scholarly journals ON FAMILIES IN DIFFERENTIAL GEOMETRY

2013 ◽  
Vol 10 (09) ◽  
pp. 1350042 ◽  
Author(s):  
GIOVANNI MORENO
Keyword(s):  
Differential Geometry ◽  
Algebraic Topology ◽  
Differential Calculus ◽  
Ad Hoc ◽  
Theoretical Physics ◽  
Homotopy Formula ◽  
Commutative Algebras ◽  
Algebraic Framework ◽  
Fundamental Theorems

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

scholarly journals Editorial

2014 ◽  
Vol 60 ◽  
pp. 1-4
Author(s):  
Trevor Stuart
Keyword(s):  
Differential Geometry ◽  
London Mathematical Society ◽  
Theoretical Physics ◽  
Chern Classes ◽  
Common Currency ◽  
Medical Sciences ◽  
The Usa ◽  
History Of ◽  
Obituary Notice ◽  
Bonnet Theorem

As is usual, the volumes of Biographical Memoirs contain much material of interest to the student of the mathematical, physical, engineering, biological and medical sciences. Often a memoir has been written in collaboration with another Academy or Society. In the present volume the memoir of Shiing-Shen Chern is an expanded version of an obituary notice by Nigel Hitchin that appeared in the Bulletin of the London Mathematical Society . Chern was a great geometer, who revolutionized differential geometry and whose mathematical tools are now common currency in geometry, topology and theoretical physics. His proof of the Gauss–Bonnet theorem, which was a pivotal event in the history of differential geometry, led to the importance of the Chern classes. Moreover. S.-S. Chern was extremely influential in the development of mathematics and geometry both in the USA, at the Institute of Advanced Study, Princeton, and Chicago and Berkeley, and in China, in Shanghai and Nankei.

scholarly journals DIFFERENTIAL GEOMETRY FROM QUANTUM FIELD THEORY

2013 ◽  
Vol 10 (04) ◽  
pp. 1350003
Author(s):  
W. F. CHEN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Differential Geometry ◽  
Historical Development ◽  
Theoretical Physics ◽  
Quantum Field ◽  
Yang Mills ◽  
And Mathematics

We review the historical development and physical ideas of topological Yang–Mills theory and explain how quantum field theory, a physical framework describing subatomic physics, can be used as a tool to study differential geometry. We further emphasize that this phenomenon demonstrates that the inter-relation between theoretical physics and mathematics have come into a new stage.

scholarly journals Shiing-Shen Chern. 26 October 1911 — 3 December 2004

2014 ◽  
Vol 60 ◽  
pp. 75-85
Author(s):  
Nigel J. Hitchin
Keyword(s):  
Differential Geometry ◽  
Qing Dynasty ◽  
Theoretical Physics ◽  
International Congress ◽  
Common Currency ◽  
The Qing Dynasty ◽  
The Common ◽  
Mathematical Sciences ◽  
Set Up ◽  
Research Institute

Shiing-Shen Chern was a towering figure in mathematics, both for his contributions to differential geometry and as a source of inspiration and encouragement for all mathematicians, and particularly those in China. Born in the final year of the Qing dynasty, and educated at a time when China was only beginning to set up Western-style universities, he lived to preside over the 2002 International Congress of Mathematicians in Beijing. He was a co-founder of the Mathematical Sciences Research Institute in Berkeley and its first Director in 1981; he also set up the Nankai Institute for Mathematics in 1985. His contributions to differential geometry were of foundational importance for the global viewpoint that developed in the postwar years, and the mathematical tools he introduced are now the common currency in geometry, topology and even aspects of theoretical physics.

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