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June Huh

June Huh (Korean: 허준이; born June 9, 1983) is an specializing in geometric combinatorics, renowned for bridging and combinatorics through innovative applications of , , and singularity theory. Born in , to parents pursuing graduate studies, Huh moved to , , at age two and grew up there, completing elementary and before dropping out of high school at 16 to pursue interests in and . He enrolled at in 2002, initially in physics and astronomy, but a pivotal 2007 course with Fields Medalist Heisuke Hironaka inspired him to switch to mathematics; he earned a B.S. in physics and astronomy in 2007, followed by an M.S. in mathematics in 2009 under advisor Young-Hoon Kiem. After beginning graduate studies at the University of at Urbana-Champaign in 2009, he transferred to the University of , completing his Ph.D. in 2014 under advisor Mircea Mustaţă, with a dissertation on combinatorial applications of . Huh's career accelerated post-Ph.D., beginning with a Clay Research Fellowship (2014–2019) and Veblen Fellowship at the (IAS) and (2014–2017), followed by visiting professorships at IAS (2017–2019 and 2019–2020 as Fernholz Visiting Professor). He served as a professor at from 2020 to 2021, then joined as a full professor in 2021, while also holding positions at the Korea Institute for Advanced Study (KIAS), including distinguished professor since 2022. His has transformed geometric by proving major s, including Read's on chromatic polynomials (2012, as a graduate student), the Dowling–Wilson for geometric lattices (with Botong Wang), the Heron–Rota–Welsh on log-concavity of characteristic polynomials (with and Eric Katz), and the strong Mason (with Petter Brändén), alongside developing the theory of Lorentzian polynomials with broad applications in and . For these contributions, he received the ProQuest Distinguished Dissertation Award (2015), New Horizons in Mathematics Prize (2019), from the (2022)—the first for a of descent—and Fellowship (2022).

Early life and education

Upbringing and early interests

June Huh was born in 1983 in to South Korean parents pursuing graduate studies there. His father, Huh Myung-hoe, later became a professor emeritus of statistics at , while his mother, Lee In-young, served as a professor emerita of Russian language and literature at . The family relocated to , , when Huh was two years old, where he grew up in an environment that emphasized , influenced by his mother's literary . As a child in , Huh attended Bangil Elementary School and Isu Middle School, developing a strong interest in and from an early age. He enrolled in Sangmoon High School but dropped out at age 16 during his first year, choosing instead to focus on writing and immersing himself in . He spent much of this time reading extensively at the National Library of , inspired by nature, music, and personal introspection to craft verses that he hoped would capture profound beauty. After a period of self-directed study and preparation through cram schools, Huh gained admission to in 2002. There, he pursued an undergraduate degree initially in physics and astronomy, though his early interests remained rooted in the ; he considered careers in or and continued writing verse, some of which was later published for the first time in 2022. He earned a B.S. in physics, astronomy, and in 2007.

Transition to mathematics

In 2002, June Huh enrolled at , initially majoring in physics and astronomy with aspirations toward , but he struggled academically and lacked interest in . This changed in his sixth and final undergraduate year around 2007–2008, when he attended a course on taught by Heisuke Hironaka, a Japanese mathematician and 1970 recipient serving as a visiting professor at the university. Despite having no prior background in advanced , Huh was captivated by Hironaka's approach, which emphasized intuitive, concrete examples over abstract formalism, prompting him to question his earlier self-perception as untalented in the subject. Hironaka recognized Huh's latent potential and took him on as a mentee, encouraging him to abandon his original career path and pursue full-time, even though Huh had barely passed his earlier math courses. Their collaboration began informally through discussions and lunches, evolving into intensive guidance where Hironaka introduced Huh to singularity theory and broader geometric concepts. Motivated by this mentorship, Huh switched his major to include , embarking on rigorous self-study of foundational topics including and to catch up with his peers. As part of his early training under Hironaka, Huh was exposed to combinatorial problems, whose tangible, visual nature—such as those involving graphs and enumerations—resonated with him and ignited a particular fascination with the field, contrasting with the more elusive abstractions he had previously avoided.

