### Seminari

### Prossimi Seminari

**Dealing with unreliable computing platforms at extreme scale**

Luc Giraud, INRIA (Inria Bordeaux – Sud-Ouest)

mercoledì 23 gennaio 2019 alle ore 14:00, Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO**Poroelasticity: Discretizations and fast solvers based on geometric multigrid methods**

Francisco José Gaspar Lorenz, Department of Applied Mathematics -Zaragoza University – Spain

giovedì 31 gennaio 2019 alle ore 14:00, Sala Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO**Application of Polyconvexity and multivariable convexity of energy potentials in nonlinear solid mechanics**

Javier Bonet, University of Greenwich

giovedì 14 febbraio 2019 alle ore 14:00, Aula Consiglio VII Piano – Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO

### Seminari Passati

**QUANTUM INCOMPATIBILITY WITNESSES**

Alessandro Toigo, Politecnico di Milano

giovedì 17 gennaio 2019 alle ore 11:30 precise, Aula Consiglio VII pianoABSTRACTWe discuss the following variant of the standard minimum error state discrimination problem: Alice picks the state she sends to Bob among one of several disjoint state ensembles, and she communicates him the chosen ensemble only at a later time. Two different scenarios then arise: either Bob is allowed to arrange his measurement set-up after Alice has announced him the chosen ensemble (pre-measurement scenario), or he is forced to perform the measurement before of Alice’s announcement (post-measurement scenario). In the latter case, he can only post-process his measurement outcome when Alice’s extra information becomes available. We demonstrate that quantum incompatibility can always be detected by means of a state discrimination task within the pre-measurement scenario. This is done by showing that only incompatible measurements allow for an efficient use of pre-measurement information in order to improve Bob’s probability of guessing the correct state. The gap between the guessing probabilities with pre- and post-measurement information is thus a witness of the incompatibility of a given collection of measurements. We prove that all linear incompatibility witnesses can be implemented as some state discrimination protocol according to this scheme.**Expected utility maximization beyond the Markovian setting**

Marina Santacroce, Politecnico di Torino

martedì 8 gennaio 2019 alle ore 16:30 precise, Aula Seminari del III pianoABSTRACTAn overview of the recent approaches used to solve

portfolio optimization problems for general market models

is given.

In particular, the focus will be on dynamic programming

techniques and on their applicability to expected utility

maximization in non-Markovian settings for classical

utilities (power, exponential or log type), including the

case of partial information. Moreover, another method

which works for general utilities is presented and

compared to recent results obtained by dynamic

programming.

This talk is based on joint works with M. Mania, R.

Tevzadze and B. Trivellato.

**The Birch-Swinnerton-Dyer conjecture, some recent progress**

Guido Kings, Università di Regensburg

lunedì 7 gennaio 2019 alle ore 16:00, Aula C, Dipartimento di Matematica, Via C. Saldini 50, MilanoABSTRACTFinding rational solutions of polynomial equations is one of the most difficult questions in arithmetic geometry. The Birch-Swinnerton-Dyer conjecture (one of the millennium problems) proposes an answer to this question in the case of elliptic curves. In the last years, using techniques like Euler systems in combination with methods involving p-adic families of modular forms, new insights and results concerning refinements of this conjecture were obtained.

In this talk we want to give an introduction to the Birch-Swinnerton-Dyer conjecture, avoiding all technicalities and review what is known about it. In the end we want to explain the ideas which lead to new results on a refinement of the Birch-Swinnerton-Dyer conjecture.**Advanced polyhedral discretization methods for poromechanical modelling**

