### Description

OVERVIEW:

A grant is requested to cover the travel and lodging expenses for speakers and participants of the program "Emphasis Year in Noncommutative Geometry" at Northwestern University in the academic year 2018-2019.

INTELLECTUAL MERIT:

The subject of non-commutative geometry dates its origin back to early 1980s. A major part of it is cyclic homology that was invented independently in 1981 by A. Conn es and B. Tsygan. Since then, non-commutative geometry has been extensively developing.

In ordinary geometry, the notion of "space" is commonly identified with that of smooth manifolds. Many problems in mathematics or physics do, however, require one to pass to more general "noncommutative

spaces". The idea of non-commutative geometry is to study geometry via algebras of functions on "non-commutative manifolds". On such a "non-commutative manifold", the relevant objects are no longer points in a space, but rather an associative algebra, which may not be commutative.

It turns out that many notions from "commutative" differential geometry do extend to noncommutative geometry. Cyclic homology is an excellent example of that kind providing for a noncommutaive replacement of the de Rham theory. On the other hand, cyclic homology is closely related to K-theory and index theory, thereby allowing one to solve various problems in these fields.

BROADER IMPACTS:

Noncommutative geometry is connected to many branches of mathematics and mathematical physics. The proposed program aims to promote Interaction between mathematicians, physicists and groups working on related areas from different aspects. The project will include introductory graduate courses, as well as workshops on recent research by leading experts.

The PIs expect that these activities will bring together established mathematicians and younger researchers working on these rapidly developing subjects. There are many bright young people working in the area of noncommutative geometry. The planned activities will be great opportunities for them to disseminate their ideas and to broaden their perspectives. The PIs anticipate inviting many mathematicians at the postdoctoral and graduate-student levels to these events. The school

will provide an excellent opportunity for young American scientists to exchange ideas with their colleagues in other countries and to jump-start collaborations.

Stimulating and encouraging young participants is one of the core goals of the Noncommutative Geometry Emphasis Year in the Northwestern University. The PIs will be particularly supportive of the participation of women and other underrepresented groups. The activities will provide opportunities for young participants to consult with senior experts.

A grant is requested to cover the travel and lodging expenses for speakers and participants of the program "Emphasis Year in Noncommutative Geometry" at Northwestern University in the academic year 2018-2019.

INTELLECTUAL MERIT:

The subject of non-commutative geometry dates its origin back to early 1980s. A major part of it is cyclic homology that was invented independently in 1981 by A. Conn es and B. Tsygan. Since then, non-commutative geometry has been extensively developing.

In ordinary geometry, the notion of "space" is commonly identified with that of smooth manifolds. Many problems in mathematics or physics do, however, require one to pass to more general "noncommutative

spaces". The idea of non-commutative geometry is to study geometry via algebras of functions on "non-commutative manifolds". On such a "non-commutative manifold", the relevant objects are no longer points in a space, but rather an associative algebra, which may not be commutative.

It turns out that many notions from "commutative" differential geometry do extend to noncommutative geometry. Cyclic homology is an excellent example of that kind providing for a noncommutaive replacement of the de Rham theory. On the other hand, cyclic homology is closely related to K-theory and index theory, thereby allowing one to solve various problems in these fields.

BROADER IMPACTS:

Noncommutative geometry is connected to many branches of mathematics and mathematical physics. The proposed program aims to promote Interaction between mathematicians, physicists and groups working on related areas from different aspects. The project will include introductory graduate courses, as well as workshops on recent research by leading experts.

The PIs expect that these activities will bring together established mathematicians and younger researchers working on these rapidly developing subjects. There are many bright young people working in the area of noncommutative geometry. The planned activities will be great opportunities for them to disseminate their ideas and to broaden their perspectives. The PIs anticipate inviting many mathematicians at the postdoctoral and graduate-student levels to these events. The school

will provide an excellent opportunity for young American scientists to exchange ideas with their colleagues in other countries and to jump-start collaborations.

Stimulating and encouraging young participants is one of the core goals of the Noncommutative Geometry Emphasis Year in the Northwestern University. The PIs will be particularly supportive of the participation of women and other underrepresented groups. The activities will provide opportunities for young participants to consult with senior experts.

Status | Active |
---|---|

Effective start/end date | 3/1/19 → 2/29/20 |

### Funding

- National Science Foundation (DMS-1839515)

### Fingerprint

Noncommutative Geometry

Cyclic Homology

Physics

Index Theory

Associative Algebra

Smooth Manifold

Differential Geometry

K-theory

Date

Replacement

Jump

Branch

Cover

Algebra

Interaction