Centre for the Book

62 Queen Victoria Street, Cape Town, 8001

27th of January 2023 

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Time: 17 - 17:30:   Welcome  from Mr. Emanuele Pollio (Consul of Italy in Cape Town)

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Time: 17:30 -- 18:15.

Speaker: Francesco Russo

Title: Some Symmetries in Music

Affiliation: University of Cape Town, Cape Town, South Africa

Abstract: The present talk will describe some techniques, which allow us to translate a musical piece in a time--frequency diagram of the plane. An application is shown for the Musical Offering of J. S. Bach. The main idea is to illustrate a mathematical model, which is based on the concept of ''equal temperament''. At the same time we will show that some mathematical functions have a natural translation in the musical language. Baroque authors, and especially J. S. Bach, made a large use of the ''equal temperament'' along with a series of interesting techniques of Harmony and Counterpoint. Indeed the techniques of Baroque composers may be naturally regarded as appropriate transformations of the Euclidean plane; for instance, compositions such as ''canons in contrario motu'' may be regarded as an appropriate ''reflection'' in the mathematical language. Therefore one can better understand the logic of composition of a musical piece, when an appropriate mathematical context is presented. Time permitting, we will note that the first proposals for the ''equal temperament'' were given by Gioseffo Zarlino and by the father of Galileo Galilei (Vincenzo Galilei) during the Italian Renaissance. Both the contributions of Gioseffo Zarlino and Vincenzo Galilei were based on the first scientific studies of  acoustics of musical instruments, so a first interesting approach to the modern physics originates from the perspective of the music.

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Time: 18:30 -- 19:15.

Speaker: Francesco Belardo

Title: The Importance of Eigenvalues and Eigenvectors in the Study of Complex Networks

Affiliation: University of Naples “Federico II”, Naples, Italy

Abstract: Real-world complex systems are modelled by means of complex networks, consisting of nodes (vertices) and links between them (edges). Complex networks (or simply graphs, as they are called by mathematicians) have been intensively studied in the recent decades by scholars from different scientific areas. A direct way to investigate complex networks is to represent them by means of matrices. The topological properties of the networks are then embedded in the algebraic properties of the corresponding (graph) matrices. Furthermore, many relevant algebraic and combinatorial properties of matrices are encoded in their invariants (known as ''eigenvalues''), which lead to the notion of ''spectrum''. Hence the spectral methods are  fundamental in studying the matrices of complex networks, and so they can help to investigate the structure of complex networks. Here, we discuss two well-known applications of the eigenvalues and corresponding eigenvectors. The first involves the use of the largest eigenvalue (the ''Perron eigenvalue'') and its unit eigenvector (the ''Perron vector'') as a tool for creating a ranking among vertices. This allows us to understand which vertices are more ``important'' than others (the notion of ''eigencentrality'' is relevant here). The second  application of the eigenvalues involves the use of the least Laplacian eigenvalues and their eigenvectors for the detection of subsets of vertices with a significant number of links among them (the notion of ''communities'' will be discussed here).

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Time: 19:30 - 21:00.  Refreshment offered by the Consul of Italy in Cape Town Mr. Emanuele Pollio

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We thank the following institutions for support and assistance: