Stevo Todorčević

Stevo Todorčević is a Canadian-French-Serbian mathematician, one of the world’s leading logicians and a world leader in set theory and its applications to pure mathematics. He is a Canada Research Chair Professor in mathematics at the University of Toronto, and a senior director of research at the Centre national de la recherche scientifique (CNRS) in Paris.

Early life and education

Todorčević was born at Ubovića Brdo, Bosnia and Herzegovina where he lived until the second grade of primary school. After, his family moved to Banatsko Novo Selo where he finished primary school. He enrolled "Uroš Predić" grammar school in Pančevo. He demonstrated his talent and affinity toward mathematics in the third and forth years of the grammar school. After finishing grammar school he enrolled Faculty of Science, Belgrade University, where he studied pure mathematics. During his undergraduate studies he attended Đuro Kurepa's advanced mathematical classes. In 1978 he enrolled graduate studies. Kurepa validated Todorčević's master thesis as good enough to be accepted as a doctoral thesis. Regardless, Todorčević wrote his doctoral thesis in 1979 with Kurepa as his advisory. In his address, preceding the oral defense of the doctoral thesis, Kurepa stressed that he was not able to find external readers of the Stevo's doctoral thesis in Yugoslavia, capable of fully understanding and evaluating Stevo's work, and turned to two university professors from England. Kurepa added that Stevo's talent was a miracle and that Stevo was the most talented out of the 40 Ph.D. students he advised in the past.

Career

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According to the Centre de Recherches Mathématiques, the Fields Institute and the Pacific Institute for Mathematical Sciences announcement, as of December 14, 2014,his work is recognized for its striking originality and technical brilliance. He was an invited speaker at the 1998 ICM in Berlin for his discovery and work on rho-functions. He made major contributions to the study of S- and L-spaces in topology, proved a remarkable classification theorem for transitive relations on the first uncountable ordinal, made a deep study of compact subsets of the Baire class 1 functions thus continuing work of Bourgain, Fremlin, Talagrand, and others in Banach space theory. Together with P. Larson he completed the solution of Katetov’s old compact spaces metrization problem. Among the most striking recent accomplishments of Todorčević (and co-authors) are major contributions to the von Neumann and Maharam problems on Boolean algebras, the theory of non-separable Banach spaces, including the solution of an old problem of Davis and Johnson, the solution of a long-standing problem of Laver, and the development of a duality theory relating finite Ramsey theory and topological dynamics.

Further, Todorčević is known for his the side-condition method in set-theoretic Forcing, the invention and development of walks on ordinals and their characteristics, and other research that bridge between different areas of mathematics.

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Todorčević's first recognized contribution to Set theory was given in his 1978 Master’s Thesis. He constructed a model of [math]\displaystyle{ MA+ \neg wKH }[/math] in a way to allow him to make the continuum any regular cardinal and derived a variety of topological consequences of [math]\displaystyle{ MA+ \neg wKH. }[/math] Here [math]\displaystyle{ MA }[/math] is an abbreviation for Martin's Axiom and [math]\displaystyle{ wKH }[/math]– for weak Kurepa Hypothesis.

Todorčević earned his doctoral degree in 1979 at the University of Belgrade with Đuro Kurepa as advisor and Keith Devlin as outside reader. Devlin attended the defense; he encouraged Todorčević to visit Jerusalem where he attended Saharon Shelah's lectures on forcing.

In the July–August 1980 Todorčević attended the six-week summer school called Settop held in Toronto. At the conference, Todorčević along with Abraham had proved the existence of rigid Aronszajn trees and the consistency of [math]\displaystyle{ MA + \neg CH }[/math] + there exists a first countable [math]\displaystyle{ S }[/math]-space. [math]\displaystyle{ CH }[/math] is an abbreviation for the Continuum Hypotesis.

He gave a survey of work on trees from combinatorial and set-theoretic perspectives, in the 1980s., and continued this work on exploring consistent possibilities for various types of trees, looking for results for trees on multiple cardinals, or with required or forbidden types of subtrees. The elegance of his presentation drew a wide audience for this work.

As to the partition calculus in 1980’s, "Todorčević proved a startling square bracket partition result for the uncountable and introduced new technology whose ramifications are still unfolding, and proved a stepping up lemma for negative square bracket partition relations."

Todorčević was a Miller Research Fellow in Berkeley from 1983 to 1985. In the 1985/6 academic year, he was a member of the Institute for Advanced Study.

