{ "id": "epfwn7recz", "slug": "peter-b-kronheimer", "title": "Peter B. Kronheimer", "updated": "2026-10-25", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists" }, { "id": "physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics" }, { "id": "physical.scientists.mathematics-statistics.topologists-and-geometers", "label": "Topologists and geometers", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.topologists-and-geometers" }, { "id": "physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "label": "Low-dimensional and knot theorists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists" } ], "geo": [ { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "label": "United States · 1946 to 2000: Low-dimensional and knot theorists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "path": [ { "id": "geo.us", "label": "United States", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us" }, { "id": "geo.us.t1946", "label": "United States · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946" }, { "id": "geo.us.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical" }, { "id": "geo.us.t1946.physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers", "label": "Topologists and geometers", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "label": "Low-dimensional and knot theorists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists" } ] } ], "excerpt": "Peter B. Kronheimer is a mathematician at Harvard working on gauge theory and low-dimensional topology, known for proving Property P and the Milnor conjecture on torus knots.", "snippet": "Peter B. Kronheimer is a mathematician at Harvard working on gauge theory and low-dimensional topology, known for proving Property P and the Milnor conjecture on torus knots.", "node": "physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "markdown": "# Peter B. Kronheimer\n\n**Peter Benedict Kronheimer** is a mathematician who works on gauge theory and low-dimensional topology at Harvard University, where he is the William Caspar Graustein Professor of Mathematics.[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) A student of [Michael Atiyah](https://www.edgepedia.org/michael-atiyah) in Oxford, he has had an enduring influence on gauge theory and low-dimensional topology: his results include the structure theorem for Donaldson's polynomial invariants, the first proof of the Milnor conjecture on torus knots, the proof of Property P, and the theorem that [Khovanov homology](https://www.edgepedia.org/khovanov-homology) detects the unknot.[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) Much of this work is joint with [Tomasz Mrowka](https://www.edgepedia.org/tomasz-mrowka).[1](https://www.ams.org/notices/201104/rtx110400606p.pdf)\n\n| Key fact | Detail |\n|---|---|\n| Education | D.Phil., University of Oxford, 1986, dissertation \"ALE Gravitational Instantons\", advisor Michael Atiyah (the AMS Notices gives 1987)[3](https://genealogy.math.ndsu.nodak.edu/id.php?id=47154) • [1](https://www.ams.org/notices/201104/rtx110400606p.pdf) |\n| Position | William Caspar Graustein Professor of Mathematics at Harvard, since 1995; previously fellow and tutor at Merton College, Oxford[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) |\n| Structure theorem | For an admissible simple-type 4-manifold with b₁ = 0 and non-zero Donaldson invariant, the invariants are exponential sums exp(Q/2) Σ βᵣ exp(Kᵣ), yielding the genus bound 2 genus(ΣS) − 2 ≥ Q(S) + Kᵣ(S)[4](https://people.math.harvard.edu/~kronheim/structhm.pdf) |\n| Milnor conjecture | Slice genus and unknotting number of the torus knot T(p,q) both equal (p−1)(q−1)/2, first proved by Kronheimer and Mrowka with singular instanton gauge theory[5](https://web.stanford.edu/~cm5/proc_ecm.pdf) |\n| Property P | +1-surgery on a non-trivial knot in S³ is never a homotopy sphere; proved in Geometry & Topology 8 (2004), 295–310[6](https://msp.org/gt/2004/8-1/gt-v8-n1-p07-s.pdf) |\n| Unknot detection | A knot is the unknot if and only if its reduced Khovanov cohomology has rank 1[7](https://pmihes.centre-mersenne.org/articles/10.1007/s10240-010-0030-y/) |\n| Prizes | Whitehead Prize, Oberwolfach Prize, Oswald Veblen Prize (with Mrowka), Joseph L. Doob Prize (2011), Leroy P. Steele Prize (2023); Fellow of the Royal Society, 1997[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) |\n\n## Life and career\n\nKronheimer took his doctorate at Oxford under Michael Atiyah, writing on ALE gravitational instantons.