mathematics

Stephen M. Gersten

Stephen M. Gersten 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 Stephen M. Gersten rather than just read about it. In short: Stephen M. Gersten (born 2 December 1940) is an American mathematician, specializing in finitely presented groups and their geometric properties.

Key takeaways

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

Reference excerpt

Stephen M. Gersten (born 2 December 1940) is an American mathematician, specializing in finitely presented groups and their geometric properties. Gersten graduated in 1961 with an AB from Princeton University and in 1965 with a PhD from Trinity College, Cambridge. His doctoral thesis was Class Groups of Supplemented Algebras written under the supervision of John R. Stallings. In the late 1960s and early 1970s he taught at Rice University. In 1972–1973 he was a visiting scholar at the Institute for Advanced Study. In 1973 he became a professor at the University of Illinois at Urbana–Champaign. In 1974 he was an Invited Speaker at the International Congress of Mathematicians in Vancouver. At the University of Utah he became a professor in 1975 and is now semi-retired there. His PhD students include Roger C. Alperin, R. Keith Dennis and Edward W. Formanek. Gersten's conjecture has motivated considerable research.

Gersten's theorem If φ is an automorphism of a finitely generated free group F then { x : x ∈ F and φ(x) = {\displaystyle =} x } is finitely generated.

Selected publications Gersten, S. M. (1972). "On the spectrum of algebraic K {\displaystyle K} -theory". Bulletin of the American Mathematical Society. 78 (2): 216–220. doi:10.1090/S0002-9904-1972-12924-0. Gersten, S. M. (1973). "Higher K {\displaystyle K} -theory for regular schemes". Bulletin of the American Mathematical Society. 79: 193–197. doi:10.1090/S0002-9904-1973-13150-7. Gersten, S. M. (1973). "Higher K-theory of rings". Higher K-Theories. Lecture Notes in Mathematics. Vol. 341. pp. 3–42. doi:10.1007/BFb0067049. ISBN 978-3-540-06434-3. Brown, Kenneth S.; Gersten, Stephen M. (1973). "Algebraic K-theory as generalized sheaf cohomology". Higher K-Theories. Lecture Notes in Mathematics. Vol. 341. pp. 266–292. doi:10.1007/BFb0067062. ISBN 978-3-540-06434-3. Gersten, S. M. (1983). "A short proof of the algebraic Weierstrass preparation theorem". Proceedings of the American Mathematical Society. 88 (4): 751–752. doi:10.1090/S0002-9939-1983-0702313-2. (See Weierstrass preparation theorem.) Gersten, S. M. (1983). "On fixed points of automorphisms of finitely generated free groups". Bulletin of the American Mathematical Society. 8 (3): 451–455. doi:10.1090/S0273-0979-1983-15116-9. (This paper presents a proof of a conjecture made by G. Peter Scott.) Gersten, S. M. (1984). "On Whitehead's algorithm". Bulletin of the American Mathematical Society. 10 (2): 281–285. doi:10.1090/S0273-0979-1984-15246-7. Gersten, S. M. (1987). "Reducible Diagrams and Equations over Groups". Essays in Group Theory. Mathematical Sciences Research Institute Publications. Vol. 8. pp. 15–73. doi:10.1007/978-1-4613-9586-7_2. ISBN 978-1-4613-9588-1. Gersten, S. M.; Short, Hamish B. (1990). "Small cancellation theory and automatic groups". Inventiones Mathematicae. 102: 305–334. Bibcode:1990InMat.102..305G. doi:10.1007/BF01233430. S2CID 120267906. Baumslag, Gilbert; Gersten, S.M.; Shapiro, Michael; Short, H. (1991). "Automatic groups and amalgams". Journal of Pure and Applied Algebra. 76 (3): 229–316. doi:10.1016/0022-4049(91)90139-S. Gersten, S. M.; Short, H. B. (1991). "Rational Subgroups of Biautomatic Groups". Annals of Mathematics. 134 (1): 125–158. doi:10.2307/2944334. JSTOR 2944334. Gersten, S. M. (1992). "Dehn Functions and l1-norms of Finite Presentations". In Baumslag G.; Miller C.F. (eds.). Algorithms and Classification in Combinatorial Group Theory. Mathematical Sciences Research Institute Publications. Vol. 23. New York: Springer. pp. 195–224. doi:10.1007/978-1-4613-9730-4_9. ISBN 978-1-4613-9732-8. ISSN 0940-4740. Gersten, S.M. (1993). "Isoperimetric and isodiametric functions of finite presentations". Geometric group theory. Vol. 1. Cambridge University Press. pp. 79–96. ISBN 9780521435291.

Worked examples

Example 1 — a first encounter with Stephen M. Gersten

Start with the simplest possible case. Write down what Stephen M. Gersten 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 Stephen M. Gersten 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 Stephen M. Gersten 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 Stephen M. Gersten

In research: Stephen M. Gersten 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 Stephen M. Gersten 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: Stephen M. Gersten is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1940 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Stephen M. Gersten 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 Stephen M. Gersten in 20 minutes

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Frequently asked questions

What is Stephen M. Gersten in simple terms?

Stephen M. Gersten (born 2 December 1940) is an American mathematician, specializing in finitely presented groups and their geometric properties.

Why does Stephen M. Gersten 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 Stephen M. Gersten?

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 Stephen M. Gersten.