Related papers: A new phantom on a rational surface
Given a collection of points in the plane, classifying which subsets are collinear is a natural problem and is related to classical geometric constructions. We consider collections of points in a projective plane over a finite field such…
This article is a revised, short and english version of my PhD thesis. First, we show a mirror theorem : the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective space is isomorphic to the one attached to…
We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular…
A new combinatorial object, called generalised nice set, is classified up to collineations of the Fano plane. This classification is necessary to find the graded contractions of all the exceptional complex Lie algebras of dimension at least…
Extending the discovery by Giles Gardam of a concrete counterexample to Kaplansky's unit conjecture in characteristic 2, a family of counterexamples for every prime characteristic is presented.
In this paper, we present a number of examples of k-nets, which are special configurations of lines and points in the projective plane. Such a configuration can be regarded as the union of k completely reducible elements of a pencil of…
We study the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and virtual (Gromov-Witten) counts are found to agree asymptotically in the Fano (and some…
We construct a new non-desarguesian projective plane from a complex analytic structure. At the same time the construction can be explained in terms of so called Hrushovski's construction. This supports the hypothesis that in general…
We prove that a generalized Schwarzschild-like ansatz can be consistently employed to construct $d$-dimensional static vacuum black hole solutions in any metric theory of gravity for which the Lagrangian is a scalar invariant constructed…
We give an explicit counterexample to the Bunkbed Conjecture introduced by Kasteleyn in 1985. The counterexample is given by a planar graph on $7222$ vertices, and is built on the recent work of Hollom (2024).
Viewing a fan as a partially ordered set (of cones) we consider a category of sheaves on the fan which corresponds to a category of equivariant sheaves on the corresponding toric variety if the fan is rational. In this category we define an…
In this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain "universal extension classes" for the general linear supergroup. Some of these classes restrict to the universal…
The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…
This is a revised version of a part of the author's preprint "On p-adic uniformization of fake projective planes" (preprint, Max-Planck-Institut fuer Mathematik, 1998 (121)). In this paper we construct explicitly a Shimura surface of…
We provide new equivalent conditions for an algebra $\Lambda$ to be $g$-finite, analogous to those established by L. Demonet, O. Iyama, and G. Jasso, but within the category of projective presentations $\mathcal{K}^{[-1,0]}(\text{proj}…
We provide a framework for abstract reconstruction problems using the $K$-theory of categories with covering families, which we then apply to reformulate the edge reconstruction conjecture in graph theory. Along the way, we state some…
Consider the Fano manifold $X$ formed by blowing up $\mathbb{P}^n$ along its linear subspace $\mathbb{P}^r$, we check the conifold conditions [3, 1] for its mirror Laurent polynomial $f$, which can imply that $X$ satisfies both Conjecture…
We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue…
Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). J\'anos Pach (1981) answered this question in the negative. We…
We present a new family of IR dualities in three space-time dimensions with eight supercharges. In contrast to 3d mirror symmetry, these dualities map Coulomb branches to Coulomb branches and Higgs branches to Higgs branches in the deep IR.…