[3](https://genealogy.math.ndsu.nodak.edu/id.php?id=47154) The Mathematics Genealogy Project dates the degree 1986, while the AMS prize notice gives 1987.[3](https://genealogy.math.ndsu.nodak.edu/id.php?id=47154) • [1](https://www.ams.org/notices/201104/rtx110400606p.pdf) He then held a fellowship and tutorship at [Merton College, Oxford](https://www.edgepedia.org/merton-college-oxford), before moving to Harvard in 1995.[1](https://www.ams.org/notices/201104/rtx110400606p.pdf)\n\nHis prizes trace the recognition of his joint program with Mrowka: the Whitehead Prize of the London Mathematical Society, the Oberwolfach Förderpreis, the Oswald Veblen Prize of the American Mathematical Society, and election to the Royal Society in 1997.[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) In 2011 the two received the AMS Joseph L. Doob Prize for their 2007 monograph *Monopoles and Three-Manifolds* ([Cambridge University Press](https://www.edgepedia.org/cambridge-university-press)), and in 2023 the Leroy P. Steele Prize.[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) The Clay Mathematics Institute ran a workshop, \"Gauge Theory and Topology\", celebrating his 60th birthday, citing among the field's applications the proof of the Thom Conjecture, the Weinstein Conjecture, the resolution of the high-dimensional Triangulation Conjecture, and new insights into the concordance and homology cobordism groups.[2](https://www.claymath.org/events/gauge-theory-and-topology-in-celebration-of-peter-kronheimerss-60th-birthday/) Among his doctoral students is Ciprian Manolescu.[15](https://mathgenealogy.org/id.php?id=47154)\n\n## Donaldson theory and the simple type theorem\n\nIn the 1980s [Simon Donaldson](https://www.edgepedia.org/simon-donaldson) developed invariants of smooth 4-manifolds. Kronheimer and Mrowka determined the algebraic structure of these invariants for a large class of manifolds. Their structure theorem states: for an admissible 4-manifold X of simple type with b₁ = 0 whose Donaldson invariants are not all zero, there are finitely many cohomology classes K₁, …, Kₛ in H²(X; Z), lifting w₂(X), and non-zero rational numbers β₁, …, βₛ, such that the invariants are exponential sums of the form exp(Q/2) Σ βᵣ exp(Kᵣ), where Q is the intersection form.[4](https://people.math.harvard.edu/~kronheim/structhm.pdf)\n\nThe theorem converts Donaldson's invariants into usable geometric bounds. If ΣS is a smoothly embedded, oriented surface in X representing a non-trivial homology class S with Q(S) ≥ 0, then its genus satisfies the lower bound 2 genus(ΣS) − 2 ≥ Q(S) + Kᵣ(S).[4](https://people.math.harvard.edu/~kronheim/structhm.pdf)\n\n**Simple type.** After Seiberg–Witten theory transformed 4-manifold topology in 1994, Kronheimer and Mrowka reformulated the condition in their monograph: a closed oriented 4-manifold X with b⁺(X) ≥ 2 has simple type if the monopole invariants m(u e²(X), s_X) vanish for all e > 0 and all spinᶜ structures.[8](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf) The condition is known to hold for many classes of 4-manifolds with b⁺ ≥ 2, including all symplectic 4-manifolds and all 4-manifolds containing a tight surface.[8](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf) The general statement remains open as [Conjecture](https://www.edgepedia.org/conjecture) 1.6.2 of the monograph: all closed oriented 4-manifolds with b⁺ ≥ 2 have simple type.[8](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf)\n\n## Knots and 3-manifolds: the Milnor conjecture and Property P\n\nThe Milnor conjecture asks for the four-ball genus of the torus knot T(p,q) for relatively prime p and q. The answer is (p−1)(q−1)/2, and a corollary is that the minimum number of crossing changes needed to untie T(p,q), its unknotting number, is the same.[5](https://web.stanford.edu/~cm5/proc_ecm.pdf) Kronheimer and Mrowka gave the first proof, using Donaldson invariants, that is, gauge theory with respect to connections on 4-manifolds that are singular along embedded surfaces.