String theory is arguably the most exciting research area in modern mathematical physics. It is known to the general public as the "Theory Of Everything", thanks to its great success in unifying Relativity and Quantum Field Theory, yielding Quantum Gravity theory. For a cool video on string theory, based on Brian Greene's bestseller book entitled "The elegant universe", click here. The impact of string theory is not just felt in physics, but it also has profound interactions with a broad spectrum of modern mathematics, including noncommutative geometry, K-theory and index theory. The pioneers of string theory were M.B. Green and J.H. Schwarz and the leading figure in the field is E. Witten.

**String theory and contemporary culture:** Vibrating strings, similar to the vibrations of violin strings, give rise to fundamental particles such as quarks and gluons, which in turn are the constituents of protons, neutrons, and electrons etc., establishing a close connection between music and string theory. Since science drives our economy, changing our culture by creating a popular awareness and understanding of science is an important goal of ours. An example is the recent consultancy role of Bouwknegt and myself on a production of the Australian Dance Theatre entitled
*Multiverse*, where the main choreographer, Garry Stewart, has used duality in string theory to generate ideas for a wonderful dance piece on both artistic and intellectual levels.
*Multiverse* premiered at the Space Theatre, Adelaide Festival Centre on
10 July 2014 and we were honoured guests.

The theory of D-branes forms an important part of string theory.
It arises as the T-dual of open strings on a circle bundle,
where the open strings in the dual theory are no longer free to move
everywhere in space, but
are endowed with Dirichlet boundary conditions so that
the endpoints are free to move only on a submanifold
known as a D-brane.
For a link describing the mathematics behind D-branes, cf.
superstrings. Such D-branes come with (Chan-Paton) vector bundles, and therefore
their charge determines an element of K-theory, as was argued by Minasian-Moore.
In the presence of a nontrivial B-field but whose
Dixmier-Douady class is a torsion element of H^{3}(M, **Z**), Witten
argued that D-branes no longer carry honest vector bundles, but they
have a twisted or gauge bundle.
In the presence of a nontrivial B-field whose
Dixmier-Douady class is a general element of H^{3}(M, **Z**), it was
proposed in **[BM]** that
D-brane charges in type IIB
string theories are measured by the twisted K-theory that was
described earlier by Rosenberg, and the twisted bundles
on the D-brane world-volumes were elements in this twisted
K-theory. In **[BCMMS]**, using bundle gerbes and their modules,
a geometric interpretation of elements of twisted K-theory
was obtained, and the the Chern-Weil representatives of the
Chern character was studied. This was further generalized to the equivariant
and the holomorphic cases in **[MS]**.
The relevance of the equivariant case to
conformal field theory was highlighted by the remarkable
result of Freed, Hopkins and Teleman that
the twisted G-equivariant K-theory of a compact
connected Lie group
G (with mild hypotheses) is graded isomorphic to the Verlinde
algebra of G, with a shift given by the dual Coxeter number
and the curvature of the B-field, where we
recall that
Verlinde algebra of a compact connected Lie group G is defined in
terms of positive energy representations of the loop group of G,
and arises naturally in physics in Chern-Simons theory which is
defined using quantum groups and conformal field theory.

Type I D-branes in the presence of an H-flux are studied
in **[MMS]**, where a geometric interpretation of H^{2}(M,**Z**_{2})
is given in terms of stable isomorphisms of real bundle gerbes,
and the twisted KO theory is interpreted geometrically in terms
of real projective vector bundles.

One development is the novel discovery in **[BEM], [BEM2]** of
T-duality isomorphisms in twisted K-theory and twisted cohomology
and the character formulae relating these.
Briefly, T-duality defines an isomorphism
between the twisted K-theory of the total space of a circle bundle,
to the twisted K-theory of the total space of a "T-dual" circle bundle
with "T-dual" twist,
and with a change of parity. Similar statements hold for twisted cohomology.
One interesting consequence is that
a spacetime and its T-dual spacetime can be *topologically different*!
To quote Edward Witten, "*There was a long history of speculation that in quantum gravity,
unlike Einstein's classical theory, it might be possible for the
topology of spacetime to change*." Thus our research work can be viewed
as realizing this speculation
in the context of T-duality in String Theory with background H-flux.

Another outcome of our work is that we can construct *fusion type products*
in twisted K-theory and twisted cohomology,
whenever the twist is a decomposable cohomology class.
Another interesting consequence of our work is that it
gives convincing evidence that a type IIA string theory A on a circle
bundle of radius R in the presence of an *H*-flux,
and a type IIB string theory B on a "T-dual" circle bundle of
radius 1/R in the presence of a "T-dual" *H*-flux, are equivalent in the
sense that the string states of string theory A are in
canonical one to one correspondence
with the string states of string theory B.
This is a fundamental property of type II string theories that was
predicted only in special cases earlier.

**[BHM]** studies the more general case of T-duality for principal torus
bundles. The new phenomenon that occurs here is that not all H-fluxes
are T-dualizable, and this paper works out the precise class of
T-dualizable H-fluxes. The isomophisms in twisted K-theory and
twisted cohomology also follow in this case.