Michele Botti , Université de Montpellier

martedì 18 dicembre 2018 alle ore 14:00, Aula Seminari ‘Saleri’ VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano – Edificio 14ABSTRACTDuring the talk, I will present analytical and numerical results for the (possibly nonlinear) coupled equations of poroelasticity describing the fluid flow in a deformable porous medium. We will focus on novel schemes based on a Hybrid High-Order discretization of the mechanics and a Symmetric Weighted Interior Penalty discontinuous Galerkin discretization of the flow. The method has several assets, including, in particular, the validity in two and three space dimensions, inf-sup stability, and the support of general polyhedral meshes, nonmatching interfaces, and arbitrary space approximation order. Our analysis delivers stability and error estimates that hold also when the constrained specific storage coefficient vanishes, and shows that the constants have only a mild dependence on the heterogeneity of the permeability coefficient. The performance of the method is extensively investigated on a complete panel of model problems using stress-strain laws corresponding to real materials. In the last part of the talk, we will consider the numerical solution of the poroelasticity problem with random physical coefficients in the context of uncertainty quantification. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution. The approximation of the stochastic partial differential system is realized by non-intrusive techniques based on polynomial chaos decompositions. We will conclude by performing a sensitivity analysis to asses the propagation of the input uncertainty on the solutions considering application-oriented test cases.

Contact: paola.antonietti@polimi.it

**Positive solutions to indefinite problems: a topological approach**

Guglielmo Feltrin, Politecnico di Torino

giovedì 6 dicembre 2018 alle ore 15:30, Aula seminari 3° pianoABSTRACTIn this seminar, we present some recent existence and multiplicity results for positive solutions of boundary value problems associated with second-order nonlinear indefinite differential equations. More precisely, we deal with the ordinary differential equation

u?? + a(t)g(u) = 0,

where a: [0,T] ? R is a Lebesgue integrable sign-changing weight and g: [0,+?[ ? [0,+?[ is a continuous nonlinearity.

We focus on the periodic boundary value problem and on functions g(u) with superlinear growth at zero and at infinity (including the classical superlinear case g(u) = up, with p > 1). Exploiting a new approach based on topological degree theory, we show that there exist 2m ? 1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. In this manner, we give a complete answer to a question raised by Butler (JDE, 1976) and we solve a conjecture by G ?omez-Ren ?asco and L ?opez-G ?omez (JDE, 2000). The method also applies to Neumann and Dirichlet boundary conditions and, furthermore, provides a topological approach to detect infinitely many subharmonic solutions and globally defined positive solutions with chaotic behaviour.

Thereafter, we illustrate other directions for the research on indefinite problems: super-sublinear problems, models in population genetics, and also problems involving more general differential oper- ators, as the Minkowski-curvature one or the one-dimensional p-Laplacian. Exact multiplicity results and indefinite problems in the PDE setting are also discussed.

The talk is based on joint works with Alberto Boscaggin (University of Torino), Elisa Sovrano (University of Porto) and Fabio Zanolin (University of Udine) and on the book “Positive Solutions to Indefinite Problems. A Topological Approach” (Frontiers in Mathematics, Birkh ?auser/Springer, 2018).**A decomposition of the Hilbert scheme given by Gröbner schemes**

Yuta Kambe, Saitama University

mercoledì 5 dicembre 2018 alle ore 11:00 precise, Aula seminari III pianoABSTRACTWe consider the Hilbert scheme H which is the scheme parameterizing all closed subschemes of the projective space P^n with Hilbert polynomial P. If we fix a monomial order < on the polynomial ring S with n+1 variables, each homogeneous ideal in S has a unique reduced Grobner basis with respect to <. Using this fact we can decompose the Hilbert scheme H into locally closed subschemes of H called the Grobner schemes. On the other hand, Bialynicki-Birula shows that any smooth projective scheme with a 1-dimensional torus action has a cell decomposition called the Bialynicki-Birula decomposition.

In this talk, I would like to explain Gröbner schemes and the decomposition. I introduce a 1-dimensional torus action on the Hilbert scheme H which is compatible with < and I show that the decomposition given by the Gröbner schemes can be constructed by such torus action in the sense of Bialynicki-Birula.

