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Seminars 2009-2010

A Brief Introduction to Singularity Theory

In this mini-course (about 3 lectures) I am planning to give a brief introduction to Singularity Theory. I will speak about singularities of plane curves, smooth functions and maps and implicit differential equations. In particular, the Division Theorem and Malgrange Preparation Theorem will be given (without proofs), and Whitney Theorem (about generic singularities of mappings of 2-manifolds) will be proved.

On the H-bubble problem

A Brief Introduction to Singularity Theory

In this mini-course (about 3 lectures) I am planning to give a brief introduction to Singularity Theory. I will speak about singularities of plane curves, smooth functions and maps and implicit differential equations. In particular, the Division Theorem and Malgrange Preparation Theorem will be given (without proofs), and Whitney Theorem (about generic singularities of mappings of 2-manifolds) will be proved.

Fields of Extremals for Multiple Integrals

The minimization problems for multiple integrals relative to the set of manifolds with fixed boundary conditions are considered. New constructions for the fields of extremals and new variants of generalized Legendre-Weyl-Caratheodory transforms are introduced and explored. The connection and the curvature of bundles where the integrals are defined, are caclulated.

Generalities on Kähler geometries I

Existence results for elliptic problems in unbounded domains

Singularities of curves

Singularities of curves

Singularities of smooth maps

Controlling mechanical systems by active constraints

Reduced basis method for viscous flows: focus on efficient geometrical parametrization for optimization

In the last decades optimal shape design problems have gained an increasing importance in many engineering fields and especially in structural mechanics and in thermo–fluid dynamics. The problems we consider, being related with optimal design and flow control, necessarily involve the study of an evolving system modelled by PDEs and the evaluation of functionals depending on the field variables, such as velocity, pressure, drag forces, temperature, energy, wall shear stress or vorticity.

Especially in the field of shape optimization, where the recursive evaluation of the field solution is required for many possible configurations, the computational costs can easily become unacceptably high. Nevertheless, the evaluation of an "input/output" relationship of the system plays a central role: a set of input parameters identifies a particular configuration of the system and they may represent design or geometrical variables, while the outputs may be expressed as functionals of the field variables associated with a set of parametrized PDEs. The rapid and reliable evaluation of many input/output relationships typically requires great computational expense, and therefore strategies to reduce the computational time and effort are being developed.

Among model order reduction strategies, reduced basis method represents a promising tool for the simulation of flow in parametrized geometries, for shape optimization or sensitivity analysis. An implementation of the reduced basis method is presented by considering different shape or domain parametrizations: from simple affine maps to non–affine ones, transforming an original parametrized domain to a reference one.

Our analysis will focus on the general properties and performance of the reduced basis method by highlighting with several examples its special suitability and considering parametrized wavy or curvy geometries. The proposed approach includes also a geometric model reduction resulting from a suitable low–dimensional parametrization of the geometry based on free–form deformations technique. We focus on the possibility of handling very generic geometric parametrizations without requiring to create "ad hoc" affine representations necessary to solve the problem efficiently, but recovering this property by an empirical interpolation method in order to take advantage of an offline–online decomposition. We present in particular some examples of reduced basis method applied to steady incompressible viscous flows for shape optimization, parameter estimation and inverse problems in cardiovascular geometries.

In collaboration with L. Iapichino, T. Lassila, A. Manzoni, and A. Quarteroni.

Numerical models for coupling free and porous–media flows

In this talk we consider the modeling and numerical simulation of coupled free/porous–media flows. We present different modeling approaches involving the Navier–Stokes equations to represent the incompressible fluid and Darcy of Fochheimer equations in the porous region. We discuss and compare different coupling strategies based on the so–called penalization method or on the application of the transmission conditions of Beavers–Joseph–Saffman.
After giving some results on the well–posedness of the resulting coupled models, we illustrate a domain decomposition framework to set up possible iterative methods to compute their finite element solution. We discuss the effectiveness and robustness of such algorithms showing several numerical results.
Finally, we present some possible applications of the models that we have studied.

Multiple solutions for Yang–Mills connections in four dimensions

Soft Materials: from Traumatic Brain Injuries to Blast and Ballistic Mitigation

Soft materials such as polymers and biological tissues have several engineering and biomechanical applications. These materials exhibit complex mechanical characteristics and the need to accurately predict their behavior has posed a tremendous challenge for scientists and engineers.
Less than a half century old, continuum computational modeling of soft materials is undoubtedly still in its infancy and its projected outlook remains promising for improving humanity's quality of life. Continued advances in computers and computational methods are increasing our ability to handle large amounts of data and to model complex phenomena. Modeling of soft materials has a vital role to play in the development of the needed mathematical models and analyses. Because of the incredible complexity of the biochemophysical aspects of soft tissues and the chemophysical aspects of polymers, this type of modeling requires increased interdisciplinary and multidisciplinary research that brings scientists from various fields together in teams, both in research and education.
In this talk, I shall present a seamless, fully variational constitutive model capable of capturing several complex mechanical characteristics exhibited by such materials. Application to traumatic brain injury (TBI) under impact/acceleration loading will be presented. Clinically observed brain damage is reproduced and a predictive capability of the distribution, intensity, and reversibility/irreversibility of brain tissue damage will be demonstrated. Another application to ballistic impact on a polyurea retrofitted DH36 steel plate is simulated and validated, and computational capability for assessing the blast performance of metal/elastomer composite shells will also be presented.
Future directions of this work may lead to the formulation of head-injury criteria for medical, governmental, and industrial applications; addressing the definition of clinical-biomechanical injury thresholds and tolerances; the simulation of a wide range of injuries, including blast-induced TBI and the effects of growing tumors; neurosurgical simulations; and the design and the assessment of effective protective devices, such as helmets including honeycomb materials, polymers, or foam padding.

L\infinity bounds for solutions of supercritical elliptic problems with finite Morse index

The talk will concern the relationship between the L\infinty norm of solutions of some nonlinear elliptic equation and their Morse index (for some nonlinearity bounds on solution are equivalent to bounds on their Morse index).
In the first part i will present what it has been done in the subcritical case.
The second part will concern the supercritical case.

Existence of H-bubbles

Compactness and quantization phenomena in conformal geometry: Some recent results and open problems

We shall recall some recent concentration-compactness and quantization results for elliptic equations having critical nonlinearity, which arise in conformal geometry and functional analysis. By looking at a model case, we shall try to discuss how these results can be used to prove existence for some important equations (mean field equation, prescribed Q–curvature equation). We shall also see several open problems.

Brownian motion and Newtonian capacity

The material can be found in: \S455 and \S\S477-479 of the Web version of his book, see here.

Hydrodynamic Synchronization in a Carpet of Microswimmers

Swimming microorganisms often exhibit cooperative motion of active elements attached on the cell membrane, such as the metachronal waves of cilia and the rotation of bundled flagella. In order to understand the role of long-range hydrodynamic interactions in their collective dynamics, we introduce a simple and generic model of active microfluidic rotors arrayed on a substrate. The model shows a variety of dynamical patterns such as global synchronization, frustrated disorder, and turbulent spiral waves. We discuss the results in connection with recent experimental observations on bacterial carpets, as well as with theoretical developments on non-locally coupled oscillators.
References: NU & R. Golestanian, arXiv:0911.4253; arXiv:1001.2701.

Some topics involving fractional powers of elliptic operators

I will describe several topics in PDE (symmetry of solutions) , conformal geometry (dimension of the singular set of metrics) and harmonic analysis (Poincaré inequalities on Lie groups), where fractional powers of elliptic operators play an important role.

Symmetry and nonexistence of low Morse index solutions to semilinear elliptic equations

I will discuss symmetry properties of a certain class of solutions to semilinear elliptic equations. These solutions are characterized variationally by bounds on their Morse index. In the case where the underlying domain is radial, the solutions do not always inherit the full radial symmetry of the domain but at least axial symmetry. In some cases, the symmetry properties lead, a posteriori, to nonexistence results.

Locally conformally flat Ricci solitons

I will discuss a classification result for locally conformally flat Ricci solitons.

Enhancement of electromagnetic fields caused by interacting subwavelength cavities

On the singular $σ_k$-Yamabe problem

We prove the existence of constant positive σk- scalar curvature metrics which are complete and conformal to the standard metric on S^n \ Λ, where Λ ⊂ S^n is a finite number of points with cardinality at least two, and n, k are positive integers such that 2 ≤ 2k < n. In general this problem is equivalent to solve a singular fully nonlinear second order elliptic equation. For k = 1 (i.e., in the case of the ordinary scalar curvature) the problem reduces to solve a semilinear elliptic equation and it has been studied by several authors (Schön, Mazzeo-Pacard et al.).

Linear Mathematical models, complexity, and simulation in medicine, sports, and the environment

Last modified: 10 November 2010

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