Dipartimento di Matematica e Informatica

Sums of squares in fields - Karim Becher (Università di Antwerp, Belgio)

Stumbo Fabio — 2017-04-03T09:00:19Z

Calendario:

martedì 4 aprile ore 15.30 aula 4
mercoledì 5 aprile ore 15.30 aula 4
giovedì 6 aprile ore 15.30 aula 5
venerdì 7 aprile ore 10.30 aula 1

Programma:

I Sums of squares in number theory
II Sums of squares in real algebra
III Basic quadratic form theory over fields
IV Local-global principles for quadratic forms

Abstract:

In this course I want to touch on several aspects of sums of squares and their role in the development of modern algebra. The focus will be on sums of squares in commutative rings and in particular in fields.
Some basics of the theory of quadratic forms over fields will be introduced. The course will only assume some general algebraic knowledge on bachelor level.

The study of sums of squares is an old topic in number theory and algebra.
In Brahmagupta’s book an identity is given for writing a product of two sums of two squares again as a sum of two squares.
In modern terms it can be obtained by using the norm form for the ring of Gaussian integers, or more generally for the quadratic extension of a given commutative ring by adjoining a square root of -1.
Euler gave a similar identity for products of sums of four squares, which can likewise be viewed as the norm form of an algebraic structure, the quaternions.
He probably did this with the goal to prove that every positive integer is a sum of four squares, which was one of Fermat’s (correct) statements.
This way the problem had been reduced to showing the claim for prime numbers, and on this basis the proof was later completed by Lagrange.

Later Degen and Cayley found independently a similar identity for sums of eight squares and a related algebraic structure, the octonions.
On the other hand, attempts to find such an identity for sums of 16 squares failed and in the end Hurwitz showed that such an identity cannot exist.
More generally, the problem is to express a product of two quadratic forms (homogeneous polynomials of degree two) as a quadratic form applied to bilinear expressions in the variables.
If this is possible then a solution to this problem is called a composition formula. Using Clifford algebras one can explain why composition formulae do not exist for certain triples of forms, just by their dimensions.

Many problems of an analytic flavour can be described by asking whether a certain real valued function takes only non-negative values.
An obviously sufficient condition is that the function can be written as a sum of squares of other functions.
For rational functions (fractions of polynomials), Hilbert asked in his 17th problem whether the converse is true, namely whether any rational function taking only non-negative values can be written as a sum of squares of rational functions.
The positive answer was found by Artin in 1927 and it opened the way to a new research area, called real algebra.
Here different versions of the problem can be considered, for example, for a polynomial taking only non-negative values, one can ask whether it can be written as a sum of squares of polynomials.
This is generally not true. But also the number of squares in such an expression can be studied. This gives rise to the definition of the Pythagoras number of a commutative ring, the smallest number p such that every sum of squares is equal to a sum of p squares.
For a field we may ask whether -1 can be written as a sum of squares in it and if so, how many squares are needed. The smallest such number s, if it exists, is called the level of the field.
Pfister’s showed that the level of a field where -1 is a sum of squares is always a 2-power and that on the other hand every 2-power is the level of some field.
The proofs of these results give a nice insight into the topics of quadratic form theory over fields.

A main feature of 20th century algebra, number theory and geometry is the local-global analysis of problems.
It started with Hensel’s construction of the p-adic integers. It is not too difficult to see that polynomials with integer coefficients can have local solutions over every p-adic completion but no solution over the integers, but specifically for quadratic forms, by the Hasse-Minkowski Theorem, the local-global way of reasoning for finding (nontrivial) solutions works. Unfortunately, the same fails for quadratic forms over most other types of fields (when taking completions with respect to discrete valuations).
Recently, however, it was proven for certain types of function fields of arithmetic curves that such a local-global principle holds. In my own research I am studying some of these situations with the particular interest for the study of sums of squares, say, over the function field of a curve over R((t)), the field of formal Laurent series with real coefficients. I will discuss some of these results in relation to the study of valuations on these fields and give some examples.

Quantitative symmetry in OBVP's and related problems in geometry - Giorgio POGGESI (Università di Firenze)

Stumbo Fabio — 2017-03-28T14:35:09Z

Alexandrov's Soap Bubble theorem states that a compact hypersurface embedded in R^N with constant mean curvature must be a sphere. We will start discussing about the stability for this celebrated result. The question is: how much a hypersurface is near to be a sphere if its mean curvature is near to be constant in some norm?
Then we will deal with the stability for the symmetry of some related overdetermined boundary value problems.
The talk is based on a joint work with R. Magnanini (Firenze).

On the boundedness of minimal models of general type - Luca Tasin (Università Roma Tre)

Stumbo Fabio — 2017-07-19T16:22:25Z

Abstract. It is well known since the beginning of the 20th century that a minimal model of a complex surface is unique and that minimal models of surfaces of general type with bounded volume form a bounded family.

