mathematics

Xinyi Yuan

Xinyi Yuan 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 Xinyi Yuan rather than just read about it. In short: Xinyi Yuan (Chinese: 袁新意; born 1981) is a Chinese mathematician who is currently a professor of mathematics at Peking University working in number theory, arithmetic geometry, and automorphic forms. In particular, his work focuses on arithmetic intersection theory, algebraic dynamics, Diophantine equations and special values of L-functions.

Key takeaways

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

Reference excerpt

Xinyi Yuan (Chinese: 袁新意; born 1981) is a Chinese mathematician who is currently a professor of mathematics at Peking University working in number theory, arithmetic geometry, and automorphic forms. In particular, his work focuses on arithmetic intersection theory, algebraic dynamics, Diophantine equations and special values of L-functions.

Early life and education Yuan is from Macheng, Huanggang, Hubei province, and graduated from Huanggang Middle School in 2000. That year, he received a gold medal at the International Mathematical Olympiad while representing China. Yuan obtained his A.B. in mathematics from Peking University in 2003 and his Ph.D. in mathematics from the Columbia University in 2008 under the direction of Shou-Wu Zhang. His article "Big Line Bundles over Arithmetic Varieties," published in Inventiones Mathematicae, demonstrates a natural sufficient condition for when the orbit under the absolute Galois group is equidistributed.

Career He spent time at the Institute for Advanced Study, Princeton University, and Harvard University before joining the Berkeley faculty in 2012. Yuan was appointed a Clay Research Fellow for a three-year term from 2008 to 2013. Together with a number of other collaborators, Yuan was profiled in Quanta Magazine and Business Insider for, among other things, his research on L-functions. Yuan left UC Berkeley to become a full professor at Peking University in 2020. In 2025, Yuan was award the ICCM Gold medal at the International Congress of Chinese Mathematicians. He is also an invited speaker at the 2026 International Congress of Mathematicians.

Research In his thesis, Yuan generalized the equidistribution theorem of Lucien Szpiro, Emmanuel Ullmo and Shou-Wu Zhang to the broader framework of adelic line bundles and in particular to algebraic dynamical systems. Together with Shou-Wu Zhang, Yuan proved the averaged Colmez conjecture which was later shown to imply the André–Oort conjecture for Siegel modular varieties by Jacob Tsimerman. With Shou-Wu Zhang, Yuan wrote a monograph generalizing the theory of adelic line bundles to quasi-projective varieties. Yuan subsequently applied these ideas to reprove the uniform Bogomolov conjecture for curves, thereby obtaining a new proof of Mazur's Conjecture B. Later, in collaboration with Jiawei Yu and Shengxuan Zhou, he derived the first explicit uniform estimate for the number of rational points on curves of genus at least two in terms only of the Jacobian rank. Yuan has also made major contributions to Diophantine geometry over function fields through joint work with Junyi Xie. Their results include proofs of the geometric Bogomolov conjecture in arbitrary characteristic and the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties.

Publications (selected) Yuan, Xinyi (2008). "Big Line Bundles over Arithmetic Varieties" (PDF). Inventiones mathematicae. 173 (3): 603–649. doi:10.1007/s00222-008-0127-9. Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei (2012). The Gross–Zagier formula on Shimura curves. Annals of Mathematics Studies. Vol. 184. Princeton University Press. Yuan, Xinyi; Zhang, Tong (2013). "Effective Bound of Linear Series on Arithmetic Surfaces" (PDF). Duke Mathematical Journal. 162 (10): 1723–1770. doi:10.1215/00127094-2322779. Yuan, Xinyi; Zhang, Shou-Wu (2018). "On the averaged Colmez conjecture". Annals of Mathematics. 187 (2): 553–638. arXiv:1507.06903. doi:10.4007/annals.2018.187.2.4. S2CID 118916754. Xie, Junyi; Yuan, Xinyi (2022). "Geometric Bogomolov conjecture in arbitrary characteristics". Inventiones Mathematicae. 229 (2): 607–637. arXiv:2108.09722. doi:10.1007/s00222-022-01112-1. Yuan, Xinyi; Zhang, Shou-Wu (2026). Adelic Line Bundles on Quasi-Projective Varieties. Annals of Mathematics Studies. Vol. 221. Princeton, NJ: Princeton University Press. ISBN 978-0-691-27172-9. Yuan, Xinyi (2026). "Arithmetic bigness and a uniform Bogomolov-type result". Annals of Mathematics. 2. 203 (1): 15–119. doi:10.4007/annals.2026.203.1.2. Xie, Junyi; Yuan, Xinyi (2023). "Partial Heights, Entire Curves, and the Geometric Bombieri–Lang Conjecture". Acta Mathematica (to appear). arXiv:2305.14789. Xie, Junyi; Yuan, Xinyi (2023). "The Geometric Bombieri–Lang Conjecture for Ramified Covers of Abelian Varieties". Peking Mathematical Journal (to appear). arXiv:2308.08117. Yu, Jiawei; Yuan, Xinyi; Zhou, Shengxuan (2026). "Quantitativity on the number of rational points in the Mordell conjecture". arXiv:2602.01820 [math.NT].

References

Illustrations

Worked examples

Example 1 — a first encounter with Xinyi Yuan

Start with the simplest possible case. Write down what Xinyi Yuan 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 Xinyi Yuan 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 Xinyi Yuan 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 Xinyi Yuan

In research: Xinyi Yuan 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 Xinyi Yuan 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: Xinyi Yuan is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1981 births, 21st-century mathematicians, Arithmetic geometers, so understanding it makes those chapters shorter.
In everyday life: Look for Xinyi Yuan 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 Xinyi Yuan in 20 minutes

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

Frequently asked questions

What is Xinyi Yuan in simple terms?

Why does Xinyi Yuan 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 Xinyi Yuan?

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 Xinyi Yuan.