Seminars 2010-2011

Wannier functions, Bloch bundles and the Marzari-Vanderbilt localization functional

Speaker: Prof. Adriano Pisante (Roma la Sapienza)
Time: Wed. July 27, 2011, 2.00 p.m.
Place: SISSA - Santorio A - room 133

Many properties of electrons in crystals are described by Schrodinger operators with periodic potentials. The construction of exponentially localized generalized eigenfunctions is very important, but for theoretical and for computational purposes. A natural base, both for single and for multiband systems is given by (composite) Wannier functions. Existence is already known by methods from obstruction theory. In a work in collaboration with Gianluca Panati, we construct exponentially localized Wannier functions by a variational approach, i.e. by minimization of the Marzari-Vanderbilt localization functional.

Brownian motion and elasticity in biological filaments and networks

Speaker: Prof. Prashant Purohit (University of Pennsylvania)
Time: Fri. July 22, 2011, 11.00 a.m.
Place: SISSA - Santorio A - room 133

It is well known that biofilaments at thermodynamic equilibrium under the action of forces and moments fluctuate around their minimum energy configuration due to Brownian motion. This remains true of filaments in networks and gels such as those of actin, spectrin, fibrin or other biopolymers. The thermal motion of these filaments at the microscopic scales manifests itself as entropic elasticity at the macroscopic scales. In this talk we present a theory to efficiently calculate the thermo-mechanical properties of fluctuating heterogeneous filaments and networks. The central problem is to evaluate the partition function and free energy of heterogeneous filaments and networks under the assumption that their energy can be expressed as a quadratic function in the kinematic variables. We analyze the effects of various types of boundary conditions on the fluctuations of filaments and show that our results are in agreement with recent work on homogeneous rods as well as experiments and simulations. We apply similar ideas to filament networks and calculate the area expansion modulus and shear modulus for hexagonal networks. We also apply our methods to study partially unfolded proteins and the consequences of unfolding on the macroscopic behavior of fibrin networks.

Willmore minimizers with prescribed isoperimetric ratio

Speaker: Dr. Johannes Schygulla (Freiburg)
Time: Wed. June 29, 2011, 3.15 p.m.
Place: SISSA - Santorio A - room 134

Variation formulas for the sub-riemannian area in contact 3 manifolds and applications

Speaker: Dr. Matteo Galli (Granada)
Time: Wed. June 29, 2011, 2.15 p.m.
Place: SISSA - Santorio A - room 134

Quasi-periodic solutions of Hamiltonian PDEs

Speaker: Prof. Massimiliano Berti (Napoli)
Time: Mon. June 27, 2011, 2.50 p.m.
Place: SISSA - Santorio A - room 134

New recent results and techniques about KAM theory for PDEs will be presented.

Existence and regularity results for stationary solutions of the pseduo-relativistic Schrodinger equation

Speaker: Prof. Vittorio Coti Zelati (Napoli)
Time: Mon. June 27, 2011, 2.00 p.m.
Place: SISSA - Santorio A - room 134

We discuss some existence and regularity results of ground states for the pseduo-relativistic Schrodinger equation. The results have been obtained in collaboration with M. Nolasco (L'Aquila).

Symplectic capacity and pseudo-holomorphic curves - III.

Speaker: Dr Pancholi Dishant Mayurbhai (ICTP)
Time: Thu. June 16, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 136

Symplectic Geometry Seminar IX - This is the last part of the seminar in which it is discussed the proof of Gromov's compactness theorem and its applications towards measuring symplectic capacity of a symplectic manifold.

Symplectic capacity and pseudo-holomorphic curves - II.

Speaker: Dr Pancholi Dishant Mayurbhai (ICTP)
Time: Thu. June 9, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 134

Symplectic Geometry Seminar VIII - This is the second part of the seminar in which it is discussed the proof of Gromov's compactness theorem and its applications towards measuring symplectic capacity of a symplectic manifold.

Symplectic capacity and pseudo-holomorphic curves.

Speaker: Dr Pancholi Dishant Mayurbhai (ICTP)
Time: Wed. June 1, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 136

Symplectic Geometry Seminar VII - We outline main ideas involved in the proof of Gromov's compactness theorem and its applications towards measuring symplectic capacity of a symplectic manifold. The seminar will be divided into two parts; the second part will be discussed next week.

Some explicit solutions to a system of implicit partial differential equations

Speaker: Prof. Paolo Marcellini
Time: Fri. May 27, 2011, 3.00 p.m.
Place: SISSA - Santorio A - room 133

Please click here for the abstract.

Lagrange multipliers and Lagrange submanifolds, Maslov index and Morse index.

Speaker: Dr Davide Barilari, Dr Antonio Lerario
Time: Thu. May 26, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Symplectic Geometry Seminar VI - In this 1 hour seminar we complete the discussion on the Maslov index. In the first part we study the set of the set of Lagrange multipliers (critical points of a constrained variational problem). In the second one we discuss the relation between the Morse index and the Maslov index.