In **[MR]**, we give a complete characterization
of T-duality on principal 2-torus-bundles with H-flux.
As noticed in **[BHM]** for instance, principal
torus bundles with H-flux do not necessarily have a T-dual which is a
torus bundle. A big puzzle has been to explain these
mysterious
``missing T-duals.'' Here we show that this problem is
resolved using noncommutative topology. It turns out that
every principal 2-torus-bundle with H-flux does
indeed have a T-dual, but in the missing cases (which we characterize),
the T-dual is non-classical and is
a bundle of *noncommutative tori*.
This suggests an unexpected
link between classical string theories and the ``noncommutative'' ones,
obtained by ``compactifying'' along noncommutative tori.

In **[MR2]**, we give a complete characterization
of T-duality for general principal torus-bundles with H-flux,
generalizing the results in **[MR]** to higher rank torus bundles.
The striking new feature in the case when the rank of the torus bundle
is greater than or equal to 3 is that not every such torus bundle has a
T-dual, either classical or nonclassical. The simplest example is
the rank 3 torus over a point. We also define the action of the
T-duality group GO(n, n, **Z**) on T-dual pairs of principal
torus bundles, where n
is the rank of torus bundle, where GO(n, n, **Z**) is the subgroup
of GL(2n, **Z**) that preserves the bilinear pairing upto sign.
All of T-dual pairs in a given orbit of GO(n, n, **Z**) define
physically equivalent type II string theories.

February 2008 - January 2012, Editor in charge of "Global Analysis,
Noncommutative Geometry, and the Mathematics of String Theory",
* Proceedings of the American Mathematical Society*.

__Recent invited talks in String Theory__

Invited Lecture-series, RIMS International Conference on
Noncommutative Geometry and Physics, RIMS, Kyoto University, Kyoto, Japan,
November 8-12, 2010.

Invited Lecture-series, Global Centre of Excellence (GCOE), Tohoku University, Sendai, Japan, November 13-20, 2010.

Invited speaker, 1st PRIMA conference, UNSW, Sydney, July 6-10, 2009.

Invited speaker, NSF/CBMS Regional Conference in the Mathematical Sciences: Topology, C*-algebras, and String Duality. Texas Christian University, Fort Worth, Texas, USA May 18-22, 2009.

Invited speaker, Center of Excellence (COE) Program on "Exploring New Science by Bridging Particle-Matter Hierarchy", Tohoku University, Sendai, Japan, 13-15 December 2007.

I gave a keynote lecture at the special session on Mathematical Physics, at the 51st Annual Meeting of the Australian Mathematical Society, 25-28 September 2007, La Trobe University, Melbourne.

I gave a couple of plenary lectures at the Nishinomiya-Yukawa Memorial Symposium held at Nishinomiya, Japan, 11-12 November 2006, and at Kyoto, Japan, 13-15 November 2006.

I organized an IGA satellite workshop to "Mathematics of String Theory 2006" 27 July 2006.

I co-organized a major international AMSI workshop entitled, Mathematics of String Theory 2006 which was held at ANU, Canberra, 13-23 July, 2006.

From March - June, 2006, I took up a post as Senior Research Fellow at the Erwin Schrodinger Institute for Mathematical Physics in Vienna, Austria. I gave a course of advanced graduate lectures entitled "K-theory applied to Physics". About half of the course was on applications of K-theory to String Theory.

I am an invited participant at the KITP (Santa Barbara) program, "Mathematical Structures in String Theory", October 17 - November 12, 2005. The link to the video and slides of my talk is here.

I gave a talk entitled "Global aspects of T-duality in string theory" at the MFO, Oberwolfach (Germany) Mini-Workshop "Gerbes, Twisted K-Theory and Conformal Field Theory", 14-20 August 2005.

I gave an invited lecture series entitled, "Twisted K-theory and global aspects of T-duality in the presence of background flux" at the 2005 summer school entitled, "Geometric and Topological methods for Quantum Field Theory" which was held at Villa de Leyva, Colombia, July 7 - July 19, 2005.

I've been an invited speaker, in May 2004 to the Erwin Schrodinger Institute for a conference entitled "Mathematical aspects of branes in Calabi-Yau spaces", in June 2004 at Luminy, Marseille for a conference entitled "Geometry of index theory and quantum field theory", in July 2004 as a principal lecturer at Bialowieza, Poland for a conference entitled "Workshop on Geometric Methods in Physics", in September 2004 to the Oberwolfach Mathematische Institut for a conference entitled "Noncommutative Geometry".

I was invited to give a lecture series entitled "K-theory and Physics", February 22 until March 3, 2003, at IUPUI, Indianapolis and also the neighbouring universities in Bloomington and Wabash.

I was an invited speaker at the "String Theory Conference 2002" in Hangzhou, China August 12-15, 2002 and in Beijing, China August 17-19, 2002. I also co-organized an international workshop on String theory in January 2002, and am a co-organizer for an international String theory conference in 2003.

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