For his proof of the partition relation [math]\displaystyle{ \aleph_{1} \vdash \sideset{}{_{\eta_{1}}^2}{[\aleph_{1}]} }[/math], Todorčević earned explicit appreciations. Paul Erdös wrote, "This certainly is an unexpected and sensational result." and Jean A. Larson added, "... (it) was a wonderful shock that introduced a wide audience to the walks on ordinalsTemplate:Refn and the oscillation function." Todorčević obtained this partition relation in September 1984, while lecturing on it in the Berkeley seminar, wrote up the notes of his lectures and circulated them in January 1985 and published the result later, in 1987. The walks on ordinals method Todorčević devised in May 1984 when he came up with a new proof of the existence of a Countryman line.

In order to establish this partition relation, Todorčević discovered an entirely new mathematical object called rho functions. Sierpinski in 1933 coloured the edges of the complete graph [math]\displaystyle{ G }[/math] whose vertices are the elements of the smallest uncountable cardinal number. He coloured the edges of [math]\displaystyle{ G }[/math] with 2 colours in such a way that each colour appears on some edge of any uncountable subgraph of [math]\displaystyle{ G }[/math]. Galvin and Shelah in 1980s had increased the number of colours from 2 to 3. Improving 3 to 4 seemed beyond any available methods. Todorčević used his newly discovered rho functions to increase the colours not just to 4, but all the way up to the smallest uncountable cardinal, which is the maximum conceivable number. This was one of the results for which he was invited to the Berlin ICM.

The discovery of rho functions (and the various applications they have found), an entirely new mathematical object, one out of the five in Set theory in the twentieth century, is celebrated as a major advance in understanding of mathematics and an extended period of exciting progress.

In 1989 Todorčević published a monograph, Partition Problems in Topology. He wrote that proof techniques developed for solving the S-space problem and the L-space problem turn out to be useful in many other problems in general topology, writing "this is so because Ramsey-type theorems are basic and so much needed in many parts of mathematics and (S) and (L) happen to be Ramsey-type properties of the uncountable most often needed by the topologist".

He became a corresponding member of the Serbian Academy of Sciences and Arts as of 1991 and a full member of the Academy in 2009.

He was invited to deliver the Tarski Lectures in 2014.

Todorčević is the Royal Society of Canada fellow. In the 2016 RSC fellowship nomination detailed appraisal it was written:

"Dr. Todorcevic has been a brilliantly creative and productive mathematician for almost forty years, and is now clearly a world leader in set theory and its applications to pure mathematics."

Advisory work

One of his Ph.D. students, Ilijas Farah, won the 1997 Sacks Prize for his Ph.D. dissertation. The Ph.D. was received on June, 1997, at the University of Toronto. Another Todorčević's Ph.D. student, Justin Tatch Moore, won the "Young Scholar's Competition" award in 2006, in Vienna, Austria. The Competition was a part of the "Horizons of Truth" celebrating the Gödel Centenary 2006.

Awards

Todorčević is the winner of

• the first prize of the Balkan Mathematical Society for 1980 and 1982,
• the 2012 CRM-Fields-PIMS prize in mathematical sciences,
• the Shoenfield prize for 2013, and
• the twenty-seventh annual Gödel lecturer (2016) of the Association for Symbolic Logic.

Sources

• [Gabbay, Kanamori and Woods, 2012] Sets and Extensions in the Twentieth Century by Dov M. Gabbay, Akihiro Kanamori, and John Woods (editors), Elsevier, 2012

Other websites

• CRM-Fileds-PIM Prize Lecture: Stevo Todorcevic on Walks on Ordinals and Their Characteristics by Assaf Rinot
• CRM Fields PIMS Prize Lecture: Prof. Stevo Todorcevic (photo album)
• CRM-Fields-PIMS Prize Lecture: Stevo Todorcevic (University of Toronto)
• Papers by Stevo Todorcevic [1]
• Stevo Todorcevic at Institut de Mathématiques de Jussieu - Paris Rive Gauche [2]
• Todorčević najcenjeniji (Todorčević most respected) [3]
• Stevo Todorcevic on Set theory axiomatic systems in Dispute over Infinity Divides Mathematicians by Natalie Wolchover, Quanta Magazine, November 26, 2013 [4]
• Institute for Advanced Study member: Stevo Todorcevic
• Prof. Todorčević Interview (in Serbian)
• Basis problems in set theory by Stevo Todorcevic, Leeds, August 3, 2016