The Speakers

Abstract

The mathematical foundations of artificial intelligence are the fundamental elements that enable both the theoretical understanding and practical success of this technology. Linear algebra forms the core of models by representing data as vectors and matrices, while calculus, gradient descent, and the chain rule govern the learning process. Probability theory and statistics, on the other hand, enable us to make decisions under uncertainty and measure confidence intervals in predictions. Information theory establishes the balance between model complexity and performance, while optimization methods allow us to train these models most efficiently.

A concrete example of the practical application of these mathematical disciplines is a machine learning model that can diagnose abnormalities in frontal chest X-rays. This model, trained using calculus-based optimization and image data processed with linear algebra, can measure diagnostic uncertainty thanks to probability theory, creating a reliable decision-support tool for healthcare professionals. In short, artificial intelligence is built on mathematical principles and could neither be understood nor developed without these disciplines. This study clearly demonstrates the close relationship between artificial intelligence and mathematics. 

https://scholar.google.com/citations?user=yTk8FeQAAAAJ&hl=tr

https://research.itu.edu.tr/tr/persons/kurulay

Abstract

An ongoing theme throughout the study of áows is the need to model and predict their detailed behavior and the phenomena that they manifest. The latest developments in mutiphase áows combine theoretical, analytical, and numerical methods to create stronger veriÖcation and validation modeling methods. Such models are explored, experimentally through equipped laboratory-sized models, theoretically using mathematical equations, or numerically exploiting advanced algorithms and powful machines to investigate the complexity of the phenomena. Here, we exploit mathematical and numerical analysis for multiphase áows in problems derived from hydrogeology and medicine, where we consider simultaneous áow of materials with different states or phases [1, 3] or materials with different chemical properties but in the same state/phase [2, 4]. 

References [1] M. Albert-Gimeno, I. Hadji, T. Joshi, L. Kimpton, T. Kwan, G. Lang, S. Moise, S. Naire, F.Z. Nouri, G. Richardson & R. Whittaker, Targeting stem cells following I.V. injection using magnetic particle based approaches, Mathematics in Medecine 2012. [2] A. Assala, N. Djedaidi & F.Z. Nouri, Dynamical behaviour of miscibles áuids in Porous Media, Int. J. Dynamical Systems and Di§erential Equations, Vol. 8, No.3, 2018, 176-188. [3] Cuc Bui, Vanessa Lleras & Olivier Pantz, ModÈlisation et simulation de la dynamique des globules, hal.archives 2015. [4] F.Z. Nouri and N. Djedaidi, Interface Dynamics for a Bi-Phasic Problem in Heterogeneous Porous Media, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 30 (2023) 21-33 

https://scholar.google.com/citations?user=MVXQdvsAAAAJ&hl=en

https://www.researchgate.net/profile/Fatma-Zohra-Nouri

Abstract

Abstract This presentation outlines the spectral theory of linear relations (multivalued operators). It covers fundamental definitions, key classes like Fredholm and demicompact relations, and their associated spectra. The talk also explores perturbation results for these classes and extends classical concepts like pseudospectra and essential spectra to the framework of linear relations. It demonstrates how these tools solve differential inclusions and analyzes the properties of resolvents and spectra for these generalized operators. 

https://scholar.google.com/citations?user=-Ps-dtwAAAAJ&hl=en

https://www.researchgate.net/profile/Aymen-Ammar