Robert Langlands

How to use this course

The Langlands programme is often called a grand unified theory of mathematics. It predicts that two very different kinds of objects carry the same information. On one side are Galois representations, which encode how prime numbers factor in number fields and how many solutions equations have modulo primes. On the other are automorphic forms, highly symmetric functions studied with harmonic analysis. Robert Langlands proposed this correspondence, and a mechanism called functoriality that organizes it, in a handwritten letter of January 1967.

This course follows his life from British Columbia to the Institute for Advanced Study and explains the main ideas: Eisenstein series, the L-group, functoriality, base change, the theorem with Jerrold Tunnell that gave Andrew Wiles his starting point, his work on percolation in statistical physics, and his late return to the geometric theory.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Many results are stated in simplified form; the references give the full hypotheses.

Timeline

British Columbia, 1936–1957

Robert Phelan Langlands was born in New Westminster, on the Fraser River east of Vancouver, on 6 October 1936. His parents were Robert Langlands and Kathleen Johanna Phelan; he was the eldest child, with two younger sisters. For several early years the family lived in a small community on the coast north of Vancouver, between Lang Bay and Myrtle Point, before returning to New Westminster, where he began school.

In 1946 the family moved to White Rock, a seaside town close to the border with Washington State. Langlands attended Semiahmoo High School and graduated in 1953. He had not planned on university. His English teacher, Crawford Vogler, told him that skipping it would betray his talents, and the advice stuck. In 2020 the school unveiled a mural in his honour.

He entered the University of British Columbia in 1953. He took his B.A. in 1957 and his M.Sc. in 1958. In 1956 he married Charlotte Lorraine Cheverie.

Yale and a thesis on semigroups, 1958–1960

Langlands went to Yale for doctoral study and finished in two years. His advisor was Cassius Ionescu Tulcea, a Romanian-born analyst. The thesis, Semi-groups and representations of Lie groups (1960), belongs to functional analysis. It studies how a representation of a Lie group on a Banach space gives rise to a semigroup of operators generated by elements of the Lie algebra, in the spirit of the Hille–Yosida theory of evolution equations.

The subject looks far from number theory. It gave him two lasting tools: the analytic theory of infinite-dimensional representations of Lie groups, and the habit of treating differential operators on a group as elements of an algebra. Both are basic to the theory of automorphic forms, where the Laplacian on the upper half-plane becomes the Casimir element of \(\mathrm{SL}_2(\mathbb R)\).

The combination of harmonic analysis on Lie groups with classical number theory became his signature.

Princeton and the letter, 1960–1967

In 1960 Langlands became an instructor at Princeton University, then lecturer (1961) and assistant and associate professor (1962). The senior analyst Salomon Bochner took an interest in him. Langlands described the pair as a timid young man and a timid old man. He spent 1962–63 as a member of the Institute for Advanced Study, where Atle Selberg and Harish-Chandra were working on the analytic theory of automorphic forms.

His first major results used Harish-Chandra’s representation theory. In 1963 he derived a dimension formula for spaces of automorphic forms on bounded symmetric domains. Selberg had sketched how Eisenstein series should give the continuous part of the spectrum on quotients of the upper half-plane; Langlands set out to treat arbitrary reductive groups. During 1964–65, as a Miller and Sloan Fellow at Berkeley, he worked out a general theory. The manuscript circulated for a decade before its publication in 1976. One quick application, published in 1966, proved Weil’s conjecture that the Tamagawa number of a split simply connected group equals 1.

The letter

Studying the constant terms of Eisenstein series, Langlands found products of new L-functions attached to representations of complex groups. By the end of 1966 he saw how these fit into a general pattern. In January 1967 he sent André Weil a handwritten letter of 17 pages, asking him to read it as pure speculation if he wished. The letter introduced what is now called the L-group, defined L-functions for automorphic representations of any reductive group, and conjectured that homomorphisms of L-groups transfer automorphic representations from one group to another. Weil had the letter typed, and copies circulated widely.

That spring Langlands gave the Yale lectures Euler Products, in which he proved the meromorphic continuation of some of the new L-functions using Eisenstein series. In 1967 he moved to Yale as a full professor.

Eisenstein series and the spectrum

For the general reader. On a sphere, every function is a sum of standard vibrations, the spherical harmonics. On a non-compact space such as the modular surface, some vibrations are not discrete notes but a continuous band. Eisenstein series are the functions that make up that band. Langlands showed how to build them for every reductive group, and found number-theoretic L-functions hidden inside them.

The case of \(\mathrm{SL}_2(\mathbb Z)\)

For \(z=x+iy\) in the upper half-plane \(\mathbb H\) and \(\operatorname{Re}s>1\), let

It is invariant under \(\mathrm{SL}_2(\mathbb Z)\) and is an eigenfunction of the hyperbolic Laplacian \(\Delta=-y^2(\partial_x^2+\partial_y^2)\) with eigenvalue \(s(1-s)\). Write \(\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)\) for the completed Riemann zeta function.

