mathematics

Nikolai Ivanov (mathematician)

Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician) rather than just read about it. In short: Nikolai V. Ivanov (Russian: Николай Владимирович Иванов; born 1954) is a Russian mathematician who works on topology, geometry and group theory (particularly, modular Teichmüller groups).

Key takeaways

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

Reference excerpt

Nikolai V. Ivanov (Russian: Николай Владимирович Иванов; born 1954) is a Russian mathematician who works on topology, geometry and group theory (particularly, modular Teichmüller groups). He is a professor at Michigan State University. He obtained his Ph.D. under the guidance of Vladimir Abramovich Rokhlin in 1980 at the Steklov Mathematical Institute. Among his contributions to mathematics are his classification of subgroups of surface mapping class groups, and the establishment that surface mapping class groups satisfy the Tits alternative. He is a fellow of the American Mathematical Society since 2012. He is the author of the 1992 book Subgroups of Teichmüller Modular Groups.

Selected publications "Automorphisms of complexes of curves and of Teichmuller spaces" (1997), International Mathematics Research Notices 14, pp. 651–666. with John D. McCarthy: "On injective homomorphisms between Teichmüller modular groups I" (1999), Inventiones mathematicae 135 (2), pp. 425–486. "On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients" (1993), Contemporary Mathematics 150, pp. 149–149.

References

External links N. V. Ivanov website Ivanov's blog

Worked examples

Example 1 — a first encounter with Nikolai Ivanov (mathematician)

Start with the simplest possible case. Write down what Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician)

In research: Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician) 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: Nikolai Ivanov (mathematician) is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1954 births, 20th-century Russian mathematicians, 21st-century Russian mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nikolai Ivanov (mathematician) 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.
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Frequently asked questions

What is Nikolai Ivanov (mathematician) in simple terms?

Why does Nikolai Ivanov (mathematician) 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 Nikolai Ivanov (mathematician)?

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 Nikolai Ivanov (mathematician).