Related papers: An Approach with Toric Varieties for Singular Lear…
The mathematical modelling of genetic-metabolic networks is of out most importance in the field of systems biology. Different formalisms and a huge variety of classical mathematical tools have been used to describe and analyse such…
Watanabe's singular learning theory provides a framework for asymptotic analysis of Bayesian model selection for statistical models with singularities, where traditional statistical regularity assumptions fail. Learning coefficients, also…
The study of hypersurfaces in a torus leads to the beautiful zoo of amoebas and their contours, whose possible configurations are seen from combinatorial data. There is a deep connection to the logarithmic Gauss map and its critical points.…
We develop a correspondence between the structure of Turing machines and the structure of singularities of real analytic functions, based on connecting the Ehrhard-Regnier derivative from linear logic with the role of geometry in Watanabe's…
This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the $u$-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the…
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 survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.
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…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…
These lecture notes are an introduction to toric geometry. Particular focus is put on the description of toric local Calabi-Yau varieties, such as needed in applications to the AdS/CFT correspondence in string theory. The point of view…
Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have…
In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which…
These lecture notes are written for a PhD mini-course I gave at the CIRM in Luminy in 2019. Their intended purpose was to present, in the context of smooth toric varieties, a relatively self-contained and elementary introduction to the…
Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly…
In the long paper "Family Blowup formula, Admissible Graphs and the Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the enumeration problem of nodal (or general singular) curve counting on algebraic surfaces by…
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component.…
Toric varieties are perhaps the most accessible class of algebraic varieties. They often arise as varieties parameterized by monomials, and their structure may be completely understood through objects from geometric combinatorics. While…
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for…
Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g_1,...,g_n of degree d in B=k[x,y] which parameterize C in a birational, base point free, manner. We study the…