Graduate studies

Following his B.S., Huh earned an M.S. in mathematics in 2009 at Seoul National University under advisor Young-Hoon Kiem. In 2009, he enrolled in the PhD program in mathematics at the University of Illinois at Urbana-Champaign. His early graduate work there included a proof of Read's conjecture on the log-concavity of coefficients in the chromatic polynomial of graphs, a longstanding problem in combinatorial graph theory that he resolved using techniques from algebraic geometry. In 2011, Huh transferred to the to continue his doctoral studies, where he worked under the advisement of Mircea Mustaţă, with a research focus bridging and . During this period, he advanced his contributions to theory, collaborating with Eric Katz and others on partial progress toward Rota's conjecture regarding the log-concavity of matroid characteristic polynomials. Huh completed his in 2014, defending a dissertation titled Rota's and Positivity of Algebraic Cycles in Permutohedral Varieties, which earned him the Distinguished Dissertation Award from the . The work culminated in a proof of Rota's for representable matroids, leveraging and positivity properties in algebraic cycles to establish key combinatorial inequalities.

Professional career

Academic positions

Following the completion of his PhD in from the University of Michigan in 2014, June Huh embarked on his postdoctoral career as a Veblen Fellow at the Institute for Advanced Study (IAS) and , holding the position from 2014 to 2017. Concurrently, he served as a with the from 2014 to 2019, a prestigious fellowship supporting early-career mathematicians in their research endeavors. From 2017 to 2019, Huh transitioned to the role of Visiting Professor at the IAS, continuing his association with the institute while deepening his ties to Princeton through joint appointments. In 2019–2020, he was appointed Fernholz Visiting Professor, jointly at the IAS and Princeton University, further solidifying his presence in the Princeton mathematical community. These positions allowed him to focus on independent research while benefiting from the collaborative environment of leading institutions. In 2020, Huh joined Stanford University as a full Professor of Mathematics, a role he held until 2021. He then returned to Princeton University as Professor of Mathematics, a position he has maintained since July 2021. Throughout his career, Huh has maintained ongoing affiliations, including as a Visiting KIAS Scholar at the Korea Institute for Advanced Study from 2015 to 2021, KIAS Professor from 2021 to 2022, and KIAS Distinguished Professor since 2022. He also served as Distinguished Visiting Professor at the IAS from 2024 to 2025. Huh continues to engage in visiting lectures and academic visits, exemplifying his role in the global mathematics community. For instance, in September 2025, he delivered the Stelson Lectures at the Georgia Institute of Technology, accompanied by a special School of Mathematics Colloquium.

Research collaborations

June Huh's research collaborations have significantly advanced the intersections of , , and , often through joint efforts that leverage diverse mathematical perspectives. Beginning around 2010, Huh partnered with Eric Katz to explore theory and its connections to , notably proving log-concavity properties of matroid characteristic polynomials using the Bergman fan. This work, published in Mathematische Annalen in 2012, resolved longstanding questions about the and log-concavity of these polynomials for realizable matroids, providing a foundational algebraic-geometric approach to combinatorial problems. A key collaboration unfolded with and Eric Katz on combinatorial Hodge theory, culminating in their 2018 paper in the . Titled "Hodge theory for combinatorial geometries," this joint effort established the hard Lefschetz theorem and Hodge-Riemann relations for the Chow rings of matroids, enabling proofs of central conjectures like the Rota conjecture on the log-concavity of the h-vector for matroids. Their approach adapted classical to discrete structures, bridging with and influencing subsequent developments in geometric combinatorics. Huh has also engaged in partnerships addressing log-concavity conjectures more broadly, building on earlier work with Katz and extending to applications in morphisms and related polynomials. For instance, in collaboration with Christopher Eur, Huh proved strong log-concavity for the generating functions of bases in morphisms in a 2020 Advances in Mathematics paper, advancing understanding of discrete convexity in combinatorial settings. These efforts highlight Huh's role in collaborative proofs of unimodality and log-concavity across varieties. Huh's involvement in workshops and programs at the Institute for Advanced Study (IAS) and has fostered ongoing collaborations. As Distinguished Visiting Professor for the 2024–2025 special year on algebraic and geometric at IAS, Huh co-organized events including the October 2024 "Geometry of " workshop, the November 2024 "Combinatorics of Fundamental Physics" workshop, and the February 2025 "Combinatorics of " workshop, which brought together researchers to explore structures, their geometric realizations, and related interdisciplinary topics. Similarly, at Princeton, Huh has participated in seminars and programs promoting interdisciplinary exchanges, such as those on , facilitating joint projects with emerging mathematicians. Up to 2024, Huh's joint papers have increasingly applied theory to combinatorial geometries. In a 2023 collaboration with Federico Ardila and Graham Denham, published in the Journal of the American Mathematical Society, they developed Lagrangian geometry for s, using singularity-inspired techniques to analyze Lagrangian subvarieties and their intersections, providing new tools for studying matroid realizability and positivity in Chow rings. This work extends earlier singularity applications from Huh's partnerships with Katz and Adiprasito, emphasizing discrete analogs of smooth geometric phenomena.

