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. ]]>

The Thom class of a complex vector bundle is a characteristic

class that contains, in some sense, more information than

the Chern class does. I give an expression of the Thom class in

the relative de Rham cohomology and show how this can be used

to prove a "universal Riemann-Roch theorem for embeddings".

Abstract

Several advanced multiscale representations, most notably curvelets and shearlets, were introduced during the last decade to overcome known limitations of wavelets and other traditional methods. In fact, even though wavelets are very efficient to handle signals with point singularities, they are suboptimal when dealing with edges and those distributed singularities which typically dominate multidimensional data. Shearlets, in particular, are specially designed to combine the power of multiscale analysis with ability to handle directional information efficiently. As a result, they offer very useful microlocal properties and optimally efficient representations, in a precise sense, for a large class of multivariate functions.

In this talk, I will illustrate the construction of shearlet frames and give a brief overview of their sparse approximation properties. Next, I will present and discuss several results illustrating the unique ability of the shearlet transform to provide a precise geometric characterization of singularities. These properties provide the theoretical underpinning for several state-of-the-art applications from signal processing and inverse problems, including data restoration, edge detection and feature extraction.

In this talk, I will illustrate the construction of shearlet frames and give a brief overview of their sparse approximation properties. Next, I will present and discuss several results illustrating the unique ability of the shearlet transform to provide a precise geometric characterization of singularities. These properties provide the theoretical underpinning for several state-of-the-art applications from signal processing and inverse problems, including data restoration, edge detection and feature extraction.

--

Il seminario, pur di ambito applicativo, include argomenti trasversali di analisi armonica e teoria dell'approssimazione.

]]>Can a fish with limited velocity capabilities reach any point in the (possibly unbounded) ocean? In a recent paper by D. Burago, S. Ivanov and A. Novikov, "A survival guide for feeble fish", an affirmative answer has been given under the condition that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This brilliant result extends some known point-to-point global controllability theorems though being substantially non constructive. We will give a fish a different recipe of how to survive in a turbulent ocean, and show how this is related to structural stability of dynamical systems by providing a constructive way to change slightly a divergence free vector field with vanishing mean drift to produce a non dissipative dynamics. This immediately leads to closing lemmas for dynamical systems, in particular to C. Pugh's closing lemma, saying also that the fish can eventually return home.

Joint work with Sergey Kryzhevich (Nova Gorica and St. Petersburg).

**Relatore**:Stefan Schreieder (Leibniz Universität Hannover)

**Luogo**: Aula 5

**Data e ora**: 29 settembre 2022, ore 14:30

The Gauss--Green formulas are of significant relevance in many areas of analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields, and thus led to the definition of the divergence-measure fields. These are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose distributional divergence is a Radon measure, and so they form a new family of function spaces, which in a sense generalize the $BV$ fields. The divergence-measure fields were introduced at first by Anzellotti in 1983 for the case $p = \infty$, and then they have been rediscovered in the early 2000s by many authors interested in various applications.

In this talk, we shall present an overview of such researches, with a particular focus on the results concerning essentially bounded divergence-measure fields. For such a family of vector fields, Silhavy (2005), Chen, Torres and Ziemer (2009) and Comi and Payne (2017) showed that Gauss--Green formulas hold for sets of finite perimeter, by proving the existence of essentially bounded interior and exterior normal traces on the reduced boundary of the given set.

Subsequent extensions of these results and generalizations to non-Euclidean geometries will be also briefly discussed.

]]>Network dynamics preserves the sum of all incoming pairwise coupling strengths and is designed to adaptively interact with system dynamics.For adaptive couplings, we use two adaptive coupling laws for the pairwise coupling strength. Kuramoto oscillators are assumed to be on the nodes of the networks.

We present frameworks that guarantee the emergence of synchronization for various coupling feedback laws. Our results generalize earlier work on the synchronization of Kuramoto oscillators in fixed and symmetric networks.

]]>- 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.