Offerta didattica avanzata – XXXIII Ciclo


Advanced teaching activites – Cycle XXXIII

Computable Analysis
Total hours: 10
SSD: MAT/01 Logica matematica

Learning objectives
This course aims to enable students to understand how one can compute with infinite objects and how this subject is linked to topology. Students should manage to use representations as a tool to handle abstract data types as they occur in analysis (real numbers, closed subsets, continuous functions). Another learning objective is to understand Weihrauch reducibility as a tool to compare computational problems. Students should be able to handle standard separation and proof techniques. Another aim is to learn how the choice problem of different spaces can be used to calibrate computational problems. Finally, students should also be able to see how this approach leads to a uniform version of reverse mathematics.

Prerequisites
Some basic knowledge in computability theory (computable functions, computable and computably enumerable sets, Gödel numberings) and in topology (metric spaces, continuous functions, open, closed and compact sets) is expected.

Course content
1. Computable functions
2. Universality
3. Represented spaces
4. Computable real numbers and metric spaces 5. Closed subsets and trees
6. The Weihrauch lattice
7. Choice and non-deterministic computations 8. Jumps and limit computations
9. Problems from analysis
10. Algorithmic randomness and probabilistic computations

References
J. Avigad and V. Brattka. Computability and analysis: the legacy of Alan Turing. In R. Downey, editor, Turing’s Legacy: Developments from Turing’s Ideas in Logic, volume 42 of Lecture Notes in Logic, pages 1–47. Cambridge University Press, Cambridge, UK, 2014.

V. Brattka. Computability and analysis, a historical approach. In A. Beckmann, L. Bienvenu, and N. Jonoska, editors, Pursuit of the Universal, volume 9709 of Lecture Notes in Computer Science, pages 45–57, Switzerland, 2016. Springer. 12th Conference on Computability in Europe, CiE 2016, Paris, France, June 27 – July 1, 2016.

V. Brattka, G. Gherardi, and A. Pauly. Weihrauch complexity in computable analysis. In in preparation. 2017.

V. Brattka, P. Hertling, and K. Weihrauch. A tutorial on computable analysis. In S. B. Cooper, B. Löwe, and A. Sorbi, editors, New Computational Paradigms: Changing Conceptions of What is Computable, pages 425–491. Springer, New York, 2008.

K. Weihrauch. Computable Analysis. Springer, Berlin, 2000.

Teacher
Vasco Brattka


On transitivity and Devaney’s chaos: autonomous and nonautonomous discrete dynamical systems
Total hours: 10
SSD: MAT/02 Algebra

Prerequisites
Some basic knowledge in mathematics.

Course content
This is intended as a brief course on topological transitivity and Devaney’s chaos for autonomous (respectively, non-autonomous) discrete dynamical systems on metric spaces. Our main goal is to give an overview of basic results on these topics. We emphasize the difference between autonomous and non-autonomous systems. The outcomes are purely “topological” and they do not reflect differentiable dynamics or ergodic theory aspects of these topics.

Teacher
Manuel Sanchis Lòpez


Software Configuration Management
Total hours: 20
SSD: INF/01 Informatica

Course content
Software Configuration Management (SCM) is one of the fundamental key process areas of the Capability Maturity Model for software development. It will have to be in place if an organization wants to progress from the initial level of maturity that is characterized by chaos and confusion.
SCM has two different focus areas: what the company needs to control changes to its products and ensure their integrity – and what a team of developers need to manage and co-ordinate their collaboration on a project. For many aspects the needs are quite similar in nature, but will be implemented in different ways. However, there are also some needs that are particular for each focus area.

In two pairs of lecture-exercises, this short course will introduce the students to the main concepts and principles of SCM for each of the two focus areas. The lectures will explain and motivate the concepts and principles and during the exercises the students will discuss how to apply and use the concepts and principles and how to relate and adapt them to different situations and contexts.
In addition, there is a computer lab where the students will have the opportunity to experiment with different work models and co-ordination mechanisms and will obtain a general template for evaluating version control tools.

Teacher
Lars Bendix