## IGA/AMSI Workshop |

Abstract

Dirac introduced his eponymous operator to describe electrons in quantum theory. It was rediscovered by Atiyah and Singer in their study of the index problem on manifolds. In these lectures we explore new theorems and applications. Several of these also involve K-theory in its recent*twisted* and *differential* variations.

Titles of individual lectures

* Loop groups and Dirac families*

I will describe the finite and infinite-dimensional versions of the construction which yields my theorem with Hopkins and Teleman. This will include some generalities about loop groups and their representations.

* Introduction to twisted K-theory*

After a review of standard K-theory I will discuss the twisted form and some properties. Then I will compute twisted K_G(G) in some cases.

* A TQFT from twisted K-theory*

I will discuss TQFTs in general and give some examples, etc. I will show how to construct the ring structure on twisted K_G(G).

* Differential K-theory and the Atiyah-Singer theorem*

An introductory lecture on differential cohomology theories in general and differential K-theory in particular. Then the differential index theorem with Lott.

* Anomalies and "categorified" index theorems*

A general lecture on some physics, then some discussion of anomalies in supersymmetric quantum mechanics and on the worldsheet of string theory.

Dirac introduced his eponymous operator to describe electrons in quantum theory. It was rediscovered by Atiyah and Singer in their study of the index problem on manifolds. In these lectures we explore new theorems and applications. Several of these also involve K-theory in its recent

Titles of individual lectures

I will describe the finite and infinite-dimensional versions of the construction which yields my theorem with Hopkins and Teleman. This will include some generalities about loop groups and their representations.

After a review of standard K-theory I will discuss the twisted form and some properties. Then I will compute twisted K_G(G) in some cases.

I will discuss TQFTs in general and give some examples, etc. I will show how to construct the ring structure on twisted K_G(G).

An introductory lecture on differential cohomology theories in general and differential K-theory in particular. Then the differential index theorem with Lott.

A general lecture on some physics, then some discussion of anomalies in supersymmetric quantum mechanics and on the worldsheet of string theory.

Michael Batanin (Macquarie University)

Arun Ram (University of Melbourne)

David Ridout (Australian National University)

Frédéric Rochon (Australian National University)

Anne Thomas (University of Sydney)

Craig Westerland (University of Melbourne)

There will be no registration fees: all are welcome. However, if you are interested in attending, kindly send an e-mail to Snigdhayan Mahanta by

Please also indicate in your e-mail whether you would like to attend the workshop dinner (Tuesday evening, 19 Oct), which will be $5 per person.

This event is co-sponsored by the Australian Mathematical Sciences Institute (AMSI). AMSI allocates a travel allowance to each of its member universities (for a list of members, click here). Students or Early Career Researchers from AMSI member universities without access to a suitable research grant or other source of funding may apply to their Head of Mathematical Sciences for subsidy of travel and accommodation out of the departmental travel allowance. No other funding is available.

Participants, other than the invited speakers, are asked to make their own travel and accommodation arrangements. For a list of budget hotels in Adelaide, please click here.

**Organisers****
**

Prof Mathai Varghese School of Mathematical Sciences University of Adelaide North Terrace, Adelaide, SA 5005 Phone: (08) 8303 4173 Fax: (08) 8303 3696 E-mail: mathai.varghese@adelaide.edu.au |
Prof Peter Bouwknegt Department of Theoretical Physics Research School of Physics and Engineering and Department of Mathematics Mathematical Sciences Institute Australian National University Canberra, ACT 0200 Phone: (02) 6125 2969 Fax: (02) 6125 4676 E-mail: peter.bouwknegt@anu.edu.au |
Dr Pedram Hekmati School of Mathematical Sciences University of Adelaide North Terrace, Adelaide, SA 5005 Phone: (08) 8303 5261 Fax: (08) 8303 3696 E-mail: pedram.hekmati@adelaide.edu.au |
Dr Snigdhayan Mahanta School of Mathematical Sciences University of Adelaide North Terrace, Adelaide, SA 5005 Phone: (08) 8303 1604 Fax: (08) 8303 3696 E-mail: snigdhayan.mahanta@adelaide.edu.au |

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