[5](https://web.stanford.edu/~cm5/proc_ecm.pdf) • [9](https://msp.org/gt/2025/29-8/gt-v29-n8-p04-s.pdf) Alternative proofs came later from Ozsváth and Szabó using [Heegaard Floer homology](https://www.edgepedia.org/heegaard-floer-homology) and from Rasmussen using Khovanov homology.[9](https://msp.org/gt/2025/29-8/gt-v29-n8-p04-s.pdf) Their proof can also be recast in terms of the concordance invariant s♯ that they introduced, motivated by Rasmussen's work and further studied by Gong.[10](https://arxiv.org/html/2209.05400)\n\n**Property P.** Property P is the statement that +1-surgery on any non-trivial knot in the 3-sphere does not yield a homotopy sphere. Kronheimer and Mrowka proved it in a 2004 Geometry & Topology paper: for a non-trivial knot K, the fundamental group of the surgered manifold Y admits a non-trivial homomorphism to SO(3), so Y is not simply connected.[6](https://msp.org/gt/2004/8-1/gt-v8-n1-p07-s.pdf) The proof combined tools from gauge theory and symplectic topology, including Taubes' non-vanishing theorem for symplectic 4-manifolds, Gabai's taut foliations, the Eliashberg–Thurston contact-structure machinery, Floer's exact triangle, and a weak version of Witten's conjecture established by Feehan and Leness.[6](https://msp.org/gt/2004/8-1/gt-v8-n1-p07-s.pdf)\n\n## Instanton Floer homology and Khovanov homology\n\nKronheimer and Mrowka built [Floer homology](https://www.edgepedia.org/floer-homology) theories for knots and for 3-manifolds with boundary. For each partial flag manifold of SU(N), they define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities along the knot, imitating Floer's construction in which the representation variety R(K) appears as the set of critical points of a Chern–Simons functional on a space of SU(2) connections.[11](https://arxiv.org/html/0806.1053)\n\nIn \"Knots, sutures and excision\" they developed monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of Juhász's sutured Heegaard Floer homology. Applications include a new proof of Property P that is independent of the Feehan–Leness work on Witten's conjecture and requires no contact or symplectic tools; they also showed that instanton homology captures the Thurston norm on an irreducible 3-manifold.[12](https://projecteuclid.org/journalArticle/Download?urlid=10.4310%2Fjdg%2F1274707316) Their sutured instanton knot homology turned out to be exactly the same as an instanton homology for knots defined by Floer twenty years earlier, and they conjectured that, over a field of characteristic zero, the knot homology groups of Ozsváth–Szabó and of Rasmussen are isomorphic to Floer's instanton knot homology.[12](https://projecteuclid.org/journalArticle/Download?urlid=10.4310%2Fjdg%2F1274707316)\n\n**The Alexander polynomial theorem.** For any knot K in S³, the Euler characteristics of the generalized eigenspace summands KHI(K, j) of their instanton Floer homology are minus the coefficients of the symmetrized [Alexander polynomial](https://www.edgepedia.org/alexander-polynomial) with Conway's normalization.[13](https://people.math.harvard.edu/~kronheim/alexander.pdf) The invariant carries a distinguished endomorphism of even degree, arising from the 2-dimensional homology class represented by a Seifert surface, and among other applications they deduced that instanton homology detects fibered knots.[13](https://people.math.harvard.edu/~kronheim/alexander.pdf)\n\n**Khovanov homology is an unknot-detector.** Kronheimer and Mrowka proved that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps: a spectral sequence beginning with reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons, and a proof that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement, an invariant already known to detect the unknot.[7](https://pmihes.centre-mersenne.org/articles/10.1007/s10240-010-0030-y/)\n\n## How it compares with other Floer theories\n\nKronheimer and Mrowka's monograph *Monopoles and Three-Manifolds* gives a detailed construction of monopole Floer homology, the Seiberg–Witten-based theory of 3-manifolds; the Doob Prize citation notes that it develops [Morse theory](https://www.edgepedia.org/morse-theory) for manifolds with boundary, sharper compactness results, and functoriality for Floer theory, with full details appearing there for the first time.