mathematics

György Elekes

György Elekes 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 György Elekes rather than just read about it. In short: György Elekes (19 May 1949 – 29 September 2008) was a Hungarian mathematician and computer scientist who specialized in Combinatorial geometry and Combinatorial set theory. He may be best known for his work in the field that would eventually be called Additive Combinatorics.

Key takeaways

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

Reference excerpt

György Elekes (19 May 1949 – 29 September 2008) was a Hungarian mathematician and computer scientist who specialized in Combinatorial geometry and Combinatorial set theory. He may be best known for his work in the field that would eventually be called Additive Combinatorics. Particularly notable was his "ingenious" application of the Szemerédi–Trotter theorem to improve the best known lower bound for the sum-product problem. He also proved that any polynomial-time algorithm approximating the volume of convex bodies must have a multiplicative error, and the error grows exponentially on the dimension. With Micha Sharir he set up a framework which eventually led Guth and Katz to the solution of the Erdős distinct distances problem. (See below.)

Life After graduating from the mathematics program at Fazekas Mihály Gimnázium (i.e., "Fazekas Mihály high school" in Budapest, which is known for its excellence, especially in mathematics), Elekes studied mathematics at the Eötvös Loránd University. Upon completing his degree, he joined the faculty in the Department of Analysis at the university. In 1984, he joined the newly forming Department of Computer Science, which was being headed by László Lovász. Elekes was promoted to full professor in 2005. He received the Doctor of Mathematical Sciences title from the Hungarian Academy of Sciences in 2001.

Work Elekes started his mathematical work in combinatorial set theory, answering some questions posed by Erdős and Hajnal. One of his results states that if the set of infinite subsets of the set of natural numbers is split into countably many parts, then in one of them, there is a solution of the equation A∪B=C. His interest later switched to another favorite topic of Erdős, discrete geometry and geometric algorithm theory. In 1986 he proved that if a deterministic polynomial algorithm computes a number V(K) for every convex body K in any Euclidean space given by a separation oracle such that V(K) always at least vol(K), the volume of K, then for every large enough dimension n, there is a convex body in the n-dimensional Euclidean space such that V(K)>20.99nvol(K). That is, any polynomial-time estimator of volume over K must be inaccurate by at least an exponential factor. Not long before his death he developed new tools in Algebraic geometry and used them to obtain results in Discrete geometry, proving Purdy's Conjecture. Micha Sharir organized, extended and published Elekes's posthumous notes on these methods. Then Nets Katz and Larry Guth used them to solve (apart from a factor of (log n) 1/2 ) the Erdős distinct distances problem, posed in 1946.

References

External links Elekes' home page

Worked examples

Example 1 — a first encounter with György Elekes

Start with the simplest possible case. Write down what György Elekes 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 György Elekes 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 György Elekes 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 György Elekes

In research: György Elekes 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 György Elekes 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: György Elekes is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1949 births, 2008 deaths, 20th-century Hungarian mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for György Elekes 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 György Elekes in 20 minutes

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

Frequently asked questions

What is György Elekes in simple terms?

Why does György Elekes 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 György Elekes?

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 György Elekes.