Computed Tomography: a real case example of inverse problem

Wednesday 15 June 2016 h. 15:00, Room 2BC30
Elena Morotti (Dip. Mat.)
"Computed Tomography: a real case example of inverse problem"

X-ray computed tomography (CT) is a well known medical imaging technique, that seeks to reveal internal structures hidden by the skin and bones. Mathematically, the CT process can be modelled as a linear system and the image reconstruction is a challenging inverse problem. In this talk I will show both phisical and mathematical basic concepts, to explain the CT process, and the two possible approaches to solve the problem (leading to analitical or iterative numerical methods). Finally, I will shortly introduce the Digital Breast Tomosynthesis (DBT) technology, that is a 3D emerging technique for the diagnosis of breast tumors, together with numerical results for a simulated problem.

Polyhedral structures in algebraic geometry

Wednesday 1 June 2016 h. 14:30, Room 2BC30
Stefano Urbinati (Dip. Mat.)
"Polyhedral structures in algebraic geometry"

Algebraic geometry studies the zero locus of polynomial equations connecting the related algebraic and geometrical structures. In several cases, nevertheless the theory is extremely precise and elegant, it is hard to read in a simple way the information behind such structures. A possible way of avoiding this problem is that of associating to polynomials some polyhedral structures that immediately give some of the information connected to the zero locus of the polynomial. In relation to this strategy I will introduce Newton-Okounkov bodies and Tropical Geometry.

Fractional Calculus: Numerical Methods and Models

Wednesday 25 May 2016 h. 14:30, Room 2BC30
Abdelsheed Ismail Gad Ameen (Dip. Mat.)
"Fractional Calculus: Numerical Methods and Models"

In this talk, we first give a short introduction of fractional calculus (FC) and its geometrical, physical interpretation. Then, we discuss the differential equations of fractional order (Caputo type) which have recently proved to be valuable tools for modeling of many biological phenomena. Most of fractional ordinary differential equations (FODEs) do not have exact analytic solutions so that numerical techniques must be used. Hence, we present the fractional Euler method to solve systems of nonlinear FODEs and show how to use this method for solving the Susceptible-Infected-Recovered (SIR) model of fractional order.

Isoperimetric inequalities in Carnot-Caratheodory spaces

Wednesday 4 May 2016 h. 14:30, Room 2BC30
Valentina Franceschi (Dip. Mat.)
"Isoperimetric inequalities in Carnot-Caratheodory spaces"

One of the most ancient mathematical problems is Dido's problem, appearing in Virgil's Aeneid: what is the shape to give to a rope in order to enclose a maximal region of land? The expected solution is of course the circle. Despite the ancient origins, a rigorous mathematical formulation and solution is quite recent, dating back to the 1950s when Caccioppoli and De Giorgi introduced the notion of perimeter in the n-dimensional Euclidean space. The latter notion led to the study of isoperimetric inequalities and to the solution of Dido's problem generalized to n dimensions. Mathematicians then generalized isoperimetric inequalities to different frameworks, such as riemannian manifolds and metric spaces. After an overview of the classical definitions, in this talk, we present isoperimetric inequalities in a class of metric spaces arising from the study of hypoelliptic differential operators, called Carnot-Caratheodory spaces. We conclude presenting the main conjecture in this framework (Pansu's conjecture) ad some related results.

Cheapest Routes with Integer Linear Programming

Wednesday 13 April 2016 h. 14:30, Room 2BC30
Michele Barbato (LIPN, Univ. Paris 13, France)
"Cheapest Routes with Integer Linear Programming"

Combinatorial Optimization deals with the optimization of a function over a finite, but huge, set of elements.
It has a great impact on real life, as several problems arising in logistics, scheduling, facility location, to cite a few, can be stated as Combinatorial Optimization problems. Often problems of this kind can be expressed as Integer Linear Programs (ILP), i.e., problems in which the function to be optimized is linear and so are the constraints that define the feasibility set. In the first part of the talk, we provide an introductory presentation of some well-established methods in Integer Linear Programming. These methods are presented through examples that, in several cases, also motivate theoretical questions (e.g., the polyhedral study). We will consider as initial case of study the Traveling Salesman Problem (TSP). The TSP consists in finding the cheapest route that visits a prescribed set of cities exactly once, before returning to the starting point. As such, the TSP is a prototype of several other problems arising in logistics. In the second part of the presentation we will talk about the Double Traveling Salesman Problem with Multiple Stacks, that combines the construction of a cheapest route with loading constraints. We will reveal links between this problem and the TSP, as well as the limitations that a purely routing-based approach has for this problem.