The Maslov index and the Morse index

Speaker: Prof. Andrei Agrachev
Time: Thu. May 19, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Symplectic Geometry Seminar V - In this seminar we continue the discussion on the Maslov index and we explore its relation with the Morse index.

Singularities of linear waves and Fermat principle

Speaker: Dr. Ilya Bogaevsky
Time: Thu. May 19, 2011, 2.30 p.m.
Place: SISSA - Santorio A - room 133

Wave fronts of a hyperbolic system of two linear partial differential equations in three- dimensional space are described by the Fermat principle of least time if the admissible velocities at any point form an ellipse.   We describe some typical singularities of such fronts. It turns out there are two absolutely different cases: elliptic and hyperbolic. In particular, the sub-Riemannian sphere appear in the elliptic case and I am going to describe what happens in the hyperbolic case.

The symplectic group and the Maslov index

Speaker: Dr. Davide Barilari
Time: Thu. May 12, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Symplectic Geometry Seminar IV - We study the structure of the symplectic group of a linear symplectic space, proving that it contains the unitary group as a maximal compact subgroup. Then we introduce the Maslov index and its relation with the Lagrange Grassmannian. The discussion on the Maslov index will be completed in the next seminar.

The geometry of the Lagrangian Grassmannian

Speaker: Dr. Davide Barilari
Time: Thu. May 5, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Symplectic Geometry Seminar III - We briefly recall some notions on symplectic groups and of linear symplectic geometry. Then we introduce the Lagrangian Grassmannian, i.e. the manifold of the Lagrangian n-planes in a 2n dimensional symplectic space, giving different geometric characterizations. The seminar is intended also for who did not attended the previous one.

Lagrangian submanifolds and applications to symplectomorphisms

Speaker: Dr. Antonio Lerario
Time: Thu. April 28, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Symplectic Geometry Seminar II - We discuss Weinstein Tubular Neighborhood theorem. We will show some examples of Lagrangian submanifolds and their link with symplectomorphisms. The seminar is intended also for who did not attended the previous one.

The regularity of the minimum time function via nonsmooth analysis and geometric measure theory

Speaker: Dr. Nguyen Tien Khai
Time: Mon. April 18, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Please click here for the abstract.

Homotopy methods in Symplectic Geometry

Speaker: Dr. Antonio Lerario
Time: Thu. April 14, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Symplectic Geometry Seminar I - We briefly discuss Moser argument and its implications for the problem of equivalence of two symplectic manifolds. (Moser relative Theorem, Darboux Theorem, Weinstein Tubular Neighborhood)

Generalized existence of isoperimetric regions in noncompact Riemannian manifolds

Speaker: Dr. Stefano Nardulli (Università di Palermo)
Time: Thu. April 14, 2011, 2.30 p.m.
Place: SISSA - Santorio A - room 133

Introduction to patchy feedback controls: motivations, ideas and techiques - IV

Speaker: Dr. Fabio Priuli
Time:Wed. March 30, 2011, 11.00 a.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 4 lectures. For the abstract see the here.

Time-variant constraints as controls in Mechanics: the holonomic and non-holonomic cases

Speaker: Dr. Franco Rampazzo (Università di Padova)
Time: Wed. March 30, 2011, 4.00 p.m.
Place: SISSA - Santorio A - room 133

The holonomic case. Let q = (q1;...; qN; qN+1;...; qN+M) be a (local) system of coordinates for a N + M-dimensional mechanical system. One can pre-determine the motion of the last coordi- nates u(t) = (qN+1; .... ; qN+M)(t) and regard it as a control (with suitable mechanical assumptions). From the mechanical point of view this is nothing but giving time-dependent state constraints. The derivative u_ of the control u appears in the equations of the free coordinates (q1;... ; qN) linearly or quadratically, depending on the relation between the kinetic metric and the foliation made of the leaves fu = cg. The non holonomic case. If one adds non-holonomic constraints to the original N+M-dimensional mechanical system the e ffect of using the coordinates u(t) = (qN+1; ...; qN+M)(t) as controls is more involved. In particular a new quadratic term in \dot u shows up in the equations of the free co- ordinates (q1; : : : ; qN). This gives an explanation to phenomena of motion caused by rapid oscilla- tions of the controls, for example in the case of the so called Roller Racer.

Heat kernel asymptotics at the cut locus

Speaker: Dr. Robert Neel (Lehigh University)
Time: March 9, 2011, 4.30 p.m.
Place: SISSA - Santorio A - room 133

We give small-time asymptotic expansions for the gradient and Hessian of the logarithm of the heat kernel at the cut locus of a Riemannian manifold. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic interpretation in terms of the Brownian bridge. In particular, we can characterize the cut locus in terms of the behavior of the log-Hessian. We also mention how the distributional asymptotics can be used to compute the distributional Hessian of the energy function.

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

Speaker: Dr. Jean-Jacques Marigo (Ecole Polythechnique – ParisTech)
Time:Wed. March 9, 2011, 3.00 p.m.
Place: SISSA - Santorio A - room 133

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we show also that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analysis is illustrated by examples.