Mathematical contributions

Proofs of major conjectures

June Huh made significant contributions to combinatorics by resolving several longstanding conjectures concerning matroids, employing techniques from algebraic geometry. In 2012, he proved Read's conjecture, which posits that the coefficients of the chromatic polynomial of any graph form a unimodal sequence, meaning they increase up to a point and then decrease. This result was established by relating the chromatic polynomial to the Milnor numbers of projective hypersurfaces, leveraging vanishing theorems in algebraic geometry to derive the necessary inequalities. The proof not only confirmed the conjecture for graphic matroids but also extended to more general settings, providing a combinatorial interpretation through geometric positivity. Building on this, Huh, in collaboration with and Eric Katz, fully resolved the Heron-Rota-Welsh conjecture in , a of Read's conjecture to arbitrary . The conjecture asserts that the coefficients of the of any matroid are log-concave, implying and stronger inequalities akin to those in the Macaulay theorem for graded posets. Their proof introduced a combinatorial for matroids, associating to each matroid a K-class in the of the and applying Hodge-Riemann relations to establish positivity and vanishing results that imply log-concavity. This approach relied on equivariant localization and decomposition theorems to handle the non-representable case, marking a breakthrough in bridging discrete and geometric structures. Huh further advanced matroid theory by proving the Dowling-Wilson for representable matroids in 2016, jointly with Botong Wang. The , formulated in the , predicts that for a geometric lattice of r, the number of of k is at least as large as the number of of r - k for all k, exhibiting a "top-heavy" distribution. Their proof utilized the geometry of the wonderful compactification of the complement of a , deriving the inequality through K-theoretic vanishing theorems on line bundles and the decomposition theorem for semi-small maps. This confirmed the conjecture for matroids realizable over fields, with implications for in arrangements. The conjecture was fully resolved for all matroids in 2023 using singular , in collaboration with , Jacob P. Matherne, Nicholas Proudfoot, and Botong Wang. These proofs have notable applications to partition functions and graph colorings. For instance, the log-concavity results from the Heron-Rota-Welsh theorem imply refined bounds on the number of proper colorings of graphs, connecting to the partition function where the coefficients encode data. Similarly, the top-heavy property aids in analyzing the distribution of color partitions, providing asymptotic estimates for chromatic polynomials via geometric interpretations. Huh's methods, particularly the use of classes and vanishing theorems, underscore the role of algebraic tools in yielding such combinatorial insights without delving into broader interdisciplinary frameworks.

