Related papers: Geometries enumeratives complexe, reelle et tropic…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of…
In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
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.
We study the topological complexity, in the sense of Smale, of three enumerative problems in algebraic geometry: finding the 27 lines on cubic surfaces, the 28 bitangents and the 24 inflection points on quartic curves. In particular, we…
In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on…
The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine $n$-space is determined by the integer tangent vectors. Tropical isomorphisms are affine transformations preserving the integer lattice of the…
This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…
This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of…
Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and…
A brief survey of real algebraic structures on topological spaces is given. This article is written for the Gokova Gemetry/Topology Conference proceedings.
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…
We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods to…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…
We study the algebraic structure of the semigroup of all $2 \times 2$ tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the…