**TITLE: **__Einstein metrics and Ricci solitons__

**ADVISOR: **Giovanni Catino, https://sites.google.com/site/homepagecatino/, giovanni.catino@polimi.it

**Abstract:** The fundamental problem of capturing the topological properties of a manifold by its metric structure opened, in the last decades, extremely fruitful areas of mathematics. From this perspective, there has been an increasing interest in the study of Riemannian manifolds endowed with metrics satisfying special structural equations, possibly involving curvatures and vector fields. These structures arise naturally not only in several different mathematical frameworks, but also in Physics, in particular in General Relativity and String Theory. More precisely, the main objective of the thesis will be the study of geometric and analytic properties of well known special Riemannian structures such as Einstein metrics and Ricci solitons and, more in general, Riemannian metrics satisfying structural conditions, involving their Ricci curvature, scalar curvature and globally defined vector fields. In particular, the aim is to prove rigidity, triviality and classification results, both in the compact and in the non-compact case for these type of structures.

**TITLE: **__Long-time dynamics of dissipative evolutions equations with memory__

**ADVISORS: **Monica Conti, monica.conti@polimi.it, Vittorino Pata, vittorino.pata@polimi.it

**Abstract:** Many physical phenomena in which delay effects occur are modeled by equations with memory, where the dynamics is influenced by the past history of the variables in play through convolution integrals against suitable *memory kernels*. An effective way to capture the dissipativity properties of those models is translating the equations within a proper semigroup framework, where the regime behavior can be described in terms of “small" sets of the phase space able to eventually capture the trajectories of the underlying semigroup/process. Our aim is the development of new theoretical tools - in the context of the Theory of Infinite Dimensional Dynamical Systems – for the characterization of the long-term behavior of dissipative processes arising from PDEs. Applications of interest include models defined on time-dependent phase spaces such as viscoelasticity in aging materials, population dynamics, heat flow in real conductors, phase transition.

**TITLE: **__Decay of linear semigroups and applications to PDEs__

**ADVISORS: **Filippo Dell’Oro, filippo.delloro@polimi.it, Vittorino Pata, vittorino.pata@polimi.it

**Abstract:** The modern theory of evolutionary partial differential equations is a notable branch of Mathematics, with an increasing number of challenging open problems. When the underlying equation is linear, the theory of semigroups of linear operators can be effectively exploited in order to investigate a broad range of remarkable questions, such as well-posedness and asymptotic stability of solutions. In this thesis, we propose to study from the theoretical viewpoint the decay properties of linear semigroups arising from partial differential equations of physical interest. Some examples include: models coming from thermoelasticity and viscoelasticity, evolution equations on time-dependent spaces and abstract partially dissipative systems.

**TITLE: **__Stability and Numerical Reconstruction of Inclusions in a Conductor Body__

**ADVISOR: **Michele Di Cristo, michele.dicristo@polimi.it,

**Abstract: **We would like to study the inverse problem of determining an inclusion in an electrical conductor by electrostatic boundary measurements. In particular we want to analyse a body that contains a region whose electrical conductivity is different from the conductivity of the surrounding material. This region may represent a unknown damage part of the body and the goal is to detect such a part. The aim of this work is to provide some stability estimates, that is how the unknown inclusion depends from the boundary data, and provide some numerical reconstruction to locate the shape and estimate the size of the defect. More specifically we would like to consider anisotropic conductivities when only a portion of the boundary is available to perform our measurements.

**TITLE: **__Multi-phase shape optimisation problems__

**ADVISORS: **Ilaria Fragalà, https://www.mate.polimi.it/?view=personale#ann, ilaria.fragala@polimi.it, Dorin Bocur, https://www.lama.univ-savoie.fr/~bucur/, dorin.bucur@univ-savoie.fr

**Abstract:** This project is focused on shape optimization problems involving as a common feature several "unknown phases". Archetypal examples are: *optimal partition problems, *in which one has to minimize some energy depending on the cells; classical *packing problems,* in which one has to find the best arrangement of a family of balls in a box; *multi-label image segmentation problems, *in which one has to reconstruct the different zones of an input image. The target may be to establish the existence of a solution, the regularity of its "free boundaries", or some of its qualitative properties, e.g. the asymptotics for a large number of cells (a fascinating conjecture is often the optimality of a hexagonal pattern, as it happens in natural wonders like the bees' honeycombs or giant's causeway).

