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.

]]>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.

]]>In this talk I first introduce a generic hard-particle system. In order to simulate its evolution, I then present two classes of time-stepping algorithms which typically involve a non-convex minimization problem in terms of the positions and velocities, respectively. This framework will then be used to study the dynamics of ballistic aggregation of hard-spheres and the evolution of a cell tissue. ]]>

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). ]]>

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.]]>

- 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.

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.

L. Pepe, *Insegnamenti matematici e ricerca scientifica*

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.

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