Development Of Algebraic Geometry Research Articles

Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects

The workshop covered recent developments in algebraic geometry in a broad sense with a special emphasis on various moduli spaces. Problems related to mirror symmetry phenomena were discussed in a number of talks as well as singularity theory in the context of the MMP in positive characteristic. Derived categories and algebraic cycles, as well as rationality questions figured prominently in the talks and the discussions.

Classical Algebraic Geometry

Progress in algebraic geometry often comes through the introduction of new tools and ideas to tackle the classical problems in the development of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as the derived category, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as the transition from varieties to schemes or from schemes to stacks. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, with a view toward their application to the classical problems.

Classical Algebraic Geometry

Progress in algebraic geometry often comes through the introduction of new tools and ideas to tackle the classical problems in the development of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as the derived category, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as the transition from varieties to schemes or from schemes to stacks. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, with a view toward their application to the classical problems.

Classical Algebraic Geometry

Progress in algebraic geometry often comes through the introduction of new tools and ideas to tackle the classical problems the development of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as the derived category, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as the transition from varieties to schemes or from schemes to stacks. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, with a view toward their application to the classical problems.

Classical Algebraic Geometry

Progress in algebraic geometry usually comes through the introduction of new tools and ideas to tackle the classical problems of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as Voisin’s “existence of decomposition of the diagonal”, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as stacks, tropical geometry, and log structures. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, in the context of their application to the classical problems.

Classical Algebraic Geometry

Progress in algebraic geometry often comes through the introduction of new tools and ideas to tackle the classical problems the development of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as the derived category, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as the transition from varieties to schemes or from schemes to stacks. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, with a view toward their application to the classical problems.

Remarks on the relations between the Italian and American schools of algebraic geometry in the first decades of the 20th century

In this paper we give an overview of the interactions between Italian and American algebraic geometers during the first decades of the 20th century. We focus on three mathematicians—Julian L. Coolidge, Solomon Lefschetz, and Oscar Zariski—whose relations with the Italian school were quite intense. More generally, we discuss the importance of this influence in the development of algebraic geometry in the first half of the 20th century.

Discrete versions of continuous isoperimetric problems

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.

Changing canons of mathematical and physical intelligibility in the later 17th century

Learning to use the new calculus in the late 17th century meant looking at quantities and configurations, and the relationships among them, in fundamentally new ways. In part, as Leibniz argued implicitly in his articles, the new concepts lay along lines established by Viète, Fermat, Descartes, and other “analysts” in their development of algebraic geometry and the theory of equations. But in part too, those concepts drew intuitive support from the new mechanics that they were being used to explicate and that was rapidly becoming the primary area of their application. So it was that the world machine that emerged from the Scientific Revolution could be both mechanically intelligible and mathematically transcendental.

The algebraist’s upper half-plane

Introduction. The purpose of this article is to introduce the general mathematical community to some recent developments in algebraic geometry and nonarchimedean analysis. Let r = p,p a rational prime. Then these developments center around the beginnings of an theory of the polynomial ring ¥r[T] over the finite field of r elements. The goal of this theory is to use nonarchimedean analysis to do for Yr[T] what classical analysis does for Z. The theory allows us to find direct analogues of many of the classical functions of arithmetic interest in a situation that, at first glance, seems as nonclassical as possible. In the process much will be learned about the polynomials. Much also will be learned about the unique properties of Z and the classical functions. One of the exciting aspects of the theory is its great generality. Indeed, we could replace ¥r[T] with much more general affine rings of curves over finite fields. More precisely, if C is a projective, smooth curve over Fr, oo a rational point and A the functions regular away from oo, then we may use A instead of ¥r[T]. Thus one can, so to speak, get a sense of what analysis might have been forced to if Z were not a unique factorization domain. Such observations can only come in the present setting since Q is the only totally real (i.e., all Galois conjugates contained in R) field with a unique absolute-value. We have chosen to stick to the polynomials in order to keep the exposition as simple as possible. The jump from the polynomials to more general rings is not terribly large and most essential features appear for Fr[T]. Another exciting aspect is that we begin to see how a given 'arithmetic* situation generates an associated harmonic analysis. As classical harmonic analysis is based on the integers, the one developed here is based on Fr[T]. In contrast to classical harmonic analysis which is multiplicative, i.e., based on the exponential function, the one here is based on addition. Throughout the paper we compare the theory here with the classical one. In this fashion we hope the reader may speedily develop a feel for the subject. One of the most surprising (and hotly contested) aspects of classical analysis is its harmonic analysis. This centered around the possibility of expanding an arbitrary singly-periodic function in terms of sines and cosines. Since sines and cosines are easily expressed in terms of the exponential function, the central role of this function is apparent. Viewed on the complex plane, e has the following very well-known properties: It is never zero, takes addition to multiplication, is invariant under z H> z + 2777 and, finally, it is its own derivative. As a consequence, e gives

The Historical Development of Algebraic Geometry

The Historical Development of Algebraic Geometry

The Early Development of Algebraic Geometry

The Early Development of Algebraic Geometry