The space \(L^2(\mathrm{SL}_2(\mathbb Z)\backslash\mathbb H)\) then splits as constants, plus cusp forms (a discrete spectrum of Maass forms), plus the integrals \(\int_{\mathbb R} g(t)E(z,\tfrac12+it)\,dt\), a continuous spectrum.

Langlands’ general theory

Let \(G\) be a reductive group over \(\mathbb Q\) and \(P=MN\) a parabolic subgroup. From a cusp form \(\phi\) on \(M\) and a complex parameter \(\lambda\), one forms an Eisenstein series \(E(\phi,\lambda,g)=\sum_{\gamma\in P(\mathbb Q)\backslash G(\mathbb Q)}\phi_\lambda(\gamma g)\), which converges for \(\lambda\) in a cone.

The residual spectrum is the hard part. Langlands obtained it by shifting contours of integration and tracking residues of the Eisenstein series, a long argument that occupies much of the book and its appendices.

Constant terms and L-functions

For \(\mathrm{SL}_2\) the constant term contains \(\xi(2s-1)/\xi(2s)\). For a general maximal parabolic \(P=MN\) and a cuspidal automorphic representation \(\pi\) of \(M\), Langlands computed the constant term as a product over primes. He found ratios of the form

where \(r_j\) are the irreducible pieces of the adjoint action of the dual group of \(M\) on the dual of the Lie algebra of \(N\). Since the Eisenstein series continues meromorphically, so do these L-functions. This is the origin of the L-group (Lesson 05) and of what Freydoon Shahidi later developed as the Langlands–Shahidi method.

The L-group and Satake parameters

For the general reader. To each symmetry group used in number theory, Langlands attached a second group, its mirror image, built from the same data with the roles of two ingredients exchanged. Local information about an automorphic form at each prime becomes a single element of this mirror group. L-functions are then defined uniformly, one for each representation of the mirror group.

Dual groups

A split connected reductive group \(G\) is determined by its root datum \((X^*,\Phi,X_*,\Phi^\vee)\): the character lattice, the roots, the cocharacter lattice and the coroots. Exchanging the two halves gives a new root datum \((X_*,\Phi^\vee,X^*,\Phi)\), which determines a complex reductive group \(\hat G\), the dual group. For a non-split group, the L-group is the semidirect product \({}^LG=\hat G\rtimes\mathrm{Gal}(\bar F/F)\), or with the Weil group in place of the Galois group.

Unramified representations

Let \(p\) be a prime, \(G\) split over \(\mathbb Q_p\), \(K=G(\mathbb Z_p)\). An irreducible representation \(\pi_p\) of \(G(\mathbb Q_p)\) is unramified if it has a non-zero \(K\)-fixed vector. The Hecke algebra \(\mathcal H(G(\mathbb Q_p),K)\) of compactly supported \(K\)-bi-invariant functions is commutative and acts on that line by a character.

Langlands’ observation in the letter to Weil was that the right home for the Satake parameter is the complex group \(\hat G\), and that any finite-dimensional representation \(r\) of \({}^LG\) gives a local factor.

He proved that the product converges in a right half-plane and conjectured meromorphic continuation and a functional equation after completing it at the places in \(S\).

The case \(\mathrm{GL}_2\)

A holomorphic newform \(f=\sum a_nq^n\) of weight \(k\), level \(N\) and trivial character gives an automorphic representation \(\pi_f\) of \(\mathrm{GL}_2(\mathbb A)\). For \(p\nmid N\) its Satake parameter is \(\mathrm{diag}(\alpha_p,\beta_p)\) with

and \(L(s,\pi_f,\mathrm{std})\) is the classical \(L(s+\tfrac{k-1}{2},f)\).

Functoriality and reciprocity

For the general reader. If two mirror groups are related by a map, Langlands conjectured that automorphic forms on the original groups should be related as well. This principle, functoriality, predicts that L-functions of automorphic forms can be multiplied, squared and twisted in every way the mirror groups allow. Its simplest case, for the group of size one, is the class field theory of the early twentieth century.

Reciprocity

Take \(H=\{1\}\). Then \({}^LH\) is the Galois group (or Weil group) of \(F\), and an L-homomorphism to \({}^L\mathrm{GL}_n\) is essentially a continuous representation \(\sigma:\mathrm{Gal}(\bar F/F)\to\mathrm{GL}_n(\mathbb C)\). Functoriality then predicts:

For \(n=1\) this is class field theory. A character of \(\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)\) factors through \(\mathrm{Gal}(\mathbb Q(\zeta_m)/\mathbb Q)\cong(\mathbb Z/m)^\times\) by the Kronecker–Weber theorem, so it is a Dirichlet character, an automorphic form on \(\mathrm{GL}_1\). Since the 1970s the conjecture has been extended to \(\ell\)-adic Galois representations attached to motives, which should correspond to algebraic automorphic representations.

