25 Ottobre, 2010 17:00

Seminario Matematico e Fisico di Milano

TRANSIZIONI DI FASE, STABILITA E SIMMETRIA

ALBERTO FARINA, Université de Picardie

Dipartimento di Matematica, Politecnico di Milano, Aula Seminari MOX VI piano

abstract

abstract

Lo scopo del seminario è quello di presentare alcuni risultati riguardanti le proprietà geometriche e di simmetria delle interfacce che compaiono nello studio di alcuni modelli variazionali introdotti per descrivere le transizioni di fase. Facendo uso di tecniche di teoria delle equazioni alle derivate parziali, di calcolo delle variazioni, e di un pò di geometria,
dimostreremo la simmetria delle interfacce, in accordo con una famosa congettura di E. De Giorgi.

20 Ottobre, 2010 15:00 oclock

Seminari FDS

Matematica per la Medicina: il caso ANEURISK

Piercesare Secchi, Dipartimento di Matematica F.Brioschi

Aula Consiglio VII piano Nave

Lo scopo del seminario è quello di presentare alcuni risultati riguardanti le proprietà geometriche e di simmetria delle interfacce che compaiono nello studio di alcuni modelli variazionali introdotti per descrivere le transizioni di fase. Facendo uso di tecniche di teoria delle equazioni alle derivate parziali, di calcolo delle variazioni, e di un pò di geometria,
dimostreremo la simmetria delle interfacce, in accordo con una famosa congettura di E. De Giorgi.

12 Ottobre, 2010 17:15 oclock

Seminari FDS

PRICE ROBUSTNESS TO MODEL RISK

GIULIA DI NUNNO, C.M.A. (Center of Mathematics for Applications) dell UNIVERSITA DI OSLO

AULA SEMINARI al TERZO PIANO

abstract

abstract

Two types of price models with jumps are considered: either we describe small variations of price dynamics with a Poisson random measure with infinite activity, or with a Brownian motion.
We prove and discuss the robustness of prices of options written on the underlying dynamics above.
As for the computation of these prices itself, we introduce the conditional density method as an alternative to the other methods for the computation of the deltas already in use. This method can be successfully applied also to multi-factor type of models, popular in the modeling of spot prices in electricity markets.
We extend the work to a jump-diffusion framework where robustness of option prices is discussed and an extension of the Malliavin method is proposed for the computation of prices.
Various examples will be given for the different aspect treated.
The presentation is based on joint works with Fred E. Benth e Asma Khedher.

6 Ottobre, 2010 14:30 oclock

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Sharp regularity theory for singular fully nonlinear PDEs

Eduardo Teixeira, Univ. Federal Ceará , Brazilian Academy of Science

Aula seminari III piano

abstract

abstract

In this talk I will present some recent results on sharp gradient estimates for viscosity solutions to certain fully non-linear PDEs
with singular potentials.
The general results do not assume sign
constraints, thus associated free boundary problems may have two-phases.
At the end, I will dicuss a class of fully nonlinear
singularly perturbed PDEs that can be interpreted as a non-variational Alt-Caffarelli theory.

web.mate.polimi.it/cdv/

4 Ottobre, 2010 16:30

Lezioni Leonardesche

Q-CURVATURE: ANALYTIC AND GEOMETRIC ASPECTS

SUN-YUNG ALICE CHANG, Princeton University

Università di Milano, Dipartimento di Matematica, Via Saldini

In this talk I will present some recent results on sharp gradient estimates for viscosity solutions to certain fully non-linear PDEs
with singular potentials.
The general results do not assume sign
constraints, thus associated free boundary problems may have two-phases.
At the end, I will dicuss a class of fully nonlinear
singularly perturbed PDEs that can be interpreted as a non-variational Alt-Caffarelli theory.

web.mate.polimi.it/cdv/

27 Settembre, 2010 17:00

Seminario Matematico e Fisico di Milano

REMARKS ON $L^1$ REGULARITY OF GRADIENT FLOWS

PAVEL KREJCI, Institute of Mathematics, Academy of Sciences of the Czech Republic

Dipartimento di Matematica, Università di Milano, Via Saldini

abstract

abstract

A collection of arguments will be presented in order to show that $L^1$ is the natural functional framework for convex gradient flows. In particular, Lipschitz continuous data dependence is not available in general except possibly for the $L^1$ norm.
Examples from solid mechanics and phase transitions illustrate how this information can be exploited for existence and uniqueness of solutions to coupled problems.

