Related papers: Phylogenetic Algebraic Geometry
We explain how the geometric Langlands program inspires some recent new prospectives of classical arithmetic Langlands program and leads to the solutions of some problems in arithmetic geometry.
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) projectively normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties.…
This paper proposes an algebraic view of trees which opens the doors to an alternative computational scheme with respect to classic algorithms. In particular, it is shown that this view is very well-suited for machine learning and…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
The Persistent-Phylogeny Model is an extension of the widely studied Perfect-Phylogeny Model, encompassing a broader range of evolutionary phenomena. Biological and algorithmic questions concerning persistent phylogeny have been intensely…
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove…
A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called…
This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…
We study plane algebraic curves defined over a field k of arbitrary characteristic as coverings of the the projective line and the problem of enumerating branched coverings of $\mathbb{P}^{1}$ by using combinatorial methods.
This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large.…
A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g. a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous…
The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories,…
Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or…
We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…
Phylogenetics uses alignments of molecular sequence data to learn about evolutionary trees relating species. Along branches, sequence evolution is modelled using a continuous-time Markov process characterised by an instantaneous rate…
In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, initiated previously by some of the authors. It consists of the use of certain objects of Discrete…
The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras…