Related papers: Graph hypersurfaces and a dichotomy in the Grothen…

We show that any pointed, preordered module map $\mathfrak{BF}_{\mathrm{gr}}(E) \to \mathfrak{BF}_{\mathrm{gr}}(F)$ between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving $\ast$-homomorphism…

We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate…

Gordan and Noether proved in their fundamental theorem that an hypersurface $X=V(F)\subseteq \mathbb{P}^n$ with $n\leq 3$ is a cone if and only if $F$ has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that…

We study the behavior of the Chern classes of graph hypersurfaces under the operation of deletion-contraction of an edge of the corresponding graph. We obtain an explicit formula when the edge satisfies two technical conditions, and prove…

We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on…

To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path…

For a scheme S with a good action of a finite abelian group G having enough roots of unity we show that the quotient map on the G-equivariant Grothendieck ring of varieties over S is well defined with image in the Grothendieck ring of…

Graph complements G(n) of cyclic graphs are circulant, vertex-transitive, claw-free, strongly regular, Hamiltonian graphs with a Z(n) symmetry, Shannon capacity 2 and known Wiener and Harary index. There is an explicit spectral zeta…

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a…

Let F be a field, let G be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k_*(F) to the graded ring gr…

Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties…

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class.…

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras…

We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all…

In the Heisenberg group $\mathbb{H}^1$, equipped with a left-invariant and not necessarily symmetric norm in the horizontal distribution, we provide examples of entire area-minimizing horizontal graphs which are locally Lipschitz in…

We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain…

There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. In this note, we show a similar connection between the geometry of the graph and the structure of a…

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect,…

We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their…

A Lefschetz module is a module over a graded algebra $A$ that satisfies analogues of Poincar\'{e} duality, the Hard Lefschetz property, and the Hodge--Riemann relations with respect to an open convex cone $\mathscr{K}$ in the degree one…