13 Settembre, 2010 17:00

Seminario Matematico e Fisico di Milano

DERIVED CATEGORIES AND BIRATIONAL INVARIANTS

ALEXANDER KUZNETSOV, Steklov Mathematical Institute, Moscow

Università di Milano, Dipartimento di Matematica, Via Saldini, Sala di Rappresentanza

abstract

abstract

I will explain how one could construct a birational invariant from the derived category of an algebraic variety. An application to characterization of rational cubic fourfolds will be discussed.

6 Settembre, 2010 17:00

Seminario Matematico e Fisico di Milano

HYPERCOMPLEX ANALYSIS AS A UNIFYING THEORY: ON SOME BASIC IDEAS.

MICHAEL SHAPIRO, Instituto Politecnico Nacional (Mexico City)

Sala del Consiglio, 7o piano, Dipartimento di Matematica, Politecnico di Milano.

abstract

abstract

Hypercomplex analysis is a generic name for those generalizations of one-dimensional complex analysis which involve hypercomplex numbers. Quaternionic analysis is the oldest and the most known version of it, so that it will be discussed, first of all, in which sense it is a "proper" or a "closest" version in low dimensions which includes, as particular cases, such classic theories as vector analysis and holomorphic mappings in two complex variables, as well as some systems of partial differential equations. This allows to one, by developing quaternionic analysis, to obtain new results for the above classic theories and to refine known ones; some applications of this approach will be presented. Some comments on Clifford analysis and its applications will be also made.

6 Luglio, 2010 14:30 oclock

Seminario Matematico e Fisico di Milano

Recent results on mixing optimization and stability of flows in channels with porous walls

Luca Cortelezzi, Dept. of Mechanical Engineering, McGill University, Montreal, Canada

Attenzione: AULA III PIANO dipartimento di Matematica - Politecnico di Milano

abstract

abstract

The first part of this talk will discuss the development of an optimal mixer. An optimal mixer can be defined as a mixing device able to deliver a uniformly optimal mixing performance over a
wide range of operating and initial conditions. We conceptually design an optimal mixer starting from a reference mixing device,the sine flow. A careful characterization shows that the
time-periodic sine flow performs poorly and erratically over most operating and initial conditions. The optimal mixer is derived in steps modifying the design of the reference mixer
and optimizing its operations. We first optimize the time-sequence of the stirring velocity fields. We obtain a mixer which performs
substantially better than the sine flow, although its performance remains suboptimal due to a deficiency of the actuating system.
Second, we improve the sine flow by implementing a new actuating system that allows optimized shifts of the stirring velocity fields in the cross-flow direction. The performance of the resulting mixer is suboptimal only at low operating conditions due to the use of a periodic stirring protocol. Finally, an optimal mixer is
obtained by coupling the time and shift optimizations. The resulting optimal mixer is able to deliver a uniform optimal performance over the entire operating range and is quite
insensitive to the geometry of the initial conditions.

The second part of this talk will discuss the three-dimensional linear stability analysis of a pressure driven, incompressible, fully developed, laminar flow in a channel delimited by rigid,
homogeneous, isotropic, porous layers. We consider porous materials of small permeability in which the maximum fluid velocity is small compared to the mean velocity in the channel
region and for which inertial effects may be neglected. We solve the fully coupled linear stability problem, arising from the adjacent channel and porous flows, using a spectral collocation technique. We find that very small amounts of wall permeability significantly affect the Orr-Sommerfeld spectrum and can dramatically decrease the stability of the channel flow. Within our assumptions, in channels with two porous walls, permeability destabilizes up to two Orr-Sommerfeld wall modes and introduces two new damped wall modes on the left branch of the spectrum.
Permeability also introduces a new class of damped modes associated with the porous regions. The size of the unstable region delimited by the neutral curve grows substantially, and the critical Reynolds number can decrease to only 10% the corresponding value for a channel flow with impermeable walls.

2 Luglio, 2010 15:00 oclock

Seminario Matematico e Fisico di Milano

Functional Data Analysis for tree-structured data objects

J. S. Marron, Dept. of Statistics and Operations Research, University of North Carolina

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The field of FDA has made a lot of progress on the statistical analysis of the variation in a population of curves. A particularly challenging extension of this set of ideas, is to populations of tree-structured objects. Deep challenges arise, which involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. These challenges, together with some first approaches to addressing them, are illustrated using a real data example, where each data point is the tree of blood vessels in one person s brain.