mathematics

Using the Borsuk–Ulam Theorem

Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem rather than just read about it. In short: Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry is a graduate-level mathematics textbook in topological combinatorics. It describes the use of results in topology, and in particular the Borsuk–Ulam theorem, to prove theorems in combinatorics and discrete geometry.

Key takeaways

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

Reference excerpt

Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry is a graduate-level mathematics textbook in topological combinatorics. It describes the use of results in topology, and in particular the Borsuk–Ulam theorem, to prove theorems in combinatorics and discrete geometry. It was written by Czech mathematician Jiří Matoušek, and published in 2003 by Springer-Verlag in their Universitext series (ISBN 978-3-540-00362-5).

Topics The topic of the book is part of a relatively new field of mathematics crossing between topology and combinatorics, now called topological combinatorics. The starting point of the field, and one of the central inspirations for the book, was a proof that László Lovász published in 1978 of a 1955 conjecture by Martin Kneser, according to which the Kneser graphs K G 2 n + k , n {\displaystyle KG_{2n+k,n}} have no graph coloring with k + 1 {\displaystyle k+1} colors. Lovász used the Borsuk–Ulam theorem in his proof, and Matoušek gathers many related results, published subsequently, to show that this connection between topology and combinatorics is not just a proof trick but an area. The book has six chapters. After two chapters reviewing the basic notions of algebraic topology, and proving the Borsuk–Ulam theorem, the applications to combinatorics and geometry begin in the third chapter, with topics including the ham sandwich theorem, the necklace splitting problem, Gale's lemma on points in hemispheres, and several results on colorings of Kneser graphs. After another chapter on more advanced topics in equivariant topology, two more chapters of applications follow, separated according to whether the equivariance is modulo two or using a more complicated group action. Topics in these chapters include the van Kampen–Flores theorem on embeddability of skeletons of simplices into lower-dimensional Euclidean spaces, and topological and multicolored variants of Radon's theorem and Tverberg's theorem on partitions into subsets with intersecting convex hulls.

Audience and reception The book is written at a graduate level, and has exercises making it suitable as a graduate textbook. Some knowledge of topology would be helpful for readers but is not necessary. Reviewer Mihaela Poplicher writes that it is not easy to read, but is "very well written, very interesting, and very informative". And reviewer Imre Bárány writes that "The book is well written, and the style is lucid and pleasant, with plenty of illustrative examples." Matoušek intended this material to become part of a broader textbook on topological combinatorics, to be written jointly with him, Anders Björner, and Günter M. Ziegler. However, this was not completed before Matoušek's untimely death in 2015.

References

Worked examples

Example 1 — a first encounter with Using the Borsuk–Ulam Theorem

Start with the simplest possible case. Write down what Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem

In research: Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem 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: Using the Borsuk–Ulam Theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics 2003 non-fiction books, Algebraic topology, Combinatorics, so understanding it makes those chapters shorter.
In everyday life: Look for Using the Borsuk–Ulam Theorem 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Using the Borsuk–Ulam Theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Using the Borsuk–Ulam Theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Using the Borsuk–Ulam Theorem in simple terms?

Why does Using the Borsuk–Ulam Theorem 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 Using the Borsuk–Ulam Theorem?

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 Using the Borsuk–Ulam Theorem.