Bridging combinatorics and algebraic geometry

June Huh has pioneered methods that forge deep connections between and , particularly by adapting geometric tools like and to discrete structures such as matroids. His approaches provide new frameworks for understanding enumerative and positivity phenomena in combinatorial settings, revealing hidden symmetries and inequalities that were previously inaccessible through purely discrete methods. A cornerstone of Huh's contributions is the development of combinatorial Hodge theory, which integrates sheaf theory into the study of matroids to define a cohomology that mimics the Hodge decomposition in algebraic geometry. In collaboration with Karim Adiprasito and Eric Katz, Huh introduced this theory for combinatorial geometries, establishing an isomorphism between the cohomology of the matroid complex and an Orlik-Solomon algebra, thereby enabling the application of Hodge-Riemann relations to combinatorial invariants. This framework extends classical to discrete objects, allowing for the proof of structural properties like and the hard Lefschetz theorem in the matroid context. Huh has also employed to address enumerative problems in , particularly in the analysis of and their realizations. By leveraging tropicalization, which translates algebraic varieties into piecewise-linear spaces, he developed tools to count and classify structures, linking discrete to geometric invariants like volumes in tropical convex hulls. For instance, in his work on the of , Huh showed how tropical linear spaces capture the combinatorial data of , facilitating computations of invariants such as the number of bases or through geometric degenerations. This perspective has illuminated connections between matroid theory and , providing algorithmic and structural insights into . Significant among Huh's advancements are his contributions to the of matroids and arrangements, where he unified these areas under the umbrella of combinatorial . For arrangements, which realize certain matroids over fields, Huh's provides a refined understanding of the of complement spaces, extending results from arrangement theory to non-realizable matroids via abstract combinatorial axioms. His work demonstrates that the of these structures satisfies key geometric properties, such as , even in the absence of an underlying . Huh's methods have profound applications to positivity questions in characteristic p and the log-concavity of combinatorial coefficients. Using combinatorial , the Hodge-Riemann relations yield the log-concavity of the coefficients of the of matroids, resolving Rota's and establishing with no internal zeros for these sequences. In characteristic p settings, his frameworks extend to modular sheaves, ensuring positivity of cycle classes and Betti numbers in arrangements over finite fields. In joint work with Petter Brändén, Huh introduced Lorentzian polynomials in 2019, a class of homogeneous polynomials that generalize polynomials and exhibit properties analogous to Lorentzian inner products in . These polynomials preserve nonnegativity under differentiation and multiplication by Lorentzian factors, linking discrete M-convex functions to continuous Lorentzian geometry via tropicalization. This theory resolves the strong Mason conjecture on the ultra-log-concavity of independence polynomials for matroids and has applications to polynomials of convex bodies and models. Building on earlier work initiated in 2020 and refined through 2023, Huh advanced singularity theory within combinatorial contexts through singular for matroids, collaborating with , Jacob P. Matherne, Nicholas Proudfoot, and Botong Wang. They developed an intersection cohomology module for singular matroids that incorporates perverse sheaves and satisfies refined Hodge-Riemann inequalities, even for non-smooth geometries. This extension handles singularities arising in tropical degenerations and complements, providing tools to study stability and in combinatorial singularity theory. As applications, it proves the nonnegativity of coefficients in the Kazhdan-Lusztig polynomials of all matroids, resolving a 2015 conjecture.

Awards and honors

Fields Medal and MacArthur Fellowship

In 2022, June Huh was awarded the by the (IMU), one of the highest honors in , recognizing outstanding achievement for existing work and the promise of future contributions. The medal was presented for his groundbreaking contributions to combinatorial , specifically for bringing the ideas of to , proving the Dowling–Wilson conjecture for geometric lattices and the Heron–Rota–Welsh conjecture for matroids (also known as Rota's conjecture), developing the theory of Lorentzian polynomials, and proving the strong Mason conjecture. The IMU citation highlights how these advancements established new connections between discrete and continuous , revolutionizing approaches to long-standing problems in the field. The ceremony took place on July 5, , during the in , , at , marking the first in-person event since the began. Huh, then 39 years old, became the first mathematician of descent to receive the award, a milestone celebrated for inspiring underrepresented communities in global . Later that year, on October 12, , Huh received the MacArthur Fellowship, often called the "Genius Grant," from the John D. and Foundation, which provides an $800,000 no-strings-attached grant over five years to support exceptional creativity. The fellowship recognized Huh's interdisciplinary breakthroughs in bridging and to prove enduring conjectures, emphasizing his ability to forge novel links across mathematical branches that yield profound insights. This dual accolade in underscored the transformative impact of his work on unifying disparate areas of .

