mathematics

Kai Behrend

Kai Behrend 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 Kai Behrend rather than just read about it. In short: Kai A. Behrend is a German mathematician.

Key takeaways

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

Reference excerpt

Kai A. Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, Gromov–Witten invariants and Donaldson–Thomas theory (cf. Behrend function). He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks. He is the recipient of the 2001 Coxeter–James Prize, the 2011 Jeffery–Williams Prize, and the 2015 CRM-Fields-PIMS Prize. He was elected to the 2018 class of fellows of the American Mathematical Society. He was also an invited speaker at the ICM 2014.

Selected publications Behrend, K.; Bryan, J.; Szendröi, B., "Motivic degree zero Donaldson-Thomas invariants", Inventiones Mathematicae. 192 (2013), no. 1, 111-160. https://doi.org/10.1007/s00222-012-0408-1 Behrend, K. (2009), "Donaldson-Thomas type invariants via microlocal geometry", Annals of Mathematics, 2nd Ser., 170 (3): 1307–1338, https://doi.org/10.4007/annals.2009.170.1307 Behrend, K. (1997). "Gromov-Witten invariants in algebraic geometry". Inventiones Mathematicae. 127 (3). Springer Science and Business Media LLC: 601–617. arXiv:alg-geom/9601011. Bibcode:1997InMat.127..601B. doi:10.1007/s002220050132. ISSN 0020-9910. S2CID 9929474. Behrend, K.; Fantechi, B. (19 March 1997). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1). Springer Science and Business Media LLC: 45–88. arXiv:alg-geom/9601010. Bibcode:1997InMat.128...45B. doi:10.1007/s002220050136. ISSN 0020-9910. S2CID 18533009. Behrend, K.; Manin, Yu. (1 October 1996). "Stacks of stable maps and Gromov-Witten invariants". Duke Mathematical Journal. 85 (1). Duke University Press. doi:10.1215/s0012-7094-96-08501-4. ISSN 0012-7094. S2CID 16512308. Behrend, Kai A. (2003). "Derived 𝑙-adic categories for algebraic stacks". Memoirs of the American Mathematical Society. 163 (774). American Mathematical Society (AMS): 0. doi:10.1090/memo/0774. ISSN 0065-9266. MR 1963494

References

External links The personal web page of Kai Behrend Kai Behrend at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Kai Behrend

Start with the simplest possible case. Write down what Kai Behrend 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 Kai Behrend 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 Kai Behrend 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 Kai Behrend

In research: Kai Behrend 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 Kai Behrend 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: Kai Behrend is common in secondary-school and first-year university syllabi. It links to neighbouring topics 20th-century German mathematicians, 21st-century German mathematicians, Academic staff of the University of British Columbia Faculty of Science, so understanding it makes those chapters shorter.
In everyday life: Look for Kai Behrend 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 Kai Behrend in 20 minutes

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

Frequently asked questions

What is Kai Behrend in simple terms?

Kai A. Behrend is a German mathematician.

Why does Kai Behrend 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 Kai Behrend?

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 Kai Behrend.