In this talk I will show how the number of minimal models of an n-dimensional manifold can be bounded in term of its volume for any n. Moreover, I will explain that in any dimension minimal models of general type and bounded volume form a bounded family.

This is based on a joint work with D. Martinelli and S. Schreieder.

Lipschitz truncations versus regularity: an overview - Bianca STROFFOLINI (Universita’ di Napoli)

Stumbo Fabio — 2017-03-13T17:39:25Z

Lipschitz continuity for energy integrals with ellipticity conditions at infinity - Michela Eleuteri (Università di Modena e Reggio Emilia)

Stumbo Fabio — 2017-03-21T08:44:42Z

Abstract:

In the general vector-valued case, we present some recent regularity results

obtained in cooperation with P. Marcellini and E. Mascolo. More precisely

we show the Lipschitz continuity of local minimizers to some integrals of

the Calculus of Variations where the Lagrangian has the form g(x, |Du|),

under some suitable p,q-growth conditions. We prove our results only for

large values of the gradient and without further structure conditions on the

integrand. We apply the regularity results to weak solutions to nonlinear

elliptic systems and to the case of some functionals with variable exponent.

We will open the seminar with a very general introduction concerning regularity

results for equations and systems in divergence form, in the framework of standard and nonstandard conditions.

Giochi a campo medio: equilibri e modelli variazionali - Filippo SANTAMBROGIO (Université Paris-Sud, FRANCIA)

Stumbo Fabio — 2017-01-15T11:28:32Z

Farò una breve introduzione alla teoria dei "Giochi a Campo Medio" (Mean Field Games), proposti qualche anno fa da J-M Lasry et P-L Lions come modello in cui si cerca l'equilibrio di Nash di una popolazione di agenti, dove ciascuno sceglie come muoversi, tenendo conto che incorre in un costo che è una funzione crescente della densità degli individui che incontra, come in classici problemi di traffico e più in generale in "giochi di congestione". Questo dà luogo a un sistema di PDE accoppiate: un'equazione di continuità che descrive il moto della popolazione, ma la cui velocità dipende dal gradiente della funzione valore del problema di controllo di ogni individuo, e un'equazione di Hamilton-Jacobi sulla funzione valore stessa, che fa apparire però anche la densità. Mi concentrerò essenzialmente sul caso dove la condizione di equilibrio corrisponde all'equazione di Eulero-Lagrange di un problema convesso del calcolo delle variazioni, e presenterò alcuni risultati recenti di regolarità che permettono di dare una formulazione rigorosa della condizione di equilibrio, altrimenti solo formale. Alla fine del talk presenterò anche una variante, dove il costo di congestione è rimpiazzato da un vincolo di capacità: in questo caso appare una variabile aggiuntiva, che gioca il ruolo di una pressione dal punto di vista fisico, ma di un prezzo dal punto di vista economico.

Distribuzioni Lagrangiane, strutture simplettiche ed operatori di Fourier su varietà asintoticamente Euclidee - Sandro Coriasco (Università di Torino)

Stumbo Fabio — 2017-01-11T09:45:13Z

Il calcolo che ne risulta è particolarmente adatto alle applicazioni su varietà compatte prive di bordo. Un'analoga teoria degli operatori integrali di Fourier su varietà non compatte è tuttora incompleta. Una scelta naturale di una classe di operatori pseudodifferenziali, che tali operatori di Fourier dovrebbero contenere come caso particolare, è quella definita dai cosiddetti simboli SG, introdotti negli anni '70 da C. Parenti e H.O. Cordes. E' quindi necessario determinare opportune classi di sottovarietà Lagrangiane, che permettano di ottenere un "buon" calcolo di operatori di Fourier associati a queste classi di simboli, globalmente definiti su R^d.

Nel seminario illustrerò recenti risultati in questo ambito, ottenuti in collaborazione con R. Schulz (Leibniz Universitat Hannover, Germania).

Lo studio delle sottovarietà Lagangiane è un argomento importante in geometria simplettica. Una delle motivazioni è il ruolo fondamentale che svolgono come "luogo delle singolarità" dei nuclei degli opeatori integrali di Fourier su varietà. Tali nuclei sono infatti distribuzioni Lagrangiane, associate a sottovarietà Lagrangiane (nel caso più semplice, date da grafici di relazioni canoniche).Il calcolo che ne risulta è particolarmente adatto alle applicazioni su varietà compatte prive di bordo. Un'analoga teoria degli operatori integrali di Fourier su varietà non compatte è tuttora incompleta. Una scelta naturale di una classe di operatori pseudodifferenziali, che tali operatori di Fourier dovrebbero contenere come caso particolare, è quella definita dai cosiddetti simboli SG, introdotti negli anni '70 da C. Parenti e H.O. Cordes. E' quindi necessario determinare opportune classi di sottovarietà Lagrangiane, che permettano di ottenere un "buon" calcolo di operatori di Fourier associati a queste classi di simboli, globalmente definiti su R^d.Nel seminario illustrerò recenti risultati in questo ambito, ottenuti in collaborazione con R. Schulz (Leibniz Universitat Hannover, Germania).

