|26/09/17||Seminario||15:00||16:00||1101 D'Antoni||Lara BOSSINGER||University of Cologne||Toric degenerations of Grassmannians: birational sequences and the tropical variety
As toric varieties are well understood due to their rich combinatorial structure, a toric degeneration allows to deduce properties of the original variety. For Grassmannians, such degenerations can be obtained from birational sequences and the tropical Grassmannian.
The first were recently introduced by Fang, Fourier, and Littelmann. They originate from the representation theory of Lie algebras and algebraic groups. In our case, we use a sequence of positive roots for the Lie algebra sln to define a valuation on the homogeneous coordinate ring of the Grassmannian. Nice properties of this valuation allow us to define a filtration whose associated graded algebra (if finitely generated) is the homogeneous coordinate ring of the toric variety.
The second was defined by Speyer and Sturmfels and is an example of a tropical variety: a discrete object (a fan) associated to the original variety that shares some of its properties and in nice cases, as the one of Grassmannians, provides toric degenerations. In this talk, I will briefly explain the two approaches and establish a connection between them.
|20/09/17||Seminario||16:00||17:00||1201 Dal Passo||Michael Magee||Durham University||Word measures on unitary groups|
I'll talk about joint work with Doron Puder (Tel Aviv University).Fix a positive integer r, and fix a word w in the freegroup on r generators.Let G be any group. One obtains a 'word map' from the product of r copies of G to G by substituting in elements of G for occurrences of generators in w. We also call this map w.The pushforward of Haar measure under w is called the w-measure on G. We are interested in the case G = U(n), the compact Lie
group of n dimensional unitary matrices. A motivating question of our work is to what extent the w-measures on U(n) determine algebraic properties of the word w. We proved in our first paper that one can detect the 'stable commutator length' of w from these measures. One of our main tools is a formula for Fourier coefficients of
w-measures, which happen for deep reasons to be rational functions of the dimension parameter n. We can now explain all the Laurent coefficients of these rational functions in topological terms. I'll explain all this in my talk, which should be broadly accessible and of general interest. I'll also outline some remaining open questions and explain what we know so far about them.
|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
|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.