Summer Research Graduate Programme 2016

Summer Research Graduate Programme 2016
con scadenza 13/04/2016 e rivolto a iscritti al terzo anno dei corsi di dottorato negli ambiti di Economia, Finanza, Statistica e Matematica.
Informazioni sul bando

Cosheaves, an introduction

Wednesday 16 March 2016 h. 14:30, Room 2BC30
Pietro Polesello (Dip. Mat.)
"Cosheaves, an introduction"

It is well known that locally defined distributions glue together, that is, they define a sheaf. In fact, this follows immediately from the fact that test functions (i.e. smooth functions with compact support) form a cosheaf, which is the dual notion of a sheaf. By definition, cosheaves on a space X and with values in category C are dual to sheaves on X with values in the opposite category C'. For this reason, cosheaves did not attract much attention, being considered as part of sheaf theory. However, passing from C to C', may cause difficulties, as in general C and C' do not share the same good properties needed for sheaf theory. Moreover, dealing with cosheaves may be more convenient, as they appear naturally in analysis (as the compactly supported sections of c-soft sheaves, such as smooth functions or distributions), in algebraic analysis (e.g. as the subanalytic cosheaf of Schwartz functions), in topology (in relation with Fox's theory of topological branched coverings), and in tops theory. Moreover, as sheaves are the natural coefficient spaces for cohomology theories, cosheaves play the same role for homology theories, such as Cech homology, and they are (hidden) ingredients of Poincare' duality (recently, cosheaves infiltrated Poincare'-Verdier duality in the context of Lurie's "higher topos theory"). In this seminar, I will give a brief introduction to cosheaves, giving examples and explaining the relation with sheaves and with Fox's theory.

Introduction to propagation of chaos for mean-field interacting particle systems

Wednesday 2 March 2016 h. 14:30, Room 2BC30
Luisa Andreis (Dip. Mat.)
"Introduction to propagation of chaos for mean-field interacting particle systems"

The purpose of this talk is to give an overview on mean-field interacting particle systems. We will focus on the notion of propagation of chaos, which aims to understand the connection between the microscopic and the macroscopic description of phenomena. Usually, an interacting particle system refers to the microscopic level and a corresponding nonlinear process describes the macroscopic one. In a great number of situations, under hypothesis on the symmetry of the system and on the type of interaction, the link between these two levels is precisely given by propagation of chaos.
Since the talk is intended for a general audience, we start by recalling basic definitions and results of Probability. Then we introduce the basic concepts of the theory, by means of classical examples as well as recent ones.

On the behavior of membranes and plates upon perturbations of shape and density

Wednesday 17 February 2016 h. 14:30, Room 2BC30
Luigi Provenzano (Dip. Mat.)
"On the behavior of membranes and plates upon perturbations of shape and density"

In this talk we consider eigenvalue problems for second and fourth order partial differential operators. Such problems arise from the study of the transverse vibrations of thin membranes and plates, respectively. We are interested in the behavior of the normal modes of vibration (i.e., the eigenvalues) upon variations of the shape and the density of the membrane/plate. In particular, we shall consider the issue of the optimization of the eigenvalues depending on such parameters, under suitable constraints (of fixed volume or mass, for example).
The talk is of introductory type and is intended for a general audience, no matter the field of expertise.

Quivers, representations of algebras and beyond

Wednesday 20 January 2016 h. 14:30, Room 2BC30
Gabriella D'Este (Univ. Milano, Dip. Mat.)
"Quivers, representations of algebras and beyond"

I will illustrate some results obtained by using techniques and general ideas coming from representation theory of finite dimensional algebras. These algebras will almost always be "path algebras" given by quivers, that is oriented graphs, with finitely many vertices and arrows. In less technical words, I will describe some results of applied linear algebra.

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