Introduction to patchy feedback controls: motivations, ideas and techiques - III

Speaker: Dr. Fabio Priuli
Time:Fri. 25 February 2011, 11.00 a.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 4 lectures. For the abstract see the here.

Introduction to patchy feedback controls: motivations, ideas and techiques - II

Speaker: Dr. Fabio Priuli
Time:Mon. 14 February 2011, 11.00 a.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 4 lectures. For the abstract see the here.

Introduction to patchy feedback controls: motivations, ideas and techiques - I

Speaker: Dr. Fabio Priuli
Time:Fri. 4 February 2011, 11.00 a.m.
Place: SISSA - Santorio A - room 133

This short course is focused on controls in feedback form which stabilize (or which are nearly optimal for) nonlinear systems of ODEs $\dot x = f(x, u(x))$. Such controls in general do not exist among continuous functions. Therefore, one has to deal with discontinuous ODEs, which in turn might not have solutions in the sense of Caratheodory. In these seminars, I will present the motivations for feedback controls and the possible approaches to deal with the resulting discontinuous ODEs. A particular attention will be devoted to patchy controls, a class of discontinuous controls which provides the required controls and ensures additional strong robustness properties.

Systems of quadratic inequalities

Speaker: Dr. Antonio Lerario
Time:Thu. 27 January 2011, 2.00 p.m.
Place: SISSA - Santorio A - room 136

Systems of quadratic inequalities are very flexible objects in mathematics, e.g any system of polynomial equations can be reduced to a system of quadratic equations by substitutions. Thus the set X of the solutions of a system of quadratic inequalities can describe a very large class of semi-algebraic sets (the complexity of X is hidden in the number of linearly independent inequalities). To study such a system we focus on the dual object: the convex hull, in the space of all real quadratic forms on R^n, of those quadratic forms involved in the system (n is the number of variables in the system). It turns out that the homology of X is determined by the arrangement of this convex hull with respect to the cone of degenerate forms. This approach allows to efficiently compute homology for a very big number of variables n as long as the number of linearly independent inequalities is limited. Moreover, it works also for systems of integral quadratic inequalities, i.e. in the infinite dimension, beyond the semi-algebraic context. The calculations are organized in a spectral sequence whose member E_2 and the differential d_2 have a simple clear geometric interpretation. This is a joint work with Prof. Agrachev

Some compactness results in Calculus of Variations and in Critical Point Theory - III

Speaker: Dr. Carlo Mercuri
Time:Fri. 17 November 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 3 lectures. Please click here for the abstract.

Some results on Liouville-type equations - IV

Speaker: Dr. Francesca De Marchis
Time:Thu. 16 December 2010, 2.00 p.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 4 lectures. Please click here for the abstract.

Fluid flow controlled by small force

Speaker: Prof. Alexander Shnirelman (Concordia University, Montreal - Canada)
Time:Wed. 15 December 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

Please click here for the abstract.

Some compactness results in Calculus of Variations and in Critical Point Theory - II

Speaker: Dr. Carlo Mercuri
Time:Mon. 13 November 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 3 lectures. Please click here for the abstract.

Some results on Liouville-type equations - III

Speaker: Dr. Francesca De Marchis
Time:Tue. 7 December 2010, 4.15 p.m.
Place: SISSA - Santorio A - room 136

This is a minicourse of 4 lectures. Please click here for the abstract.

Some compactness results in Calculus of Variations and in Critical Point Theory - I

Speaker: Dr. Carlo Mercuri
Time:Mon. 6 November 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 3 lectures. Please click here for the abstract.

Some results on Liouville-type equations - II

Speaker: Dr. Francesca De Marchis
Time:Tue. 30 November 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 4 lectures. Please click here for the abstract.

Some results on Liouville-type equations - I

Speaker: Dr. Francesca De Marchis
Time:Tue. 23 November 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

This is a minicourse of 4 lectures. Please click here for the abstract.

A notion of Ricci curvature in sub-Riemannian geometry, Li-Yau inequalities and volume comparison theorems

Speaker: Prof. Nicola Garofalo (Purdue University, USA, and Università di Padova)
Time:Thu. 18 November 2010, 4.00 p.m.
Place: SISSA - Santorio A - room 133

I will discuss recent joint works with Fabrice Baudoin and with F. Baudoin and Michel Bonnefont. In the former we introduced a sub-Riemannian notion of Ricci curvature and with this notion proved various results connecting the geometry of the manifold to global properties of the relevant heat semigroup (such as Li-Yau gradient bounds, Harnack inequality, off-diagonal upper Gaussian bounds), and to topological properties (such as compactness of the manifold itself). In the more recent latter work we establish various entropic estimates which lead to a global volume comparison theorem, lower Gaussian bounds and a uniform Poincare' inequality.

Mean curvature flow in Kähler Manifolds

Speaker: Prof. Jiayu Li (Chinese Academy of Science, Beijing)
Time:Thu. 11 November 2010, 4.30 p.m.
Place: SISSA - Santorio A - room 134