Geometric Methods in Degree Theory for Equivariant Maps

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.

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Geometric Methods for Infinite-Dimensional Dynamical Systems

10.21236/ada566477 ◽

2012 ◽

Author(s):

Bjorn Sandstede

Keyword(s):

Dynamical Systems ◽

Infinite Dimensional ◽

Infinite Dimensional Dynamical Systems ◽

Geometric Methods

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Arithmetic L-Functions and Differential Geometric Methods

10.1007/978-3-030-65203-6 ◽

2021 ◽

Keyword(s):

Geometric Methods

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Existence of solutions for functional boundary value problems at resonance on the half-line

Boundary Value Problems ◽

10.1186/s13661-020-01459-5 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Bingzhi Sun ◽

Weihua Jiang

Keyword(s):

Boundary Value Problems ◽

Banach Spaces ◽

Existence Of Solutions ◽

Boundary Value ◽

Coincidence Degree ◽

Degree Theory ◽

Half Line ◽

Projection Scheme ◽

At Resonance ◽

Functional Boundary

Abstract By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with $\operatorname{dim}\operatorname{Ker}L = 1$ dim Ker L = 1 . And an example is given to show that our result here is valid.

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