Department of Mathematics colloquium - Sibe Mardešić presents a sequence of lectures given by leading experts in the fields of theoretical and applied mathematics. The lectures take place at the Department of Mathematics, Faculty of Science, Bijenička cesta 30, Zagreb.
Lecture announcement
About the lecturer: Ilijas Farah received Ph.D. in 1997 from the University of Toronto under the supervision of Stevo Todorčević. He was a Postdoctoral Fellow at York University, a Hill Assistant Professor at Rutgers University, and a professor at CUNY graduate center and College of Staten Island. He is currently the Research Chair in Foundations of Operator Algebras at York University in Toronto and a professor at the Mathematical Institute of the Serbian Academy of Sciences and Arts. He received numerous awards for his work, which focuses on applications of logic in the theory of operator algebras, such as the Sacks prize for the best doctorate in mathematical logic, Governor General's gold medal (also for his doctorate), dean's award for outstanding research, and the Faculty Excellence in Research award at York University. In 2014 he was an invited speaker at the International Congress of Mathematicians in Seoul. Math genealogy webpage lists 7 of his graduate students.
Lecture abstract: In the early years of the 20th century, Weyl initiated study of compact perturbations of pseudo-differential operators. The Weyl–von Neumann theorem asserts that two self-adjoint operators on a complex Hilbert space are unitarily equivalent modulo compact perturbations if and only if their essential spectra coincide. Berg and Sikonia (independently) extended this result to normal operators. New impetus to the subject was given in 1970s by Brown, Douglas, and Fillmore, who replaced single operators with (separable) C*-algebras and realized that compact perturbations can be considered as extensions by the ideal of compact operators. After passing to the quotient (the Calkin algebra, Q) and identifying an extension with a *-homomorhism into Q, analytic methods had been supplemented with methods from algebraic topology, homological algebra, and (most recently) logic. Around the same time, Shelah proved one of his many influential results, by showing that the assertion “all automorphisms of l∞/c0 are trivial” is relatively consistent with ZFC. Surprisingly, these two directions of research are intimately connected. This talk will be about rigidity of quotient structures, and it is partially based on the preprint Corona rigidity (2022, arXiv:2201.11618) coauthored with Ghasemi, Vaccaro, and Vignati, and some more recent results.
Lecture announcement
About the lecturer: Benny Sudakov received Ph.D. in 1999 from Tel Aviv University under the supervision of Noga Alon. He was an instructor and an assistant professor at Princeton University and a professor at University of California, Los Angeles. He is currently a Professor of Mathematics at ETH in Zürich. He has versatile mathematical interests, solving open problems and publishing papers in extremal combinatorics, graph and hypergraph theory, Ramsey theory, random structures, and applications of combinatorics in computer science. He received many awards and distinctions for his work. He was an invited speaker at the International Congress of Mathematicians in Hyderabad in 2010. He is a Fellow of the AMS (since 2012) and a member of the Academia Europaea (since 2019). He will also be a plenary speaker at the European Congress of Mathematics in Sevilla this year. The arXiv server hosts 275 of his research papers and preprints, his Google scholar page counts more than 10,000 citations, while Math genealogy list 16 of his finished graduate students, several of them born in Croatia.
Lecture abstract: Every large system, chaotic as it may be, contains a well-organized subsystem. This phenomenon is truly ubiquitous and manifests itself in different mathematical areas. One of the central problems in extremal combinatorics, which was extensively studied in the last hundred years, is to estimate how large a graph/hypergraph needs to be to guarantee the emergence of such well-organized substructures. In the first part of this talk we will give an introduction to this topic, mentioning some classical results as well as a few applications to other areas of mathematics. Then we discuss the recent solution (with Oliver Janzer) of the following fundamental problem, posed by Erdős and Sauer about 50 years ago: How many edges on n vertices force the existence of an r-regular subgraph (r>2)? Our proof uses algebraic and probabilistic tools, building on earlier works by Alon, Friedland, Kalai, Pyber, Rödl and Szemerédi.