Remus Nicoara
 
Associate Professor
UTK Mathematics

 


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Remus Nicoara - Research


 

    In the 1930's, John von Neumann discovered that certain algebras of operators on a Hilbert space are the natural framework for understanding symmetries of quantum physical systems. His ideas play an important role in quantum mechanics, and fundamental laws of nature such as the Heisenberg uncertainty principle appear as a natural consequence of von Neumann's abstract theory.

   
    In the early 1980's, Vaughan Jones introduced the theory of subfactors, as a Galois theory for inclusions of von Neumann algebras. Subfactor theory quickly became one of the most flourishing branches of operator algebra theory, with a multitude of deep connections in knot theory, representation theory, 3-manifolds, quantum groups, ergodic theory, integrable systems in statistical mechanics and conformal field theory.


    A subfactor can be viewed as a group-like object that encodes the symmetries of a quantum physical or mathematical situation. To decode this information, one computes the higher relative commutants, a system of inclusions of finite dimensional C*-algebras (matrix algebras) naturally associated to the subfactor. This object, called the Standard Invariant,
was axiomatically described through the work of Sorin Popa. The Standard Invariant has an extraordinarily rich algebraic-combinatorial structure, generalizing finitely generated groups, finite dimensional Hopf C*-algebras and other large classes of quantum groups.


    My main research interest lies in the study of subfactors, especially through their algebraic-combinatorial invariants such as the so-called commuting squares. These are squares of inclusions of finite dimensional C*-algebras that arise naturally in the standard invariant of a subfactor. Commuting squares can also be used as construction data for subfactors, and the most explicit examples of subfactors have been obtained this way. A particular class of subfactors arises from the so-called spin models, which are commuting squares based on complex Hadamard matrices. In the recent years Hadamard matrices have found applications in several areas of mathematics and physics, such as quantum information theory, operator algebras, error correcting codes, spectral sets and Fuglede's conjecture.


    Another related direction of my research concerns applications of Kazhdan's property (T) and of Sorin Popa's Deformation-Rigidity techniques in the context of the standard invariant of a subfactor.