### Seminari

### Prossimi Seminari

**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, Milano**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

### Seminari Passati

**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.**Emodinamica della circolazione epatica: problemi e nuove acquisizioni**

Massimiliano Tuveri, Azienda Ospedaliera Universitaria Integrata, Verona, Italy

giovedì 29 novembre 2018 alle ore 11:30, aula consiglio VII pianoABSTRACT: Il fegato ha una vascolarizzazione peculiare comparata con gli altri organi. Ha infatti afferente vascolari di tipo arterioso (arteria epatica) e venoso (vena porta) che confluiscono nel sistema sinusoidale del fegato. Da qui si diparte una efferenza unica venosa rappresentata dalle vene sovraepatiche. La vascolarizzazione intraparenchimale ha una struttura complessa: si divide in due lobi distinti e autonomi ognuno dei quali presenta dei segmenti (8 in tutto) che sono perfusi indipendentemente. Questo riveste una particolare importanza dal punto di vista clinico, specialmente chirurgico, quando si tratta di rimuovere parte del fegato. La quantità di fegato necessaria per la sopravvivenza del paziente, quando per motivi oncologici debba essere asportata ampia parte del fegato, è difficile da predire. Il rischio di insufficienza epatica acuta è infatti molto alto. Il paziente inoltre sviluppa un particolare quadro emodinamico rappresentato da uno stato circolator!

io ipercinetico con aumento della gittata cardiaca e basse pressioni periferiche, aumento delle resistenze intraparenchimali epatiche e aumento della pressione portale con sviluppo di vie collaterali portosistemiche. I meccanismi alla base della rigenerazione epatica, che permettono al restante parenchima di rimanere vitale e assicurare le funzioni vitali, sono fondamentalmente regolati dallo shear stress arterioso ma sopratutto venoso. Il severo aumento della shear stress è stato posto anche in relazione al danno sinusoidale per l’espressione patologica di sostanze vasoattive.

Un fattore chiave appare la comprensione dell’entità e del comportamento dello shear stress nel sistema portale. Vi è come detto una correlazione diretta tra la variazione e il gradiente dello shear stress e la modificazione della microcircolazione epatica e la secrezione endoteliale di sostanze vasoattive. La modellazione del fegato e delle sue componenti emodinamiche appare quindi un fattore cruciale per comprendere il funzionamento e eventuali fattori correttivi da apportare in sede clinica. Un fegato computazionale paziente-specifico appare quindi uno strumento estremamente utile per predire le modificazioni emodinamiche e pianificare nel futuro possibili strategie terapeutiche.

**Characterization of Attraction Domains for Generic Quantum Semigroups**

Damiano Poletti, Politecnico di Milano

giovedì 29 novembre 2018 alle ore 14:30 precise, Aula Seminari III pianoABSTRACTIn my talk I will present the topics covered during my theses work, concerning problems linked to decoherence and asymptotic evolution of Quantum Markov Semigroups. More precisely I will introduce the subclass of Generic Quantum Semigroups and show existing results characterizing their decoherence-free subalgebra and their invariant states. Eventually I will show the results we obtained regarding attraction domains of invariant states, a topic closely related to the asymptotic behaviour of any state undergoing system evolution.**The Quantitative Alexandrov Theorem in Space forms**

Luigi Vezzoni, Università degli Studi di Torino

martedì 27 novembre 2018 alle ore 15:15, Aula seminari 3° pianoABSTRACTThe talk focuses on a recent generalization of a classical result of Alexandrov. The celebrated Alexandrov’s Soap Bubble Theorem states that the spheres are the only closed (i.e. compact and without boundary) constant mean curvature hypersurfaces embedded in the Euclidean space. The theorem has been generalized to the hyperbolic space and to the hemisphere and to a large class of curvature operators. The main result of the talk is a quantitative version of Alexandrov’s theorem which I’ve obtained in collaboration with Giulio Ciraolo and Alberto Roncoroni by using a quantitative study of the method of the moving planes. The theorem implies a new pinching Theorem for hypersurfaces in space forms.**First Principles Determination of Reaction Rates**

Carlo Cavallotti, Dipartimento di Chimica, Materiali e Ingegneria Chimica, “G. Natta”, Politecnico di Milano

martedì 27 novembre 2018 alle ore 10:30, aula Saleri VI pianoABSTRACTThe central focus of chemical kinetics is the determination of the rate at which one or more molecules react to transform into new molecules through the rearrangements of chemical bonds.

Theoretically, chemical kinetics lies at the interface between quantum and classical mechanics, with the explicit formulation of rate laws requiring an extensive use of statistical thermodynamics.

In the last years a priori rate calculations for gas phase reactions have undergone a gradual but dramatic transformation, with current predictions often rivaling the accuracy of the best available experimental data.

In this seminar I will talk about the current status of ab initio chemical kinetics and of the challenges that must still be met in order to transform this branch of physical chemistry from a qualitative to a quantitative predictive science. A few example of research areas were contributions are necessary are the following: 1) the efficient determination of absolute and relative minima as well as first order saddle points on multidimensional potential energy surfaces for which gradient and, if necessary, Hessian information is available; 2) the determination of the number of energy states for a collection of non harmonic quantum oscillators; 3) the determination of a large number of eigenvalues for the quantum Hamiltonian of multidimensional rotors; 4) efficient approaches for the integration of the energy and momentum resolved master equation for chemically reactive systems.