Alessandra Pluda

Jacopo Schino (Institute of Mathematics, Polish Academy of Sciences)- 24th June '19

A semilinear curl-curl problem in R^3.

We look for nontrivial solutions to the semilinear elliptic problem:
curl curl u = f(x,u) in R^3, where f is Z^3-periodic in x. We give sucient conditions on the nonlinearity which provide a least energy solution and infitely many Z^3-distinct solutions. The growth and asymptotic behaviour of the nonlinearity are described by an N-function which allows us to consider other model problems than the classical power type or double-power type.
After building the proper function space where to look for solutions and showing its main characteristics, we develop an abstract critical point theory, providing results that we use to solve our equation and may be applied to other problems. The main diculties are due to working in an unbounded domain and the infite dimension of the kernel of the map from u to curl u, i.e. the space of gradient vector fields. We overcome the former using a concentration-compactness argument. At the end, we show how to solve an elliptic Schrödinger-type equation using the abstract critical point theory.
This talk is based on a joint work with Jaroslaw Mederski and Andrzej Szulkin.

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Simone Floreani (Technische Universiteit Delft)- 21st June 2019

Interacting Particle Systems in Random Environmentand PDEs.

In this talk we will derive the diffusion equation starting from a microscopic description of a system of interacting particles in an inhomogeneous media. The dynamics of the particle system will be stochastic and, more precisely, Markovian. However, due to the non-trivial interaction which particles undergo, the evolution of each particle is not Markovian in itself.
We will look at the evolution in time of the empirical density field describing how particles are located in the domain. After a so-called "diffusive" rescaling of time and space, we will show that the interaction is, roughly speaking, "lost" and that the density of particles evolves according to the linear diffusion equation. Key tool for proving such a result, also known as hydrodynamic limit, will be stochastic duality. Stochastic duality is a technique that allows to study properties of a certain stochastic process in terms of those of a - possibly simpler - dual one. In our case, for the study of empirical density fields, the dual process will be a random walk evolving in the same inhomogeneous media. This connection allows us to reduce the convergence of the empirical density fields to convergence of this random walk to Brownian motion. In other words, we derive the many-particle hydrodynamic limit from a one-particle functional central limit theorem (also called invariance principle).
 

Silvia Ghinassi (Institute for Advanced Studies, Princeton)- 18th June 2019

On the Analyst's Traveling Salesman Theorem (and Reifenberg flat sets)

Peter Jones in 1990 introduced the so-called beta-numbers, in order to provide a geometric characterization of subsets ofrectifiable curves. These numbers measure, at a given scale andlocation, how far a set is far from being a line. His motivations camefrom harmonic analysis, more specifically the study of the Cauchytransform on Lipschitz graphs, and questions on harmonic measure. Wewill start with the definition of beta-numbers, and try to motivateand justify Peter Jones's theorem and many others that have followed inthe last 30 years. If time allows, we will jump back in time and discussReifenberg flat sets, first introduced by Reifenberg in 1960 in thecontext of solving Plateau problem for higher dimensional surfaces.
 

Anna Kausamo (University of Jyväskylä)- 30th May 2019

A story about multi-marginal optimal transport

Once upon a time there was a French mathematician called Gaspard Monge who set out to explore the problem of transporting mass from one place to another place in an optimal way.  More than 100 years later, a Russian mathematician called Leonid Kantorovich studied the
duality between minimizing the cost and maximizing the benets of the transport. Today we study 'the Monge problem', 'the Kantorovich Duality', and 'The Monge-Kantorovich problem', named in honor of the two founding fathers of the field. In the most classical formulation of the
problem, we move mass from one place (formally: from one 'marginal') to another one, and the transporting gets more expensive when the transportation distance increases. But what happens if we have more than two marginals? What changes if the cost function is repulsive, i.e.
increases when the distance of the points to be coupled decreases? Why, in particular, does the Monge problem become so difficult when we move
from two to many marginals? And what is this Monge problem in the first
place...?

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Mauruzia Rossi (Università di Pisa) - 22nd May 2019

Nodal lengths of random spherical harmonics

We investigate the geometry of random eigenfunctions on manifolds. In particular, we study the asymptotic behavior, in the high-energy limit, of the nodal length of random spherical harmonics.
This talk is mainly based on a joint work with D. Marinucci (Università
di Roma "Tor Vergata") and I. Wigman (King's College London).

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Alberto Roncoroni (Università di Pavia) - 17th May 2019

Symmetry results for critical p-Laplace equations

We consider the following critical p-Laplace equation:

(1)           ∆^p u + u^(p*−1) = 0 in R^n ,
with n ≥ 2 and 1 < p < n. Equation (1) has been largely studied in the PDE’s and geometric analysis’ communities, since extremals of Sobolev inequality solve (1) and, for p = 2, the equation is related to the Yamabe’s problem. In particular it has been recently shown, exploiting the moving planes method, that positive solutions to (1) such that u ∈ L^(p*) (R n ) and ∇u ∈ L^p (R^n ) can be completely classified. Since the moving plane method strongly relies on the symmetries of the equation and of the domain, in the seminar a new approach to this problem will be presented. In particular this approach gives a complete classification of the solutions in an anisotropic setting. More precisely, we characterize solutions to the critical p-Laplace equation induced by a smooth norm inside any convex cone.
This is a joint work with G. Ciraolo and A. Figalli.
 

