Higher Line Bundles in Geometry and Physics. This project seeks to develop a theory of geometric objects, `higher line bundles', which realise elements of higher dimensional cohomology groups. In particular this project will develop a theory of differential geometry for these objects, allowing one to interpret differential forms representing cohomology classes as the `curvature' of a higher line bundle. This will have applications in quantum field theory and string/brane theory.
Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught ....Global aspects of dualities in String Theory in the presence of background fluxes. String Theory, known to the general public as the "Theory of Everything', is currently an extremely active area of research internationally. It has not only stimulated considerable interaction between mathematical physicists and mathematicians, but also increased public interest in science through television programs and books. Unfortunately, the majority of the Australian scientific community has not yet caught up with these developments. Our recent papers, all published in premier journals in this field, have not only received widespread international attention but have also increased the profile of String Theory amongst Australia's mathematicians and mathematical physicists. The proposed project is expected to continue this trend.Read moreRead less
Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ult ....Twisted K-theory and its application to String Theory and Conformal Field Theory. String Theory is, at present, the only consistent theory of quantum gravity. Recently, twisted K-theory was proposed as the algebraic structure underlying the classification of D-branes, i.e. solitonic extended objects, in certain closed string backgrounds. In this project we aim to advance our understanding of the properties of twisted K-theory in the context of String Theory and Conformal Field Theory. The ultimate goal is to find the appropriate K-theory classifying D-branes in arbitrary closed string backgrounds or, similarly, classifying boundary Conformal Field Theories. It has already emerged that the K-theory of C*-algebras will play an important role.Read moreRead less
Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspect ....Dualities in String Theory and Conformal Field Theory in the context of the Geometric Langlands Program. The Langlands program ties together seemingly unrelated areas of Mathematics. Recently, in the context of the Geometric Langlands correspondence, novel connections with Theoretical Physics have emerged, thus becoming one of the most active areas of research in both Mathematics and Theoretical Physics. Australia has a number of world-renowned experts, including the two CI's, in various aspects of the Langlands program, and is therefore well-placed to make seminal contributions. Being involved in these new developments is of crucial importance to the health of Mathematics and Theoretical Physics in Australia. An integral part of this proposal is student involvement and postgraduate training.Read moreRead less
Fractional analytic index theory. Atiyah-Singer index theory, for which M.F. Atiyah and I.M. Singer received the 2004 Abel Prize, has stimulated considerable interaction between mathematicians and mathematical physicists. An extension of this theory is Fractional Index Theory, co-invented by R.B. Melrose, I.M. Singer and myself, which has received international attention, having solved a fundamental open problem. A central aim in my research project is to extend our theory to elliptic boundary ....Fractional analytic index theory. Atiyah-Singer index theory, for which M.F. Atiyah and I.M. Singer received the 2004 Abel Prize, has stimulated considerable interaction between mathematicians and mathematical physicists. An extension of this theory is Fractional Index Theory, co-invented by R.B. Melrose, I.M. Singer and myself, which has received international attention, having solved a fundamental open problem. A central aim in my research project is to extend our theory to elliptic boundary value problems. I will assist beginners to navigate to the cutting edge of research through workshops, spring-schools and supervision. Benefits include the enhancement of Australia's position in the forefront of international research.Read moreRead less