Other prestigious recognitions

In addition to the Fields Medal and MacArthur Fellowship, June Huh has received several other distinguished awards recognizing his innovative contributions to . In 2019, he was awarded the in Mathematics Prize by the Breakthrough Prize Foundation, sharing the honor with and Eric Katz for their proof of the Heron–Rota–Welsh (also known as Rota's ) on the log-concavity of characteristic polynomials, a long-standing problem in . This prize highlights early-career achievements with potential for transformative impact in the field. Huh's work has also been honored through major fellowships and investigator awards. He received the Blavatnik Regional Award for Young Scientists in Physical Sciences and Engineering from the in 2017, acknowledging his solutions to longstanding problems in using techniques. In 2021, the selected him as a , providing five years of flexible funding to support his research on discrete structures via geometric methods, such as applications to combinatorial problems. That same year, he was awarded the Samsung Ho-Am in Science by the Ho-Am Foundation for his advancements in physics and , particularly bridging and . In 2023, he was elected a member of the Korean Academy of Science and Technology and an honorary member of the , Republic of Korea. Early in his career, Huh held the Clay Research Fellowship from the from 2014 to 2019, which supported his postdoctoral research at the Institute for Advanced Study and facilitated key developments in his field. His growing influence is further evidenced by invitations to deliver plenary and invited lectures at major international events. He presented a plenary address at the in in 2022, discussing interactions between combinatorics and . More recently, in 2024–2025, he served as Distinguished Visiting Professor at the Institute for Advanced Study, and he gave a plenary lecture at the Mathematical Congress of the Americas in in 2025. These speaking engagements underscore his role as a leading figure in .

Personal life

Family and residence

June Huh is married to Nayoung Kim, a mathematician specializing in number theory and elliptic curves, whom he met while pursuing his master's degree at Seoul National University. Kim earned her PhD in mathematics from Seoul National University in 2014. The couple wed in 2014, the same year they relocated to Princeton, New Jersey, where both began positions at the Institute for Advanced Study. Huh and Kim have two sons; their first, Dan, was born in 2014, shortly after their marriage, and their second arrived around 2021. Huh has credited fatherhood with teaching him greater life balance, noting that raising his eldest son helped him develop more practical routines amid his intense mathematical pursuits. His family provided crucial support during his early career shifts, including his parents' encouragement when he left high school to pursue poetry in the early 2000s before pivoting to mathematics. The family has resided in Princeton since 2014, where Huh joined the as a Veblen Fellow and later became a professor at in 2021. They maintain ties to through Huh's role as a at the Korea Institute for Advanced Study, involving periodic visits, such as summer stays. This arrangement reflects Huh's dual cultural identity, shaped by his birth in the United States and upbringing in after his family returned there when he was about two years old.

Interests outside mathematics

Despite transitioning to mathematics later in life, June Huh has continued to write , including the publication of four original poems in the Korean science magazine Math Donga in August 2022. These works, appearing alongside coverage of his achievement, reflect his sustained interest in poetic expression even after establishing a prominent career in academia. In interviews, Huh has described how his early poetic pursuits inform his mathematical creativity, likening the search for elegant proofs to an artist's quest for beauty and deeper meaning. He has noted that mathematicians, much like poets, aim to uncover profound structures in their respective domains, a perspective shaped by his background in literature. Huh's hobbies include reading , which he approaches through the meditative practice of hand-transcribing passages into notebooks, and engaging in cultural activities tied to his heritage, such as contributing to Korean-language publications. He also enjoys long afternoon walks in natural settings around Princeton, where he observes and allows his mind to wander freely. Huh has delivered public talks exploring the intersections of arts and sciences, including a 2019 address on the pursuit of beauty as a motivating force in , drawing parallels to aesthetic experiences in and other creative fields. To prevent , he adheres to a disciplined routine that limits intensive mathematical work to approximately three hours daily, supplemented by restorative activities like walks and repetitive tasks such as cleaning or cooking simple dishes. This balance, highlighted in 2022 profiles, allows him to sustain creativity across his pursuits.

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