Marco Bresciani (Università di Pavia) - 6th May 2019

The renormalized energy for a system of edge dislocations with multiple Burgers vectors

Dislocations are 1-dimensional defects of the crystal lattice and their motion is considered the main cause of plastic deformations in metals. In this talk, we study a variational model that describes a system of straight and parallel edge dislocations in an elastic body. These are represented by singularity points of the strain fields, which are defined on a cross-section orthogonal to the dislocation lines. In the analysis, we adopt the so-called core-region approach. We introduce the renormalized energy as a function of the empirical measure of the dislocations, then we prove a Γ-convergence result, as the number of dislocations goes to infinity and the core-radius goes to zero, in the case of two different Burgers vectors with positive scalar product. Finally, we establish a characterization of the class of measures with finite energy.

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Francesca Oronzio (Università degli studi di Napoli Federico II) - 18th April 2019

Some relations between curvature and topology via distance functions

The aim of the talk is to present new proofs of the Cartan-Hadamard and Bonnet-Myers theorems based on the analysis of the distance functions on Riemannian manifolds, following a proposal of P. Petersen. Such proofs are alternative to the classical ones and more geometric in the spirit.

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Susanna Risa (Università di Roma Tor Vergata) - 11th April 2019

Spherical rigidity of pinched ancient flows of hypersurfaces

A solution of a parabolic differential equation is ancient if it is defined for all negative times. They model the asymptotic shape of a compact submanifold evolving by a function of the extrinsic curvature as tangent flows if the second fundamental form blows up at a spherical scale. In 2014, Huisken and Sinestrari have shown that a uniform pinching condition on the curvature of a convex compact ancient solution of Mean Curvature Flow is sufficient to ensure it is a sphere shrinking by homotheties. An analogous result holds for more general fully nonlinear curvature flows: in particular, we consider spherical rigidity for evolutions by 1-homogeneous speeds and by powers of the Gaussian curvature.

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Mattia Fogagnolo (Università di Trento) - 28th March 2019

Classical and new problems in geometric analysis

In the first part of the talk, we will review the classical Alexandrov Theorem, the Isoperimetric Inequality and the Willmore Inequality in Euclidean spaces, trying to highlight their mutual connections and to discuss how they can be proved through PDE's techniques. In the second part of the talk we will consider these problems in relevant families of Riemannian manifolds, approaching the frontier of modern research on these topics.

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Gioacchino Antonelli (SNS Pisa) - 14th March 2019

How can armonic functions split your manifold?

We will start by giving the statement of the classical splitting theorem by Cheeger and Gromoll: if a complete Riemannian manifold M with non-negative Ricci curvature contains a line, then M ∼= R × N . Such a splitting is granted by the existence of a special harmonic function b on M , namely the Busemann function associated to the line. The harmonicity of b, the constancy of |∇b| and the Bochner equality will imply that ∇b is a parallel vector field. In general the existence of a function b on M with parallel ∇b - which we call a splitting function - is sufficient to obtain a splitting of the manifold. This remark will give the motivation to state and prove some classical rigidity results, proved by Li and Schoen, about harmonic and subharmonic functions on manifolds - which are interesting per se - the leading question being: what do we need to add to the harmonicity of a function b in order to let it become a splitting function on M with non-negative Ricci curvature? In particular I will state that on a complete Riemannian manifold with non-negative Ricci curvature, a positive subharmonic function which is in L p with p ∈ (0, +∞) is constant. I will prove this theorem in a simple case using an argument based on Caccioppoli inequality. I will finish discussing how these results change substituting positive subharmonic with harmonic, in particular discussing the case p = +∞.

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Andrea Vaccaro (Università di Pisa) - 28th Februar 2019

An introduction to the counterexamples to Naimark's problem and their traces spaces.

A representation (f,H) of a C*-algebra A is a *-homomorphism f from A into B(H), the latter being the C*-algebra of all linear bounded operators from a complex Hilbert space H into itself. A representation (f,H) is irreducible if the only subspaces K of H such that f[A]K is contained in K, are the trivial ones. Two representations (f,H) and (g,H) are unitarily equivalent if there is a unitary operator U from H into K such that UfU* = g.

It is well-known that there exists a unique irreducible representation of K(H) (the C*-algebra of the compact operators on H) up to unitary equivalence, namely the identity. In 1951 Naimark asked whether this property characterizes K(H) up to isomorphism. In 2004 a major breakthrough was made by Akemann and Weaver. They showed that, assuming the set theoretic principle known as Jensen's diamond (a strengthening of the continuum hypothesis), it is possible to built a "counterexample to Naimark's problem". This is a C*-algebra which is not isomorphic to K(H) (for any H), yet still has only one irreducible representation up to unitary equivalence.
In this seminar I will introduce Naimark's problem, I will discuss the modern C*-algebraic/set theoretic perspective on this topic and present some recent results concerning the trace space of these counterexamples.

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Marta Leocata (Università di Pisa) - 18th Februar 2019

The NSVP system as a scaling limit of particles in a fluid

The PDEs system Navier-Stokes-Vlasov-Fokker-Planck (NSVFP) is a model describing particles in a fluid, where the interaction particles-fluid is described by a drag force called Stokes drag force. In the talk I will present a particle system interacting with a fluid that converges in a suitable probabilistic sense to the (NSVFP) system. The talk is based on a recent work in collaboration with Franco Flandoli and Cristiano Ricci.