October 16 (Russo) We prove that an operator space is completely isometric to a ternary ring of operators if and only if the open unit balls of all of its matrix spaces are bounded symmetric domains. A holomorphic characterization of C*-algebras is also given. (Joint work with Matthew Neal)

January 22 (Schmalz)There are two types of 6-dimensional CR manifolds of CR-dimension 2 with stable Levi form: hyperbolic and elliptic manifolds. They carry a rich geometric structure that can be studied by Cartan connections and normal forms. In difference to the well-understood hypersurface case
a new sort of invariants, called torsion, appears. This torsion admits an interpretation as obstructions against integrability of almost complex and direct product structures.
 

May 14 (Gioev) The Strong Szego Limit Theorem (SSLT) the determinant of a large Toeplitz matrix. We obtain a third order generalization of SSLT
for a pseudodifferential operator on the unit circle and, more generally, on a Zoll manifold of any dimension. A particular case is a Szego type
asymptotics for an operator of multiplication by a smooth function on the standard sphere of any dimension. This is a refinement of a result by V.Guillemin and K.Okikiolu who have established a second order generalization.The proof uses the method of Guillemin and Okikiolu
and proceeds in the spirit of the combinatorial proof of the classical SSLT by M.Kac. An important role in the proof is played by
a certain combinatorial identity which generalizes the formula of G.A.Hunt and F.J.Dyson to an arbitrary natural power. The original Hunt--Dyson combinatorial formula, for the power one, has been used by M.Kac in the mentioned above proof of the classical SSLT, and also in a computation of the expected value of the maximum of a random walk with independent identically distributed (i.i.d.) steps.  It turns out that the generalized Hunt--Dyson formula is another form of a combinatorial theorem by H.F.Bohnenblust which allowed F.Spitzer to compute the characteristic function of the maximum of a random walk with i.i.d. steps.
 
 
 

May 28 (Friedman) Symmetry is an expression of laws in nature. The principle of relativity defines symmetry between the descriptions of events in two inertial  systems. Without assuming constancy of light, this symmetry implies the Lorenz transformation between these system, interval and some fixed speed conservation and that the ball of all possible velocities is a Bounded Symmetric Domain with respect to the Lorenz group. The axis of this symmetry is determined by so-called symmetric velocity, that is its relativistic half of the relative velocity between the systems. The Lorenz group is represented on the ball of all possible velocities by the group of projective automorphisms of the ball turning this ball into a  Cartan  domain of type I, while on the ball of all possible symmetric velocities the group is represented by the group of conformal  automorphisms of the ball turning this ball into a  Cartan  domain of type
 IV, called also spin factor. In both representations, the generators of this group could be identified with relativistic fields, for instance with
 the electromagnetic field.

The triple product on the spin factor is related to the Geometric product (called also the Clifford product). The latter simplifies
significantly the Maxwell equation, the Dirac equation and other equations
of physics.

The measuring process defines geometry of a state space of a Quantum system. This geometry leads to a ternary product, converting the state
space into a bounded symmetric domain.

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