This grant is for research in applications of Noncommutative Geometry (NCG) to models in theoretical physics. One part focuses on applications to quantum Hall systems and quasicrystals. The goal is to extend the NCG approach to fractional Hall effects. For quasicrystal structures, the aim is to combine algebro-geometric techniques and NCG methods. A second part of the research deals with the use of NCG techniques in spin foams models, as developed in recent work of the PI, with the aim of combining this method with the almost commutative geometries used in the NCG approach to particle physics models. A third part of the proposal deals with extending models of particle physics and cosmology based on NCG, and to connect these models of low energy limits of String Theory. The last part proposes a new point of view, based on the correspondence between BPS states counting and statistical physics model of crystal melting recently developed in String Theory.
Broader impact: The PI was invited speaker in the Mathematical Physics session of the International Congress of Mathematicians, Hyderabad 2010, and plenary speaker in the European Congress of Mathematics, Amsterdam 2008. The PI graduated twelve PhD students over the past five years (eight belonging to ethnic minorities) and is currently supervising seven other graduate students. The PI has also supervised eleven undergraduates at Caltech (four belonging to underrepresented groups) on summer research projects. In the past two years, seven papers were coauthored by the PI with Caltech undergraduates. The PI is also mentoring two postdocs (one from an underrepresented group). For outreach the PI has authored four research monographs and has edited eight volumes arising from conferences, workshops and special programs that she co-organized over the past few years. In addition to these publications and to frequent lectures at workshops and conferences, the PI also delivered five general audience lectures on her research over the past three years.
