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"

Abstract
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"

Abstract
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"

Abstract
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"

Abstract
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"

Abstract
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"

Abstract
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.

Computational social choice: between AI and Economics

Thursday 17 December 2015 h. 14:30, Room 2BC30
Andrea Loreggia (Padova, Dip. Mat.)
"Computational social choice: between AI and Economics"

Abstract
During the last decades, the trend has been for disciplines to converge on common techniques to be used in similar problems, besides focusing on specific techniques to be used in narrow domains. AI is one of the best examples: the cross-fertilisation process leads to a very fascinating solutions. Consider for example genetic algorithms, which mimic evolutionary mechanisms to solve search and optimization problems. The individualistic approach of problem solving becomes insufficient: concepts, techniques and experts need to collaborate to get a better understanding of the problems they would like to solve. The techniques that AI makes available are being used by many other disciplines. AI nowadays inundates our everyday life with tools and methods that are hidden in our household electrical devices, smartphones and much more. Starting from the field of multi-agent systems, researchers in AI recently considered the use of models and problems from economics. Notable examples are voting systems used to aggregate the results of several search engines, game theoretic methods that analyse the complex interaction of autonomous agents, and matching procedures implemented on large-scale problems such as the coordination of kidneys transplants and the assignment of students to schools. In this scenario, a number of research lines federated under the name of computational social choice. The need for a computational study of collective decision procedures is clear. On the one hand, from crowdsourcing to university admission ranking, many real-life applications apply existing social choice methods to large scale problems. On the other hand, collective decision-making is not a prerogative of human societies, and multi-agent systems can use these methods to coordinate their actions when facing complex situations. In this talk, we would like to focus on two examples that highlight the impact of a computational approach to classical problems of collective choice. First, by studying repeated decisions (think of opinion polls that precede an election) to evaluate the quality of the result, and, second, by devising innovative procedures to predict the preferences of a collection of individuals.

A simple mathematical model for micro-swimmers

Wednesday 2 December 2015 h. 14:30, Room 2BC30
Marta Zoppello (Padova, Dip. Mat.)
"A simple mathematical model for micro-swimmers"

Abstract
What does it mean swimming? How can mathematics treat this problem? What is the best strategy to move in a certain direction?
The study of the swimming strategies of micro-organisms is attracting increasing attention in the recent literature. One of the main difficulties is the complexity of the hydrodynamic forces exerted by the fluid on the swimmer as a reaction to its shape changes.
We show that there exists an optimal swimming strategy which leads to minimize the time to reach a desired target. Numerical simulations performed are in good agreement with theoretical predictions and suggest that the optimal strategy is periodic, i.e. composed of a sequence of identical strokes.

Learning with Kernels

Wednesday 18 November 2015 h. 14:30, Room 2BC30
Michele Donini (Padova, Dip. Mat.)
"Learning with Kernels"

Abstract
To solve a problem on a computer, we need an algorithm, which is a sequence of instructions that should be carried out to transform the input into the output. For some tasks, we do not have an algorithm: we know what the input is, we know what the output should be but we do not know how to transform the input into the output. What we lack in knowledge, we make up for in data. We can exploit data to "learn" using a Machine Learning algorithm, that is able to extract automatically the algorithm for the task.
In this talk, we give an introduction to a family of Machine Learning algorithms called Kernel Methods, starting from a general introduction to the Machine Learning problems and its purposes. After building up the fundamental tools of learning with kernels, we will introduce the principal ideas behind this family of algorithms and its ability to learn automatically using data.

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