ImagePinhas Grossman

Email: pinhas AT impa DOT br DOT dontwritethis

I am currently a postdoctoral fellow at IMPA in beautiful Rio de Janeiro, Brazil.

I received my Ph.D. in 2006 at UC Berkeley under the supervision of Vaughan Jones. From 2006-2009 I was an NSF RTG postdoctoral fellow at Vanderbilt University with the group in Noncommutative Geometry and Operator Algebras. In 2009-2010 I was a Marie Curie Research Fellow at the Cardiff node of the EU network in Noncommutative Geometry and Physics.

My work has also been supported by a National Science Foundation grant from the Division of Mathematical Sciences' Analysis Program.

News:

"The Brauer-Picard group of the Asaeda-Haagerup fusion categories", with Noah Snyder

We show that the Brauer-Picard group of Morita auto-equivalences of the fusion categories coming from the Asaeda-Haagerup subfactor is the Klein 4-group. We compute the groupoid of bimodule categories associated to the three fusion categories which appear in the Asaeda-Haagerup subfactor or its index two extension. There are 24 bimodule categories (up to duality) in this groupoid, and about 75 subfactors/planar algebras which realize them.

In another paper with David Jordan, we will show using the extension theory of Etingof-Nikshych-Ostrik that each of the 9 non-trivial auto-equivalences in this groupoid gives a new Z/2Z-graded fusion category whose zero-graded part is one of the original three Asaeda-Haagerup fusion categories.

Supplementary data:

Lists of fusion modules and associated graphs for the fusion rings AH1, AH2, AH3
Lists of fusion bimodules over AHi-AHj, 1<=i,j<=3
Lists of multiplicatively compatible triples of fusion bimodules
Lists of multiplicatively compatible triples of the form right module/bimodule/right module

Gauge transformation data for some connections associated to AH+1
States on certain intertwiner diagrams associated to the algebra object for AH+2

Research:
 
I study von Neumann algebras - more specifically subfactors, planar algebras, and fusion categories.

Algebras of operators on Hilbert space were introduced by von Neumann in the early twentieth century to provide a mathematical framework for quantum mechanics. In the early 1980's Vaughan Jones examined symmetries associated to certain inclusions of von Neumann algebras (subfactors) and found a powerful new knot invariant. This astonishing discovery introduced a new geometric dimension to the theory and revealed deep connections to low dimensional topology, quantum groups, and statistical mechanics.

A subfactor can be viewed as a quantum analogue of a group and is described by a standard invariant, which is a category of bimodule representations. The standard invariant has a rich algebraic and combinatorial structure, but also a strong geometric component: the standard invariant has the structure of a planar algebra, which is an algebra over the operad of planar tangles.

In some cases the invariants of a subfactor also have the structure of fusion categories. A fusion category is a "categorification" of a ring, and its representation theory is described by module categories, or "quantum subgroups."

More generally, I am interested in exploring the planar algebra approach to subfactors and its connections to topology and mathematical physics. I am also interested in quantum algebra, especially fusion categories.
Publications:

The Brauer-Picard group of the Asaeda-Haagerup fusion categories, with Noah Snyder, preprint 2011

Quantum subgroups of the Haagerup fusion categories, with Noah Snyder, accepted by Communications in Mathematical Physics

A quadrilateral in the Asaeda-Haagerup category, with Marta Asaeda, Journal of Quantum Topology, 2, (2011), 269-300

Strong singularity for subfactors, with Alan Wiggins, Bulletin of the LMS, 42, (2010), 607-620

Classification of noncommuting quadrilaterals of factors, with Masaki Izumi, International Journal of Mathematics, 19 (2008), 557-643

Forked Temperley-Lieb algebras and intermediate subfactors, Journal of Functional Analysis, 247 (2007), 477-491

Intermediate subfactors with no extra structure, with Vaughan F. R. Jones, Journal of the AMS, 20 (2007), 219-265

Intermediate subfactors with small index, UC Berkeley doctoral dissertation, 2006

Teaching:
I have taught a number of courses at Vanderbilt University, including applied statistics, differential equations and linear algebra for engineers, probability theory, and calculus. I was also a teaching assistant for several years at UC Berkeley; courses for which I was a TA included calculus (both for majors and for non-majors), complex analysis, and Banach algebras and spectral theory.