|26/07/17||Seminario||16:00||17:00||1201 Dal Passo||Marco Oppio||Universita' di Trento||Quantum theory in real or quaternionic Hilbert space: How the
complex Hilbert space structure emerges from Poincare'
Joint work with: Valter Moretti
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Soler's theorem (1995) closing a long standing problem that can be traced back to von Neumann's mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems (in Wigner's approach) cannot be described in real/quaternionic Hilbert spaces as a consequence of some peculiarity of continuous unitary projective representations of SL(2,C) related with the theory of polar decomposition of operators. Indeed such a "naive" attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the "naive" result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. In conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while there exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is completely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators with quaternionic coeff
|19/07/17||Seminario||14:00||15:00||1201 Dal Passo||Francesco Polizzi||Universita' della Calabria||A pair of rigid surfaces with p_g=q=2 and K^2=8 whose universal
cover is not the bidisk.|
We construct two complex-conjugated rigid surfaces with p_g=q=2 and K^2=8, whose universal cover is not biholomorphic to the bidisk. We show that these are the unique surfaces with these invariants and Albanese map of degree 2, apart a family of product-quotient surfaces constructed by Penegini. This is a joint work with C. Rito and X. Roulleau.
|04/07/17||Seminario||16:30||17:30||1101 D'Antoni||Junjiro Noguchi||Tokyo University||On a weak coherence theorem and Levi problem (Oka VI, IX)
I will begin with some simplifications of the proofs of Oka-Cartan theory after my last lecture at Tor Vergata. Then I will discuss more simplification in Oka's coherence by formulating a ''weak coherence theorem'', which can be proved simply by power series expansions, which goes up to Cousin I, II and Levi problems. I would like also to discuss some historical developements of the solution of Levi problem.
|28/06/17||Colloquium||15:00||16:00||1201 Dal Passo||Tristan Rivière||ETH Zurich||How much does it cost...to turn the sphere inside out ?
How much does it cost...to knot a closed simple curve ? To cover the sphere twice ? to realize such or such homotopy class ? ...etc.
All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis.
In this talk we will concentrate on the operation consisting of everting the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem.
We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.
|20/06/17||Seminario||14:30||15:30||1201 Dal Passo||Livia Corsi||Georgia Tech (Atlanta, USA)||Billiards and rigid rotations|
Probably one of the most famous open problems concerning billiard systems is the Birkhoff conjecture: "If a billiard map is integrabile than the boundary of the billiard table is an ellipse". Recently Treschev conjectured that there might exist analytic billiards, different from ellipses, for which the dynamics in the neighborhood of the period-2 orbit is conjugated to a rigid rotation, suggesting a very interesting example of local integrability for billiard tables different from ellipses. However the result of Treschev is only formal in the sense that he finds only a formal power series. Our aim is to prove the convergence of such series.
This is a joint work (in progress) with M. Procesi.
|20/06/17||Seminario||11:00||13:00||1201 Dal Passo||Rick Miranda||Colorado State University||Matrix reduction approaches to interpolation problems: a review with remarks|
About ten years ago M Dumnicki developed some techniques for interpolation problems related to a careful study of the rank of the matrices involved. Using these techniques he was able to extend the state of the art at the time, bringing certain problems into the range of computer analyses; this enabled him to prove that linear systems in the plane satisfied the SGHH conjecture for homogeneous multiplicities up to 42, the record then. We'll review the method, and consider some refinements, and applications to systems with ten points. The talk should be accessible to non-experts.