**TITLE: **__Mathematical Models for Suspension Bridges__

**ADVISOR: **Filippo Gazzola, http://www1.mate.polimi.it/~gazzola/ filippo.gazzola@polimi.it

**Abstract:** The main goal of this research project is to obtain quantitative answers concerning the structural instability of suspension bridges in the presence or wind, that is, the analytical study of the interaction between internal resonance and aeroelastic phenomena. Skills in PDE’s, ODE’s, fluid-mechanics will be intensively used.

**TITLE: ** __Nonlinear elliptic and parabolic equations on the Euclidean space and on manifolds.__

**ADVISORS:**

Gabriele Grillo, https://sites.google.com/site/homeggrillo/ gabriele.grillo@polimi.it

Matteo Muratori, https://www.mate.polimi.it/?view=pp&id=566&lg=it#ann matteo.muratori@polimi.it,

Fabio Punzo, https://www.mate.polimi.it/?view=pp&id=565&lg=it#ann fabio.punzo@polimi.it

**Abstract. **We consider nonlinear elliptic equations whose model is the Emden-Fowler equation and nonlinear evolution equations, of diffusive type, whose models are the porous medium equation or the fast diffusion equation. Among the topics we propose to investigate we mention existence, or nonexistence, of solutions, uniqueness issues, asymptotic behavior and qualitative properties of solutions, often in relation with the validity of suitable functional inequalities. These topics are being actively investigated in the Euclidean framework but a number of challenging open problems exist. Extension to non-Euclidean settings, like the manifold one, on which we recently proved the first existing results, can be dealt with and are so far almost completely open.

**TITLE: ** __Nonlocal, nonlinear elliptic and parabolic equations.__

**ADVISORS:**

Gabriele Grillo, https://sites.google.com/site/homeggrillo/ gabriele.grillo@polimi.it

Matteo Muratori, https://www.mate.polimi.it/?view=pp&id=566&lg=it#ann matteo.muratori@polimi.it,

Fabio Punzo, https://www.mate.polimi.it/?view=pp&id=565&lg=it#ann fabio.punzo@polimi.it

**Abstract.**Nonlocal diffusion operators are being widely studied since the recent, pioneering works of Caffarelli, Silvestre, Vázquez. They involve a number of possible generalization of the porous medium equation in which either the generator or the pressure are fractional powers of the Laplacian. Weighted versions of these equations, corresponding to spatially inhomogeneous media, have also been studied. Crucial issues are still open, we mention in particular uniqueness for bounded distributional solutions. Extensions to more general integro-differential operators are also planned.

**TITLE: **__Pattern formation: reaction-diffusion PDEs systems with strong competition__

**ADVISORS:**

Gianmaria Verzini, https://www.mate.polimi.it/?view=pp&id=116&lg=en, gianmaria.verzini@polimi.it,

Susanna Terracini, https://sites.google.com/site/susannaterracini/, susanna.terracini@unito.it.

**Abstract:** Several phenomena can be described by a certain number of densities (of mass, population, probability, …) distributed in a domain and subject to diffusion, reaction, and competitive interaction. When the competition is the prevailing aspect, one can reasonably expect pattern formation: the densities should not coexist in the same region, and they should tend to segregate, hence determining a partition of the domain. Prototype models include:

- competition between different species in population dynamics;

- phase separation in mixtures of Bose-Einstein condensates;

- segregation in multi-population competitive Mean Field Games system.

The main questions involve the regularity properties and the geometric structure of the final (optimal) partition and of the free boundary.

**TITLE: **__Scalable splitting algorithms for large scale optimization__

**ADVISOR:** Silvia Villa https://www.mate.polimi.it/view=pp&id=552&lg=it#ann silvia.villa@polimi.it,

**Abstract: **Recently, convex optimization has been revolutionized by data availability and data nature: problems of an unprecedented scale need to be solved, and the computation of the solution must be robust with respect to perturbations, since data are affected by noise. These two aspects fueled the design of provably convergent first-order optimization methods, which represent the state of the art to solve large scale problems. However, their implementation still faces challenges that render them often inapplicable in their original form. The objective of this thesis is to derive more scalable algorithms, by exploiting the structural properties of data driven optimization problems. The goal is to design new methods adapted to the modeling assumptions and to derive a sharper convergence analysis.