Review Lecture Periods of integrals and topology of algebraic varieties

1984 ◽  
Vol 391 (1801) ◽  
pp. 231-254
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Topology ◽  
Major Theme ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Algebraic Functions ◽  
Several Variables ◽  
And Topology ◽  
Satisfactory Theory

A major theme of nineteenth century mathematics was the study of integrals of algebraic functions of one variable. This culminated in Riemann’s introduction of the surfaces that bear his name and analysis of periods of integrals on cycles on the surface. The creation of a correspondingly satisfactory theory for functions of several variables had to wait on the development of algebraic topology and its application by Lefschetz to algebraic varieties. These results were refined by Hodge’s theory of harmonic integrals. A closer analysis of Hodge structures by P. A. Griffiths and P. Deligne in recent years has led to unexpectedly strong restrictions on the topology of the variety and to a diversity of other applications. This advance is closely linked to the study of variation of integrals under deformations, particularly in the neighbourhood of a singular point.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Optimization of Utility Functions in an Admissible Space of Higher Dimension

2016 ◽  
pp. 102-113
Author(s):  
German Almanza ◽  
Victor M. Carrillo ◽  
Cely C. Ronquillo
Keyword(s):  
Graduate Students ◽  
Linear Algebra ◽  
Algebraic Topology ◽  
Morse Theory ◽  
Utility Functions ◽  
Higher Dimension ◽  
Differential Topology ◽  
Several Variables

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.

Nets of Conics in the Euclidean Plane and an Associated Representational Geometry

1973 ◽  
Vol 16 (4) ◽  
pp. 479-495
Author(s):  
R. Blum ◽  
A. P. Guinand
Keyword(s):  
Nineteenth Century ◽  
Euclidean Plane ◽  
Algebraic Curves ◽  
Algebraic Varieties

The study of systems of conies and other algebraic curves was initiated in the middle of the nineteenth century by Cayley, Hesse, Cremona, and others. Most of the investigations from that time to the present have been concerned with extensions to algebraic varieties and systems of higher orders or dimensions, or with associated algebraic curves such as Jacobians and Hessians.

scholarly journals On singular sets of flat holomorphic mappings with isolated singularities

1978 ◽  
Vol 70 ◽  
pp. 47-80
Author(s):  
Hideo Omoto
Keyword(s):  
Algebraic Geometry ◽  
Singular Point ◽  
Critical Points ◽  
Euler Number ◽  
Holomorphic Mappings ◽  
Algebraic Varieties ◽  
Isolated Singularities ◽  
General Fiber ◽  
Singular Sets ◽  
Algebraic Maps

In [4] B. Iversen studied critical points of algebraic mappings, using algebraic-geometry methods. In particular when algebraic maps have only isolated singularities, he shows the following relation; Let V and S be compact connected non-singular algebraic varieties of dimcV = n, and dimc S = 1, respectively. Suppose f is an algebraic map of V onto S with isolated singularities. Then it follows thatwhere χ denotes the Euler number, μf(p) is the Milnor number of f at the singular point p, and F is the general fiber of f : V → S.

C-Nodal Surfaces of Order Three

1983 ◽  
Vol 35 (1) ◽  
pp. 68-100
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Surface ◽  
Partial Classification ◽  
Differentiability Condition ◽  
Nodal Surfaces

The problem of describing a surface of order three can be said to originate in the mid-nineteenth century when A. Cayley discovered that a non-ruled cubic (algebraic surface of order three) may contain up to twenty-seven lines. Besides a classification of cubics, not much progress was made on the problem until A. Marchaud introduced his theory of synthetic surfaces of order three in [9]. While his theory resulted in a partial classification of a now larger class of surfaces, it was too general to permit a global description. In [1], we added a differentiability condition to Marchaud's definition. This resulted in a partial classification and description of surfaces of order three with exactly one singular point in [2]-[5]. In the present paper, we examine C-nodal surfaces and thus complete this survey.

scholarly journals Topological complexity of collision-free motion planning on surfaces

2010 ◽  
Vol 147 (2) ◽  
pp. 649-660 ◽  
Author(s):  
Daniel C. Cohen ◽  
Michael Farber
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Main Tool ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Topological Complexity ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Lusternik Schnirelmann ◽  
Mixed Hodge Structures

AbstractThe topological complexity$\mathsf {TC}(X)$is a numerical homotopy invariant of a topological spaceXwhich is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category ofX. Given a mechanical system with configuration spaceX, the invariant$\mathsf {TC}(X)$measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space ofndistinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.

5. Flavours of topology

2019 ◽  
pp. 90-114
Author(s):  
Richard Earl
Keyword(s):  
19Th Century ◽  
Set Theory ◽  
Algebraic Topology ◽  
General Topology ◽  
Major Theme ◽  
Henri Poincaré ◽  
Combinatorial Topology ◽  
Differential Topology ◽  
French Mathematician ◽  
Modern Mathematics

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.

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