**Waring loci and decompositions of low rank symmetric tensors**

Alessandro Oneto , Barcelona Graduate School of Mathematics

mercoledì 5 dicembre 2018 alle ore 12:00 precise, Aula seminari III pianoABSTRACTGiven a symmetric tensor, i.e., a homogeneous polynomial, a Waring decomposition is an expression as sum of symmetric decomposable tensors, i.e., powers of linear forms. We call Waring rank of a homogenous polynomial the smallest length of such a Waring decomposition. Apolarity theory provides a very powerful algebraic tool to study Waring decompositions of a homogeneous polynomial by studying sets of points apolar to the polynomial, i.e., sets of points whose defining ideal is contained in the so-called apolar ideal of the polynomial. In this talk, I want to introduce the concept of Waring locus of a homogeneous polynomial, i.e., the locus of linear forms which may appear in a minimal Waring decomposition. Then, after showing some example on how Waring loci can be computed in specific cases via apolarity theory. I explain how they may be used to construct minimal Waring decompositions. These results are from recent joint works with E. Carlini, M.V. Catalisano, and B. Mourrain.**Turbulence versus Mathematics and vice versa**

Arkady Tsinober, Tel Aviv University

martedì 4 dicembre 2018 alle ore 16:00, Aula 3015 del Dipartimento di Matematica e Applicazioni dell’Università di Milano – BicoccaABSTRACTIt is much easier to present nice rational linear analysis than it is to wade into the morass that is our understanding of turbulence dynamics. With the analysis, professor and students feel more comfortable; even the reputation of turbulence may be improved, since the students will find it not as bad as they had expected. A discussion of turbulence dynamics would create only anxiety and a perception that the field is put together out of folklore and arm waving.” John Lumley, 1987.

From the outset I have to confess that I find myself 99% in agreement with John Lumley’s concern on “theories of turbulence”. This includes the first premise – i.e. the absence of a theory based on first principles. The second aspect concerns the importance of experiments and observations (both physical and numerical), below referred to as evidence. This lecture is intended to be, first and foremost, a critical presentation and examination of some fundamentally important issues.

* What do we really mean by ‘conventionally defined inertial range’ (CDIR)? Are its properties really independent of (the nature of) dissipation and/or large-scale forcing? Thus, is the inertial range a well defined concept or is it a mis-conception? Who is the guilty party for dissipation anomaly in turbulent flows? And what about the role of the self-amplification processes of vorticity, strain and super-helicity? Also, how well-defined and meaningful is the so-called ‘decomposition’ of energy in inertial and dissipative ranges?

* Is the ‘anomalous scaling’ an attribute of the inertial range? And of passive turbulence?

* Is the ‘4/5 law’ a purely inertial relation?

* Why should one expect that in the CDIR at very high Reynolds numbers the Navier–Stokes equations (NS) are invariant under infinitely many scaling groups (like the Euler equations), in the statistical sense of K41 labeled by an arbitrary real scaling exponent h? And more generally, should one expect to restore in some sense all the symmetries of Euler equations in the CDIR? And why necessarily Euler?

* Are weak solutions of Euler equations going to describe adequately a turbulent flow? Is the inviscid limit of NS always independent of the nature of dissipation and viscosity? Is it possible that the Reynolds dependence differs, but the limit (in distributional sense) remains the same? What does it happen to the solenoidal part of the acceleration as viscosity goes to zero? Could the ‘real’ inertial range of turbulence be adequately described by a suitable singular solution of some sort of Euler-like equations?

* About the concept of ‘non-locality’ of turbulence: is ‘cascade’ a well defined concept and is there a cascade in physical space? Is ‘cascade’ Eulerian, Lagrangian or what? These and other related questions will be briefly touched upon depending on the discussion and interest.

References

TSINOBER, A. 2009 An Informal Conceptual Introduction to Turbulence, Springer-Verlag.

TSINOBER, A. 2018 The Essence of Turbulence as a Physical Phenomenon. II edition (in press), Springer-Verlag.