mathematics

Mark Sapir

Mark Sapir is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mark Sapir rather than just read about it. In short: Mark Sapir (February 12, 1957 - October 8, 2022) was a U.S. and Russian mathematician working in geometric group theory, semigroup theory and combinatorial algebra. He was a Centennial Professor of Mathematics in the Department of Mathematics at Vanderbilt University.

Key takeaways

Mark Sapir belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mark Sapir to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mark Sapir from memory before moving on to harder problems.

Reference excerpt

Mark Sapir (February 12, 1957 - October 8, 2022) was a U.S. and Russian mathematician working in geometric group theory, semigroup theory and combinatorial algebra. He was a Centennial Professor of Mathematics in the Department of Mathematics at Vanderbilt University.

Education and career Sapir received his undergraduate degree in mathematics (diploma of higher education) from the Ural State University in Yekaterinburg (then called Sverdlovsk), Russia, in 1978. He received his PhD in mathematics (Candidate of Sciences) degree, joint from the Ural State University and Moscow State Pedagogical Institute in 1983, with Lev Shevrin as the advisor. Afterwards Sapir held faculty appointments at the Ural State University, Sverdlovsk Pedagogical Institute, University of Nebraska–Lincoln, before coming as a professor of mathematics to Vanderbilt University in 1997. He was appointed a Centennial Professor of Mathematics at Vanderbilt in 2001. Sapir gave an invited talk at the International Congress of Mathematicians in Madrid in 2006. He gave an AMS Invited Address at the American Mathematical Society Sectional Meeting in Huntsville, Alabama in October 2008. He gave a plenary talk at the December 2008 Winter Meeting of the Canadian Mathematical Society. Sapir gave the 33d William J. Spencer Lecture at the Kansas State University in November 2008. He gave the 75th KAM Mathematical Colloquium lecture at the Charles University in Prague in June 2010. Sapir became a member of the inaugural class of Fellows of the American Mathematical Society in 2012. Sapir founded the Journal of Combinatorial Algebra, published by the European Mathematical Society, and served as its founding editor-in-chief starting in 2016. He also was an editorial board member for the journals Groups, Complexity, Cryptology and Algebra and Discrete Mathematics. His past editorial board positions include Journal of Pure and Applied Algebra, Groups, Geometry, and Dynamics, Algebra Universalis, and International Journal of Algebra and Computation (as Managing Editor). A special mathematical conference in honor of Sapir's 60th birthday took place at the University of Illinois at Urbana–Champaign in May 2017.

Personal life Mark Sapir's elder daughter, Jenya Sapir, is also a mathematician; she was Maryam Mirzakhani's first (out of two) students. Currently, she is an assistant professor in the Department of Mathematics of Binghamton University. Mark Sapir and his wife Olga Sapir became naturalized U.S. citizens in July 2003, after suing the BCIS in federal court over a multi-year delay of their citizenship application originally filed in 1999.

Mathematical contributions Sapir's early mathematical work concerned mostly semigroup theory. In geometric group theory his most well-known and significant results were obtained in two papers published in the Annals of Mathematics in 2002, the first joint with Jean-Camille Birget and Eliyahu Rips, and the second joint with Birget, Rips and Aleksandr Olshansky. The first paper provided an essentially complete description of all the possible growth types of Dehn functions of finitely presented groups. The second paper proved that a finitely presented group has the word problem solvable in non-deterministic polynomial time (NP) if and only if this group embeds as a subgroup of a finitely presented group with polynomial Dehn function. A combined featured review of these two papers in Mathematical Reviews characterized them as ``remarkable foundational results regarding isoperimetric functions of finitely presented groups and their connections with the complexity of the word problem". Sapir was also known for his work, mostly joint with Cornelia Druţu, on developing the asymptotic cone approach to the study of relatively hyperbolic groups. A 2002 paper of Sapir and Olshansky constructed the first known finitely presented counter-examples to the Von Neumann conjecture. Sapir also introduced, in a 1993 paper with Meakin, the notion of a diagram group, based on finite semigroup presentations. He further developed this notion in subsequent joint papers with Guba. Diagram groups provided a new approach to the study of Thompson groups, which appear as important examples of diagram groups.

Selected publications Guba, Victor; Sapir, Mark (1997). "Diagram groups". Memoirs of the American Mathematical Society. 130 (620). doi:10.1090/memo/0620. MR 1396957. Sapir, Mark V.; Birget, Jean-Camille; Rips, Eliyahu (2002). "Isoperimetric and isodiametric functions of groups". Annals of Mathematics. Second Series. 156 (2): 345–466. arXiv:math/9811106. doi:10.2307/3597196. JSTOR 3597196. MR 1933723. S2CID 14155715. Birget, Jean-Camille; Ol'shanskii, Alexander Yu.; Rips, Eliyahu; Sapir, Mark V. (2002). "Isoperimetric functions of groups and computational complexity of the word problem". Annals of Mathematics. Second Series. 156 (2): 467–518. arXiv:math/9811105. doi:10.2307/3597195. JSTOR 3597195. MR 1933724. S2CID 119728458. Olʹshanskii, Alexander Yu.; Sapir, Mark V. (2002). "Non-amenable finitely presented torsion-by-cyclic groups". Publications Mathématiques de l'IHÉS. 96 (2003): 43–169. arXiv:math/0208237. doi:10.1007/s10240-002-0006-7. MR 1985031. S2CID 122990460. Borisov, Alexander; Sapir, Mark (2005). "Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms". Inventiones Mathematicae. 160 (2): 341–356. arXiv:math/0309121. doi:10.1007/s00222-004-0411-2. MR 2138070. S2CID 6210319. Druţu, Cornelia; Sapir, Mark (2008). "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups". Advances in Mathematics. 217 (3): 1313–1367. doi:10.1016/j.aim.2007.08.012. MR 2383901. S2CID 10461978.

See also List of International Congresses of Mathematicians Plenary and Invited Speakers

References

External links Mark Sapir's homepage at Vanderbilt University Mark Sapir's mathematical blog at WordPress Mark Sapir's entry at the Mathematics Genealogy Project

Worked examples

Example 1 — a first encounter with Mark Sapir

Start with the simplest possible case. Write down what Mark Sapir claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mark Sapir before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mark Sapir ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mark Sapir

In research: Mark Sapir appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mark Sapir in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mark Sapir is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1957 births, 2022 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Mark Sapir outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Mark Sapir in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mark Sapir means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mark Sapir out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mark Sapir in simple terms?

Why does Mark Sapir matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mark Sapir?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mark Sapir.