Extension fields, and classes in the genus of a lattice

Wednesday 14 December 2016 h.14:30, Room 2BC30
Frances Odumodu (Padova, Dip. Mat.)
“Extension fields, and classes in the genus of a lattice“

In this talk, which will be accessible to a large audience, a first part will be devoted to a basic reminder on extension fields with examples, and a second part to the more specific framework of number fields, i.e. finite degree extensions of rational numbers. Concerning the latter part, the Hasse-Minkowski local-global theorem for quadratic forms fails in general at the integral level, hence there are two levels of classification, the genus (local) and the integral class (global): we shall focus on some results concerning the classes in the genus of a lattice and in particular the trace form.

Biologically inspired deduction of Optimal Transport Problems

Wednesday 30 November 2016 h.14:30, Room 2BC30
Enrico Facca (Padova, Dip. Mat.)
“Biologically inspired deduction of Optimal Transport Problems”

In this talk, after a brief introduction on the Optimal Transport Problems and some PDEs based formulation, we will present a recently developed approach, based on an extension of a model proposed by Tero et al (2007), for the simulation of the dynamics of Physarum Polycephalum, a unicellular slime mold showing surprising optimization ability, like finding the shortest path connecting two food sources in a maze.
We conjecture that this model is an original formulation of the PDE-based OT problems. We show some theoretical and numerical evidences supporting our thesis.

Products of elementary and idempotent matrices and non-euclidean pids

Wednesday 16 November 2016 h. 14:30, Room 2BC30
Laura Cossu (Padova, Dip. Mat.)
“Products of elementary and idempotent matrices and non-euclidean pids”

It is well known that Gauss Elimination produces a factorization into elementary matrices of any invertible matrix over a field. Is it possible to characterize integral domains different from fields that satisfy the same property? As a partial answer, in 1993, Ruitenburg proved that in the class of Bézout domains, any invertible matrix can be written as a product of elementary matrices if and only if any singular matrix can be written as a product of idempotents.
In this seminar we present some classical results on these factorization properties and we focus, in particular, on their connection with the notion of weak-Euclidean algorithm. We then conclude with a conjecture on non-Euclidean principal ideal domains, rare and interesting objects in commutative algebra, and some related results.
In order to make the talk understandable to a general audience, we will recall basic definitions of Commutative Ring theory and provide easy examples of the objects involved.

Doctoral School in Mathematical Science - Opening Day 2016/2017

The opening day of the Doctoral School in Mathematics will take place on October 5, 2016, at
in room 1BC45.

15:30-16:15: presentation of the activities/courses of PhD Programme 2016/2017
16:15-17:15: talk of Veronica Dal Sasso "Integer Linear Programming to solve Large-Scale problems"
17:30: refreshments at the common room of 7th floor

For further information feel free to contact Pierpaolo Soravia.

Integer Linear Programming to solve Large-Scale problems

Our first seminar of 2016/17 will be held as a special event
inside the Opening Day of the Doctoral School (15:30, 1BC45)

Wednesday 5 October 2016 h.16:15, Room 1BC45
Veronica Dal Sasso (Padova, Dip. Mat.)
“Integer Linear Programming to solve Large-Scale problems"

Integer linear programming is widely used to find optimal solutions to problems that arouse in the real world and are related to logistics, planning, management, biology and so on. However, if from a theoretical point of view it is easy to give a formulation for these problems, from a computational point of view their implementation can be impractical due to the high number of constraints and variables involved.
During this seminar I will present classical results for dealing with large-scale integer linear programs and their application to a particular bioinformatic problem, related to the study of the human genome, that helps recovering information useful to study diseases and populations' behaviours.

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.

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