01 · A life in mathematics
The person behind the work
June Huh was born in Stanford, California, and grew up in South Korea. His route into research crossed from an early interest in poetry to mathematics. A professor at Princeton when he received the 2022 Fields Medal, he works in combinatorics using ideas inspired by algebraic geometry. With collaborators, he developed Hodge theory for matroids and the theory of Lorentzian polynomials, proving longstanding inequalities for counting sequences. His work shows that finite objects, including abstract systems of independence, can obey geometric principles even when no conventional geometric realization exists. This creates powerful connections between counting, positivity and curvature-like constraints.
The contribution, explained
A matroid records which subsets of a finite collection are independent, generalizing both independent vectors and cycle-free edge sets of a graph. Many natural counting sequences associated with matroids were conjectured to be log-concave: each middle term squared is at least the product of its neighbors. Huh and his collaborators built an analogue of Hodge theory for these combinatorial objects, even when no ordinary algebraic variety realizes them. Positivity and duality in the resulting algebra translate into inequalities for counting coefficients. The approach explains why a finite enumeration problem can have a hidden geometric organization rather than a merely accidental numerical pattern.
Why it matters
This work proved major conjectures about matroids and geometric lattices and helped create the theory of Lorentzian polynomials. It gives combinatorics a systematic way to derive inequalities from geometric-style positivity and structural constraints.