University of Rome Sapienza

In the first part of the seminar I will introduce shallow neural-networks from a statistical-mechanics perspective, focusing on simple cases and on a naive scenario where information to be learnt is structureless. Then, inspired by biological information processing, I will enrich this framework by accounting for structured datasets and by making the network able to perform challenging tasks like generalization or even "taking a nap”. Results presented are both analytical and numerical.

Soft active materials are largely employed to realize devices (actuators), where deformations and displacements are triggered by a wide range of external stimuli such as electric field, pH, temperature, and solvent absorption. The effectiveness of these actuators critically depends on the capability of achieving prescribed changes in their shape and size and on the rate of changes. In particular, in gel–based actuators, the shape of the structures can be related to the spatial distribution of the solvent inside the gel, to the magnitude and the rate of solvent uptake.

In the talk, I am going to discuss some results obtained by my group regarding surface patterns arising in the transient dynamics of swelling gels [1,2], based on the stress diffusion model we presented a few years ago [3]. I am also going to show our extended stress diffusion model suited for investigating swelling processes in fiber gels, and to discuss shape formation issues in presence of fiber gels [4-6].

[1]   A. Lucantonio, M. Rochè, PN, H.A. Stone. Buckling dynamics of a solvent-stimulated stretched elastomeric sheet. Soft Matter 10, 2014.

[2]   M. Curatolo, PN, E. Puntel, L. Teresi. Full computational analysis of transient surface patterns in swelling hydrogels. Submitted, 2017.

[3]   A. Lucantonio, PN, L. Teresi. Transient analysis of swelling-induced large deformations in polymer gels. JMPS 61, 2013.

[4]   PN, M. Pezzulla, L. Teresi. Anisotropic swelling of thin gel sheets. Soft Matter 11, 2015.

[5]   PN, M. Pezzulla, L. Teresi. Steady and transient analysis of anisotropic swelling in fibered gels. JAP 118, 2015.

[6]   PN, L. Teresi. Actuation performances of anisotropic gels. JAP 120, 2016.

According to Witten a gauge theory with a mass gap contains a possibly trivial Topological Field Theory  (TFT) in the infrared.  We show that in SU(N) YM it there exists a trivial TFT defined by   twistor Wilson loops whose v.e.v. is 1 in the large-N limit for any shape of the loops supported on certain Lagrangian submanifolds of space-time that lift to Lagrangian submanifolds of twistor space.

We derive a new version of the Makeenko-Migdal loop equation for the topological twistor Wilson loops, the holomorphic loop equation, that involves the change of variables in the YM functional integral from the connection to the anti-selfdual part of the curvature and the choice of a holomorphic gauge.

Employing the holomorphic loop equation and viewing Floer homology the other way around,
we associate to arcs asymptotic in both directions to the cusps of the Lagrangian submanifolds the critical points of an effective action implied by the holomorphic loop equation. The critical points of the effective action, being associated to the homology of the punctured Lagrangian submanifolds, consist of surface operators of the YM theory, supported on the punctures.  The correlators of surface operators in the TFT satisfy for large momentum the constraint that follows by the renormalization group and by the asymptotic freedom and they are saturated by an infinite sum of pure poles of scalar and pseudoscalar glueballs, whose joint spectrum is exactly linear in the mass squared.

For several physical purposes we outline  a related construction of a twistorial Topological String Theory dual to the TFT, that involves the Chern-Simons action on Lagrangian submanifolds of  
twistor space.

We discuss new symmetry results for nonconstant entire local minimizers of the standard Ginzburg-Landau functional  for maps in ${H}^{1}_{\rm{loc}}(\mathbb{R}^3;\mathbb{R}^3)$ satisfying a natural energy bound.

Up to  translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit map equivariant under the action of the orthogonal group.

More generally, for any $N\geq 3$ we  characterize the $O(N)-$equivariant vortex solution for Ginzburg-Landau type equations in the $N-$dimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional.