Related papers: On the Singular Structure of Graph Hypersurfaces
In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375],…
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…
We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or…
Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological…
Configuration polynomials generalize the classical Kirchhoff polynomial defined by a graph. Their study sheds light on certain polynomials appearing in Feynman integrands. Contact equivalence provides a way to study the associated…
We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have…
We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of…
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main…
Mather and Yau showed that an isolated complex hypersurface singularity is completely determined by its moduli algebra. It is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and,…
Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to…
The sparsity order of a (simple undirected) graph is the highest possible rank (over ${\mathbb R}$ or ${\mathbb C}$) of the extremal elements in the matrix cone that consists of positive semidefinite matrices with prescribed zeros on the…
Let $K$ be a field of characteristic $0$. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial $f$ under the linear action of $\mathrm{GL}_n$ or $\mathrm{SL}_n$, returns a polynomial differing from…
In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary…
The notion of graph cover, also known as locally bijective homomorphism, is a discretization of covering spaces known from general topology. It is a pair of incidence-preserving vertex- and edge-mappings between two graphs, the…
There is a well-established procedure of assigning a strong homotopy Lie algebra of local observables to a multisymplectic manifold which can be regarded as part of a categorified Poisson structure. For a 2-plectic manifold, the resulting…
Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear…
We study hypergraphs which are uniquely determined by their chromatic, independence and matching polynomials. B. Bollob\'as, L. Pebody and O. Riordan (2000) conjectured (BPR-conjecture) that almost all graphs are uniquely determined by…
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…
A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples…