|14/11/17||Seminario||14:30||15:30||1201 Dal Passo||Annalisa Massaccesi||Universitat Zurich||Partial regularity for the hyperdissipative Navier-Stokes equations
In this joint work with Maria Colombo and Camillo De Lellis we prove a space-time partial regularity result à la Caffarelli-Kohn-Nirenberg for suitable weak solutions of the hyperdissipative Navier-Stokes equations.
|07/11/17||Seminario||14:30||15:30||1201 Dal Passo||Paolo Caldiroli||Universita' di Torino||Embedded tori with prescribed mean curvature |
According to a famous result by A.D. Alexandrov, the only embedded, oriented, compact, constant mean curvature (CMC) surfaces in the Euclidean 3-space are round spheres. In particular there is no CMC embedded torus. We investigate the problem of embedded tori for a class of radially symmetric, prescribed mean curvature functions converging to a constant at infinity. Under suitable conditions, we construct a sequence of embedded tori. Such surfaces are close to sections of unduloids with small necksize, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov-Schmidt reduction method and some variational argument. This is a joint work with Monica Musso (Pontificia Universidad Católica de Chile).
|24/10/17||Seminario||14:30||15:30||1201 Dal Passo||Daniele Castorina||John Cabot University (Roma)||Ancient solutions of superlinear heat equations on Riemannian manifolds.|
We study the qualitative properties of ancient solutions of superlinear heat equations in a Riemannian manifold, with particular attention to positivity and triviality in space. This is joint work with Carlo Mantegazza (Napoli ''Federico II'')
|17/10/17||Seminario||14:30||15:30||1201 Dal Passo||Carlo Nitsch||Universita' di Napoli ||Problemi di ottimizzazione in isolamento termico
In questo seminario affrontiamo il problema di isolare
termicamente un corpo conduttore di calore dall'ambiente circostante,
avendo a disposizione una certa quantità di isolante da distribuire sulla
sua superficie. La definizione di "isolamento ottimo" dipenderà dalla
presenza o meno di una sorgente di calore e darà luogo a due formulazioni
distinte. Dimostreremo che la maniera migliore di disporre l'isolante
sulla superficie del conduttore non è sempre ovvia e a volte addirittura
controintuitiva. Anche nel caso più semplice, in cui il conduttore sia una
palla, e ci troviamo in assenza di sorgente di calore, distribuire
l'isolante uniformemente sulla superficie sferica non é sempre la cosa
migliore da fare.
|10/10/17||Seminario||14:30||15:30||1201 Dal Passo||Roberto Peirone||Universita' di Roma "Tor Vergata"||Existence of self-similar energies on finitely ramified fractals|
An important problem in analysis on fractals is the construction of a self-similar energy on the fractal. An old conjecture is whether on a specific important class of finitely ramified self-similar fractals there exists a self-similar energy.
In this talk, it is shown a recent example where we have no self-similar energy. On the other hand, a self-similar energy always exists if the
self-similarity is considered with respect to a suitable set of maps that define the fractal.
|10/10/17||Seminario||13:00||14:00||1201 Dal Passo||Israel Vainsencher||Universidade Federal de Minas Gerais||Counting singular surfaces
How many surfaces of a given degree present singularities of some specified type and pass through an appropriate number of points?
We focus on counting singular surfaces with certain non isolated singularities: e.g., Whitney's umbrella, quartics singular along atwisted cubic, etc. We give a proof for the polynomial nature of the formulae and make it explicit in a few cases. Conjecturally the degree of the formula is twice the dimension of the family of curves imposed in the singular locus. We manage to bound it by thrice that dimension. We draw essentially from previous joint work with Angelo Lopez and Fernando Cukierman.
|03/10/17||Seminario||16:00||17:00||1101 D'Antoni||Peter Heinzner||Ruhr-Universitaet Bochum||KAEHLERIAN REDUCTION|
In this talk we will consider Hamiltonian actions of groups of holomorphic Kaehler
isometries on Kaehlerian manifolds. In the rare cases where the orbit spaces are
smooth it is well known that the corresponding quotient spaces in the sense of Marsen
Weinstein are Kaehler manifolds as well.
We will show that in the general case the quotient has a natural structure of a
Kaehler space. The main tool in the proof is the construction of invariant Kaehler
potentials for not necessarily compact group actions.
|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