The Index Theorem for Homogeneous Differential Operators

1994 ◽  
pp. 163-183
Author(s):  
Raoul Bott
Keyword(s):  
Differential Operators ◽  
Index Theorem

scholarly journals Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space

1994 ◽  
Vol 135 ◽  
pp. 165-196 ◽  
Author(s):  
Masatake Miyake ◽  
Masafumi Yoshino
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Index Theorem ◽  
Fredholm Property ◽  
Goursat Problem ◽  
Point Of View ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Gevrey Spaces

In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

scholarly journals Operator Symbols and Operator Indices

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 64
Author(s):  
Vladimir Vasilyev
Keyword(s):  
Banach Spaces ◽  
Differential Operators ◽  
Index Theorem ◽  
Symbolic Calculus ◽  
Smooth Manifolds ◽  
Bounded Operators ◽  
Pseudo Differential Operators ◽  
Operator Symbols ◽  
Special Classes

We suggest a certain variant of symbolic calculus for special classes of linear bounded operators acting in Banach spaces. According to the calculus we formulate an index theorem and give applications to elliptic pseudo-differential operators on smooth manifolds with non-smooth boundaries.

scholarly journals Sir Michael Atiyah OM. 22 April 1929–11 January 2019

2020 ◽  
Author(s):  
Nigel Hitchin
Keyword(s):  
Differential Operators ◽  
Global Analysis ◽  
Index Theorem ◽  
Theoretical Physics ◽  
Yang Mills ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Fields Medal ◽  
Elliptic Differential Operators ◽  
Mathematical Sciences

Michael Atiyah was the dominant figure in UK mathematics in the latter half of the twentieth century. He made outstanding contributions to geometry, topology, global analysis and, particularly over the last 30 years, to theoretical physics. Not only was he held in high esteem at a worldwide level, winning a Fields Medal in 1966, the Abel Prize in 2004 and innumerable other international awards, but his irrepressible energy and broad interests led him to take on many national roles too, including the presidency of the Royal Society, the mastership of Trinity College, Cambridge, and the founding directorship of the Isaac Newton Institute for Mathematical Sciences. His most notable mathematical achievement, with Isadore Singer, is the index theorem, which occupied him for over 20 years, generating results in topology, geometry and number theory using the analysis of elliptic differential operators. Then, in mid life, he learned that theoretical physicists also needed the theorem and this opened the door to an interaction between the two disciplines that he pursued energetically until the end of his life. It led him not only to mathematical results on the Yang--Mills equations that the physicists were seeking, but also to encouraging the importation of concepts from quantum field theory into pure mathematics.

scholarly journals An index theorem for end-periodic operators

2015 ◽  
Vol 152 (2) ◽  
pp. 399-444 ◽  
Author(s):  
Tomasz Mrowka ◽  
Daniel Ruberman ◽  
Nikolai Saveliev
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Differential Operators ◽  
Riemannian Metrics ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Fredholm Theory ◽  
Eta Invariant ◽  
First Order ◽  
Periodic Operators

We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

RENORMALIZED TRACES AS A LOOKING GLASS INTO INFINITE-DIMENSIONAL GEOMETRY

2001 ◽  
Vol 04 (02) ◽  
pp. 221-266 ◽  
Author(s):  
SYLVIE PAYCHA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Differential Operators ◽  
Index Theorem ◽  
Wodzicki Residue ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Geometric Concepts ◽  
The One ◽  
Definition Of

This paper, based on results obtained in recent years with various coauthors,1–3,13,53 presents a proposal to extend some classical geometric concepts to a class of infinite-dimensional manifolds such as current groups and to a class of infinite-dimensional bundles including the ones arising in the family index theorem. The basic idea is to extend the notion of trace underlying many geometric concepts using renormalized traces which are linear functionals on pseudo-differential operators. The definition of "renormalized traces" involves extra data on the manifolds or vector bundles, namely a weight given by a field of elliptic operators which becomes part of the geometric data, leading to the notion of weighted manifold and weighted vector bundle. This weight is a source of anomaly arising typically as a Wodzicki residue of some pseudo-differential operator. We investigate the anomalies that arise when trying to extend to the infinite-dimensional setting classical results of finite-dimensional geometry such as a Weitzenböck formula, Chern–Weil invariants or the relation between the first Chern form on a complex vector bundle and the curvature on the associated determinant bundle. When comparable, we relate our approach to the one adopted for similar problems in noncommutative geometry.

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