Abstract: Abstract. In this talk, a recent computational methodology is described (see [1,2]). It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (a new kind of supercomputer patented in several countries) working numerically with infinite and infinitesimal numbers that can be written in a positional system with an infinite radix. On a number of examples (numerical differentiation, divergent series, ordinary differential equations, fractals, set theory, etc.) it is shown that the new approach can be useful from both theoretical and computational points of view. The main attention is dedicated to applications in optimization (local, global, and multi-objective) (see [1,2-7]). The accuracy of the obtained results is continuously compared with results obtained by traditional tools used to work with mathematical objects involving infinity. The Infinity Calculator working with infinities and infinitesimals numerically is shown during the lecture.

For more information see the dedicated web page http://www.theinfinitycomputer.com and this survey: The web page developed at the University of East Anglia, UK is dedicated to teaching the methodology: https://www.numericalinfinities.com/

References
1. Sergeyev Ya.D. and De Leone R., eds, Numerical Infinities and Infinitesimals in Optimization. Springer, Cham, 2022.
2. Sergeyev Ya.D. Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 2017, 4(2), 219–320.
3. De Leone R., Egidi N., Fatone L. The use of grossone in elastic net regularization and sparse support vector machines, Soft Computing, 24, 17669-17677, 2020.
4. Astorino A., Fuduli A. Spherical separation with infinitely far center, Soft Computing, 24, 17751-17759, 2020.
5. De Leone R., Fasano G., Roma M., Sergeyev Y.D. Iterative grossone-based computation of negative curvature directions in large-scale optimization, Journal of Optimization Theory and Applications, 186(2), 554-589, 2020.
6. De Leone R., Fasano G., Sergeyev Ya.D. Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming, Computational Optimization and Applications, 71(1), 73-93, 2018.
7. Cococcioni M., Pappalardo M., Sergeyev Ya.D. Lexicographic multiobjective linear programming using grossone methodology: Theory and algorithm, Applied Mathematics and Computation, 318, 298–311, 2018.

Biography: Yaroslav D. Sergeyev is Distinguished Professor at the University of Calabria, Italy (chiamata diretta per chiara fama) and Head of Numerical Calculus Laboratory at the same university. He is also Affiliated Researcher at the Institute of High Performance Computing and Networking of the Italian National Research Council, and Affiliated Faculty at the Center for Applied Optimization, University of Florida, Gainesville, USA.

He was awarded his Ph.D. (1990) from Lobachevski Gorky State University and his D.Sc. degree (1996) from Lomonosov State University, Moscow (this degree is Habilitation for the Full Professorship in Russian universities). In 2013, he was awarded Degree of Honorary Doctor from Glushkov Institute of Cybernetics of The National Academy of Sciences of Ukraine, Kiev.

His research interests include global optimization (he was President of the International Society of Global Optimization, 2017-2021), infinity computing and calculus (the field he has founded), numerical computations, scientific computing, philosophy of computations, set theory, number theory, fractals, parallel computing, and interval analysis. Prof. Sergeyev is included in the lists “Top Italian Mathematicians”. He was awarded several research prizes (Khwarizmi International Award, 2017; Pythagoras International Prize in Mathematics, 2010; EUROPT Fellow, 2016; Outstanding Achievement Award from the 2015 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Honorary Fellowship, the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015; The 2015 Journal of Global Optimization (Springer) Best Paper Award; Lagrange Lecture, Turin University, Italy, 2010; MAIK Prize for the best scientific monograph published in Russian, Moscow, 2008, etc.). In 2020, he was elected corresponding member of Accademia Peloritana dei Pericolanti in Messina, Italy. Since 2020 he is included in the rating “Highly cited authors in Scopus”.

His list of publications contains more than 300 items (among them 6 authored and 10 edited books and more than 130 articles in international journals). He is a member of editorial boards of one book series, 12 international and 3 national journals (no open access) and co-editor of 14 special issues. He delivered more than 90 plenary and keynote lectures and tutorials at prestigious international congresses. He was Chairman of 8 and Co-Chairman of 6 international conferences and a member of Scientific Committees of more than 70 international congresses. He is Coordinator of numerous national and international research and educational projects, supervises master and Ph.D. theses, takes part of Ph.D. commissions in Italy and abroad. Numerous magazines, newspapers, TV and radio channels have dedicated a lot of space to his research.

For more information see his web page https://www.yaroslavsergeyev.com/