[1](https://www.ams.org/notices/201104/rtx110400606p.pdf) • [14](https://ar5iv.labs.arxiv.org/html/1706.07729) The conjectural equivalence between Heegaard Floer theory and Seiberg–Witten/Floer theory in dimension 3 was verified by Kutluhan–Lee–Taubes and by Colin–Ghiggini–Honda.[5](https://web.stanford.edu/~cm5/proc_ecm.pdf) • [14](https://ar5iv.labs.arxiv.org/html/1706.07729)\n\nInstanton methods retain distinctive power where the others are weaker: they capture the Thurston norm on irreducible 3-manifolds[12](https://projecteuclid.org/journalArticle/Download?urlid=10.4310%2Fjdg%2F1274707316) and, through the spectral sequence above, they are the tool by which Khovanov homology was proved to detect the unknot.[7](https://pmihes.centre-mersenne.org/articles/10.1007/s10240-010-0030-y/)\n\n## References\n\n1. [Peter Kronheimer and Tomasz Mrowka receive the 2011 Joseph L. Doob Prize, AMS Notices 58(4), April 2011](https://www.ams.org/notices/201104/rtx110400606p.pdf)\n2. [Gauge Theory and Topology: in Celebration of Peter Kronheimer's 60th Birthday, Clay Mathematics Institute](https://www.claymath.org/events/gauge-theory-and-topology-in-celebration-of-peter-kronheimerss-60th-birthday/)\n3. [Peter Kronheimer, The Mathematics Genealogy Project](https://genealogy.math.ndsu.nodak.edu/id.php?id=47154)\n4. [P. Kronheimer and T. Mrowka, Embedded Surfaces and the Structure of Donaldson's Polynomial Invariants](https://people.math.harvard.edu/~kronheim/structhm.pdf)\n5. [Survey on combinatorial Heegaard Floer theory, ECM proceedings](https://web.stanford.edu/~cm5/proc_ecm.pdf)\n6. [P. Kronheimer and T. Mrowka, Witten's conjecture and Property P, Geometry & Topology 8 (2004) 295–310](https://msp.org/gt/2004/8-1/gt-v8-n1-p07-s.pdf)\n7. [P. Kronheimer and T. Mrowka, Khovanov homology is an unknot-detector, Publications mathématiques de l'IHÉS](https://pmihes.centre-mersenne.org/articles/10.1007/s10240-010-0030-y/)\n8. [P. Kronheimer and T. Mrowka, Monopoles and Three-Manifolds (Cambridge University Press, 2007)](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf)\n9. [Instantons, special cycles and knot concordance, Geometry & Topology 29 (2025)](https://msp.org/gt/2025/29-8/gt-v29-n8-p04-s.pdf)\n10. [Instantons, special cycles, and knot concordance, arXiv:2209.05400](https://arxiv.org/html/2209.05400)\n11. [P. Kronheimer and T. Mrowka, Knot homology groups from instantons, arXiv:0806.1053](https://arxiv.org/html/0806.1053)\n12. [P. Kronheimer and T. Mrowka, Knots, sutures and excision, Journal of Differential Geometry](https://projecteuclid.org/journalArticle/Download?urlid=10.4310%2Fjdg%2F1274707316)\n13. [P. Kronheimer and T. Mrowka, Instanton Floer homology and the Alexander polynomial](https://people.math.harvard.edu/~kronheim/alexander.pdf)\n14. [An overview of knot Floer homology, arXiv:1706.07729](https://ar5iv.labs.arxiv.org/html/1706.07729)\n15. [mathgenealogy.org](https://mathgenealogy.org/id.php?id=47154)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Topologists and geometers › Low-dimensional and knot theorists*\n\n*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026; Oct 25, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://people.math.harvard.edu/~kronheim/structhm.pdf", "https://web.stanford.edu/~cm5/proc_ecm.pdf", "https://people.math.harvard.edu/~kronheim/alexander.pdf" ], "url": "https://www.edgepedia.org/peter-b-kronheimer", "markdown_url": "https://www.edgepedia.org/peter-b-kronheimer.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Peter B. Kronheimer\", Edgepedia (EdgeChat), https://www.edgepedia.org/peter-b-kronheimer. Edgepedia Community License 1.0.", "credit_md": "\"[Peter B. Kronheimer](https://www.edgepedia.org/peter-b-kronheimer)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/peter-b-kronheimer](https://www.edgepedia.org/peter-b-kronheimer). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Peter B. Kronheimer\", Edgepedia (EdgeChat), https://www.edgepedia.org/peter-b-kronheimer. Edgepedia Community License 1.0.", "speakable": "Peter B. Kronheimer is a mathematician at Harvard working on gauge theory and low-dimensional topology, known for proving Property P and the Milnor conjecture on torus knots." }