Related papers: Topographs for binary quadratic forms and class nu…
In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on…
We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named \c{c}ark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the \c{c}ark.…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…
A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We…
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…
Recently a new family of enumerative invariants called leaky Hurwitz numbers was introduced by Cavalieri-Markwig-Ranganathan in the context of logarithmic intersection theory. They admit an interpretation via tropical covers where the…
On conformally compact manifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and extend aspects of already known…
We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent…
In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch…
We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
We discuss here geometric structures of condensed matters by means of a fundamental topological method. Any geometric pattern can be universally represented by a decomposition space of a topological space consisting of the infinite product…
Special generic maps are generalizations of Morse functions with exactly two singular points on spheres and canonical projections of unit spheres. They restrict the manifolds of the domains strongly in considerable cases and are important…
Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in…
We develop various aspects of classical enumerative geometry, including Euler characteristics and formulas for counting degenerate fibres in a pencil, with the classical numerical formulas being replaced by identitites in the…