Related papers: Persistence and the Sheaf-Function Correspondence

In order to define a geometric Fourier transform, one usually works with either $\ell$-adic sheaves in characteristic $p>0$ or with $D$-modules in characteristic 0. If one considers $\ell$-adic sheaves on the stack quotient of a vector…

This writeup describes ongoing work on designing and testing a certain family of correspondences between compact metric spaces that we call \emph{embedding-projection correspondences} (EPCs). Of particular interest are EPCs between spheres…

We formulate a series of conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in $d>2$ spacetime dimensions. We focus on conformal manifolds with limiting points at infinite…

In geometric representation theory, one often wishes to describe representations realized on spaces of invariant functions as trace functions of equivariant perverse sheaves. In the case of principal series representations of a connected…

We find a remarkably simple relationship between the following two models of the tangent space to the Universal Teichm\"uller Space: (1) The real-analytic model consisting of Zygmund class vector fields on the unit circle; (2) The…

The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height…

A morphism from a diagonalizable group $G$ to the torus of a toric variety $X$ induces an action of $G$ on $X$. We prove the category of ind-coherent sheaves on the quotient stack is equivalent to the category of sheaves on a cover of a…

Let $M$ be a complete Riemannian manifold and $F\subset M$ a set with a nonempty interior. For every $x\in M$, let $D_x$ denote the function on $F\times F$ defined by $D_x(y,z)=d(x,y)-d(x,z)$ where $d$ is the geodesic distance in $M$. The…

For a given base scheme, a correspondence is established between a class of sheaves on curves over this base scheme and certain points of infinite Grassmannians. This equivalence extends to a characterization of commutative algebras of…

Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…

A torsion-free sheaf $E$ on a projective variety $X$ is called quasi-trivial if $E^{\vee\vee}=\mathcal{O}_{X}^{\oplus r}$. While such sheaves are always $\mu$-semistable, they may not be semistable. We study the Gieseker--Maruyama moduli…

We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish…

Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The…

The paper deals with partial and weak preference relations defined on infinite-dimensional vector spaces and compatible with algebraic operations. By a partial preference we mean an asymmetric and transitive binary relation, while a weak…

We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the…

We analyse the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of…

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli…

By invoking the reflection functors introduced by Bernstein, Gelfand, and Ponomarev in 1973, in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the…

Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_\Sigma)$…

We prove that various morphisms related to the six Grothendieck operations on sheaves become isomorphisms when restricted to (weakly) constructible sheaves. To this end, we first study some properties of weakly cohomologically constructible…