Density of bounded maps in Sobolev spaces into closed manifold - Pierre BOUSQUET (Univ. Toulouse, Francia)

Stumbo Fabio — 2017-04-11T06:26:17Z

ABSTRACT . Given a submanifold N embedded in an Euclidean
space R ν , the Sobolev space W 1,p (B m ; N ) is the set of those maps
in W 1,p (B m ; R ν ) which are constrained to take their values into
N . Such maps exhibit some specific singularities related to this
constraint. When N is compact, these singularities are closely
linked to the topology of N . In this talk, we investigate the case
where N is assumed to be closed but not necessarily compact. The
main novelty is that the geometry of N now plays a crucial role in
the analysis of the singularities. This is a joint work with Augusto
Ponce and Jean Van Schaftingen.

Approximation methods for nonlinear eigenvalue problems - Erik Lindgren (KTH Royal Institute of Technology, Sweden)

Stumbo Fabio — 2017-05-08T14:58:44Z

Abstract

In this talk, I will discuss two novel methods for approximating extremals
of "nonlinear" Rayleigh quotients. The first approximation scheme is based on the
method of inverse iteration for square matrices. The second method is based
on the large time behavior of solutions of a doubly nonlinear evolution, corresponding
to the heat equation in the case of the eigenvalue problem for
the Laplace operator. Both schemes have the property that the
Rayleigh quotient is nonincreasing along solutions and that properly scaled
solutions converge to an extremal of the Rayleigh quotient. I will focus on
concrete examples in Sobolev spaces where our results apply.

The talk is based on joint work with Ryan Hynd (University of Pennsylvania).

Apache Hadoop e il suo ecosistema: strumenti opensource al servizio dei Big del BigData - Filippo Natali

Stumbo Fabio — 2017-05-23T15:09:01Z

Apache Hadoop è il principale framework opensource utilizzato dai colossi del Big Data per lo storage e l'analisi di grandi data set in ambiente distribuito.

L'intervento offrirà una panoramica su Hadoop, l'ecosistema di applicazioni che ne estendono le funzionalità, esempi di utilizzo e problematiche di gestione su larga scala.

A construction of big Maximal Cohen Macaulay modules - Dolors Herbera (Universitat Autònoma de Barcelona)

Stumbo Fabio — 2017-05-31T07:14:33Z

Exchange models in the study of economic inequality and social mobility

Marangon Sara — 2019-05-28T06:35:21Z

Abstract: In this seminar a model expressed by a system of differential equations for the description of micro-processes of money exchange, taxation and redistribution in a closed market society will be presented. The approach within which the model is developed aims to derive observable collective features like the income profile as the result of the interplay of a whole of economic interactions. Issues as the effect of different fiscal policies are investigated, with a specific focus on the emergence of economic inequality.

Signature tensors and the Rough Veronese variety - Francesco Galuppi (MPI Lipsia)

Stumbo Fabio — 2018-06-19T08:39:04Z

Abstract: Signature tensors of paths are a classical topic in stochastic analysis. From a geometric viewpoint, it is interesting to look at the varieties described by these signatures inside the space of tensors. After reviewing some results on piecewise linear and polynomial signature varieties, we introduce the signature variety of the class of rough paths. This shows some similarites with the Veronese variety

Su alcuni rapporti tra matematica e scale musicali - Stefano Isola (Università degli Studi di Camerino)

Stumbo Fabio — 2018-05-27T18:19:06Z

Abstract: Si discuteranno alcune soluzioni “antiche" del problema della divisione dell’ottava per la costruzione delle scale musicali e alcuni moderni “temperamenti” necessari per adattarle a nuovi contesti musicali. Tempo permettendo, si discuteranno anche alcuni aspetti della teoria delle scale ben formate e delle sue connessioni con le frazioni continue e la combinatoria delle parole.

Mean Field Modeling of a Financial Market (T. Trimborn)

Foschi Damiano — 2015-09-13T16:44:26Z

Cari colleghi,

mercoledì 16 settembre alle ore 15 in un aula del Dipartimento il Dott. Tosten Trimborn (Aachen) terrà il seguente seminario:

Mean Field Modeling of a Financial Market

Abstract: Many classical financial-market models fail to reproduce stock price anomalies, such as fat tails in asset returns. Modern econophysical models indicate that irrational behavior of investors might be a reason for these price anomalies. Mean field modeling seems to be a promising path describing such models on a mesoscopic scale and investigating the origins of these price anomalies. In this talk I will introduce the econophysical market model of Cross et al. (Physica A, 2005) and will present the corresponding mean field model.

Cordialmente,

Lorenzo