About

Benson Farb is a professor in the Department of Mathematics at The University of Chicago. He earned his PhD from Princeton University in 1994 under the supervision of W.P. Thurston. His research encompasses a wide range of mathematical fields including geometric group theory, low-dimensional topology, dynamical systems, differential geometry, Teichmuller theory, cohomology of arithmetic groups, representation theory, algebraic geometry, and 4-manifold theory, as well as the connections among these topics. Farb has received notable recognition for his contributions, including a named professorship in July 2024, the 2024 AMS Steele Prize announced in November 2023, and election to the American Academy of Arts and Sciences in April 2021. His work and influence extend across various areas of mathematics, reflecting a broad and impactful research portfolio.

Research topics

• Mathematics
• Pure mathematics
• Geology
• Geography

Selected publications

• Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group

  arXiv (Cornell University) · 2026-05-15

  preprintOpen access

  Let $\mathcal{C}$ be the moduli space of smooth complex cubic surfaces and let $π_1(\mathcal{C})$ be its (orbifold) fundamental group. We prove that the ``divisor subgroup'' of $π_1(\mathcal{C})$ is characteristic. This can be interpreted as saying that the group theory of $π_1(\mathcal{C})$ ``remembers'' the divisor of nodal cubic surfaces. We deduce from this group-theoretic result and some basic complex analysis that $\mathcal{C}$ has no nontrivial biholomorphic automorphisms as complex analytic orbifold.

  PublisherDOI

• Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group

  ArXiv.org · 2026-05-15

  articleOpen access

  Let $\mathcal{C}$ be the moduli space of smooth complex cubic surfaces and let $π_1(\mathcal{C})$ be its (orbifold) fundamental group. We prove that the ``divisor subgroup'' of $π_1(\mathcal{C})$ is characteristic. This can be interpreted as saying that the group theory of $π_1(\mathcal{C})$ ``remembers'' the divisor of nodal cubic surfaces. We deduce from this group-theoretic result and some basic complex analysis that $\mathcal{C}$ has no nontrivial biholomorphic automorphisms as complex analytic orbifold.

  PublisherOA PDF

• The smooth Mordell–Weil group and mapping class groups of elliptic surfaces

  Algebraic geometry · 2025-10-24

  articleOpen access1st authorCorresponding

  This is a paper in smooth 4-manifold topology, inspired by the Lang-Nron theorem in number theory.More precisely, we prove that a smooth version MW() of the Mordell-Weil group of an elliptic fibration : M P 1 is finitely generated.We compute MW( d ) explicitly for elliptic fibrations d : M d P 1 , where M d is a simply connected complex surface of arithmetic genus d 1 and all fibers of d are nodal.We prove in this case that the fibered structure is unique up to topological isotopy.By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of M d with d 3 is topologically isotopic to a diffeomorphism taking fibers to fibers.

  PublisherDOI

• Entropy-minimizing diffeomorphisms of pseudo-Anosov type on K3 surfaces

  ArXiv.org · 2025-07-17

  preprintOpen access1st authorCorresponding

  We construct diffeomorphisms of ``pseudo-Anosov type'' on K3 surfaces M. In particular we obtain infinitely many examples of such diffeomorphisms that minimize entropy in their homotopy class, and for which neither the diffeomorphism nor any diffeomorphism homotopic to it preserves any complex structure on M.

  PublisherOA PDFDOI

• Moduli spaces and period mappings of genus one fibered K3 surfaces

  Journal of Differential Geometry · 2025-09-08

  article1st authorCorresponding

  In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $\pi: M \rightarrow \mathbb{P}^1$ with generic fiber a complex torus (e.g., an elliptic fibration). Examples include moduli spaces of such maps with primitive fibers; with reduced, irreducible fibers; equipped with a section. Such spaces are closely related to the moduli space of Ricci-flat metrics on $M$ . We construct period mappings relating these moduli spaces to locally symmetric spaces, and use these to compute their orbifold fundamental groups. These results lie in contrast to, and exhibit different behavior than, the well-studied case of moduli spaces of polarized K3 surfaces, and are more useful for applications to the mapping class group $\operatorname{Mod}(M)$ . Indeed, we apply our results on moduli space to give two applications to the smooth mapping class group of $M$ .

  PublisherDOI

• Irrationality of the general smooth quartic 3-fold using intermediate Jacobians

  Advances in Mathematics · 2025-02-14

  article1st authorCorresponding
  PublisherDOI

• The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

  arXiv (Cornell University) · 2024-03-23

  preprintOpen access1st authorCorresponding

  This is a paper in smooth $4$-manifold topology, inspired by the Néron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(π)$ of Mordell-Weil group of an elliptic fibration $π:M\to\Pb^1$ is finitely generated. We compute $\MW(π_d)$ explicitly for elliptic fibrations $π_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\geq 1$ and all fibers of $π_d$ are nodal. We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.

  PublisherOA PDFDOI

• The Nielsen realization problem for K3 surfaces

  Journal of Differential Geometry · 2024-06-01 · 6 citations

  articleOpen access1st authorCorresponding

  The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version for involutions. Unlike the case of 2-manifolds, some $G$ are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant $L_G$ that determines in many cases whether $G$ is realizable or not, and apply this invariant to construct an $S_4$ action by isometries of some Ricci-flat metric on $M$ that preserves no complex structure. We also show that the subgroups of $\mathrm{Diff}(M)$ of a given prime order $p$ which fix pointwise some positive-definite 3-plane in $H^2(M; \mathbb{R})$ and preserve some complex structure on $M$ form a single conjugacy class in $\mathrm{Diff}(M)$ (it is known that then $p \in \{2, 3, 5, 7\}$).

  PublisherOA PDFDOI

• Essential dimension via prismatic cohomology

  Duke Mathematical Journal · 2024-10-15

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Rigidity of moduli spaces and algebro-geometric constructions

  arXiv (Cornell University) · 2023-02-13

  preprintOpen access1st authorCorresponding

  In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and holomorphic aspects, and so they also provide motivation for natural problems in geometric group theory and topology.

  PublisherOA PDFDOI

Recent grants

• Stability and Instability in Topology

  NSF · $576k · 2014–2019

• Geometry, Rigidity, and Group Actions

  NSF · $24k · 2007–2008

• Representation Theory and Homological Stability in Topology

  NSF · $441k · 2011–2015

• Geometry and Dynamics of the group of Hamiltonian diffeomorphisms of a surface

  NSF · $128k · 2009–2012

• Topics at the Intersection of Geometry, Topology and Group Theory

  NSF · $399k · 2006–2012

Frequent coauthors

• Eduard Looijenga

  42 shared

• Thomas M. Church

  University of Notre Dame

  26 shared

• Jesse Wolfson

  22 shared

• Dan Margalit

  22 shared

• Andrew Putman

  University of Notre Dame

  12 shared

• Jordan S. Ellenberg

  11 shared

• R. Keith Dennis

  11 shared

• Mark Kisin

  9 shared

Labs

• Farb LabPI

Education

• Ph.D.

  Princeton University

  1994

Awards & honors

• Named Professorship (2024)
• 2024 AMS Steele Prize
• American Academy of Arts and Sciences (2021)