Lefschetz property for generic character varieties of Riemann surfaces conjectured by Hausel, Letellier and Rodriguez-Villegas. Our main tool applies directly in the case when there is at least one puncture where the…
Inspired by the work of Bell on the dynamical Mordell-Lang conjecture, and by family Floer cohomology, we construct p-adic analytic families of bimodules on the Fukaya category of a monotone or negatively monotone symplectic manifold,…
This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection…
The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional $p$-sub-Laplacian over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the…
Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\Delta(F)$ on the barycentric subdivision $\Delta(L)$. The constructed function $\Delta(F)$ "behaves the same way" as $F$, i. e. has…
We study the real spectrum compactification of character varieties of finitely generated groups in semisimple Lie groups. This provides a compactification with good topological properties, and we interpret the boundary points in terms of…
We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are…
We prove a relative version of the Picard-Lefschetz theorem, describing the variation of relative homology groups $H_d(Y_t \setminus A_t,B_t\setminus A_t)$ in the fibers of a smooth fiber bundle $Y \to T$ of complex manifolds with $A\cup B…
We investigate smooth approximations of functions, with prescribed gradient behavior on a distinguished stratified subset of the domain. As an application, we outline how our results yield important consequences for a recently introduced…
We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge--Amp\`ere masses. They are characterized in terms of Green functions…
Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical…
We consider hypoelliptic symbols over a very regular Lie group and discuss monodromy for a spectral stratification using results of Nilsson and B\"acklund.