Examples of functoriality

• Symmetric powers. \(\mathrm{Sym}^n:\mathrm{GL}_2(\mathbb C)\to\mathrm{GL}_{n+1}(\mathbb C)\). The case \(n=2\) is due to Stephen Gelbart and Hervé Jacquet (1978); \(n=3,4\) to Henry Kim and Freydoon Shahidi (2002–03); all \(n\) for holomorphic modular forms to James Newton and Jack Thorne (2021).
• Base change. For an extension \(E/F\), the diagonal map \({}^L\mathrm{GL}_n{}_{/F}\to{}^L(\mathrm{Res}_{E/F}\mathrm{GL}_n)\) (Lesson 07).
• Automorphic induction and tensor products, such as \(\mathrm{GL}_2\times\mathrm{GL}_2\to\mathrm{GL}_4\) (Dinakar Ramakrishnan, 2000).
• Endoscopy. Inclusions \(\hat H\hookrightarrow\hat G\) of centralizers of semisimple elements, studied by Langlands, Diana Shelstad and Robert Kottwitz, and made usable by Ngô Bao Châu’s proof of the fundamental lemma (2008).

The symmetric power case had a famous consequence. Combined with the potential automorphy theorems of Richard Taylor and collaborators, it led to the proof of the Sato–Tate conjecture for elliptic curves over \(\mathbb Q\) without complex multiplication (2008–2011).

GL(2): Jacquet–Langlands and base change

For the general reader. Conjectures need test cases. In the decade after the letter, Langlands proved functoriality in several cases for the group of 2 × 2 matrices, the setting of classical modular forms. His main tool was the trace formula, which equates a sum over automorphic forms with a sum over geometric data such as conjugacy classes.

The trace formula

For a compact quotient \(\Gamma\backslash G\) and a test function \(f\), the operator \(R(f)\phi(x)=\int_Gf(y)\phi(xy)\,dy\) on \(L^2(\Gamma\backslash G)\) is trace class, and computing its trace in two ways gives Selberg’s trace formula:

The left side is spectral; the right side is a sum of orbital integrals over conjugacy classes. For non-compact quotients Eisenstein series contribute continuous terms, and James Arthur, Langlands’ student, spent decades making the general trace formula precise. Langlands’ strategy was to compare the geometric sides of trace formulas for two groups, match orbital integrals locally, and deduce an equality of spectral sides.

The proof in Automorphic Forms on GL(2) (1970) used the converse theorem and Hecke theory; the trace formula gave a second proof. The correspondence explains why modular forms appear in the arithmetic of quaternion algebras, for instance in counting representations by quaternary quadratic forms.

Base change

Hideo Saito and Takuro Shintani had treated base change for holomorphic forms over cyclic extensions of totally real fields. Langlands developed a twisted trace formula and proved the general result for \(\mathrm{GL}_2\).

Arthur and Laurent Clozel extended cyclic base change to \(\mathrm{GL}_n\) in 1989.

Langlands–Tunnell and Fermat

For the general reader. Artin conjectured in the 1920s that certain L-functions built from Galois groups have no poles. Base change let Langlands prove the conjecture for two-dimensional representations whose image looks like the symmetries of a tetrahedron, and Jerrold Tunnell extended it to the octahedron. Fifteen years later this result gave Andrew Wiles the first step in the proof of Fermat’s Last Theorem.

Projective images

Let \(\rho:\mathrm{Gal}(\bar F/F)\to\mathrm{GL}_2(\mathbb C)\) be continuous and irreducible. Its image is finite, and so is the image \(\bar G\) of the composition with \(\mathrm{GL}_2(\mathbb C)\to\mathrm{PGL}_2(\mathbb C)\). Up to conjugacy, the finite subgroups of \(\mathrm{PGL}_2(\mathbb C)\) are cyclic, dihedral, \(A_4\) (tetrahedral), \(S_4\) (octahedral) or \(A_5\) (icosahedral). Irreducibility excludes the cyclic case. In the dihedral case \(\rho\) is induced from a character of a quadratic extension, and Erich Hecke and Hans Maass had shown its L-function is automorphic.

The idea of the proof

Tetrahedral. \(A_4\) has a normal Klein four-group \(V_4\) with quotient \(\mathbb Z/3\). Let \(E/F\) be the cyclic cubic extension cut out by this quotient. Restricted to \(E\), \(\rho\) has projective image \(V_4\), which is dihedral, so \(\rho|_E\) is automorphic by Hecke theory. That automorphic representation \(\Pi\) is \(\mathrm{Gal}(E/F)\)-invariant, so by base change it descends to some \(\pi\) on \(\mathrm{GL}_2(\mathbb A_F)\). A careful comparison of local factors, using the three possible twists, picks out the \(\pi\) matching \(\rho\).

Octahedral. \(S_4\) maps onto \(\{\pm1\}\) with kernel \(A_4\). Over the quadratic extension cut out by the sign, \(\rho\) becomes tetrahedral, hence automorphic by Langlands over that field, and quadratic base change brings it back down. Matching the correct descent needed more: Tunnell also used the Gelbart–Jacquet lift to \(\mathrm{GL}_3\) and the non-normal cubic base change of Jacquet, Ilya Piatetski-Shapiro and Joseph Shalika. The icosahedral case is not solvable, and base change cannot reach it; over \(\mathbb Q\) odd icosahedral representations are now known to be modular through the work on Serre’s conjecture by Chandrashekhar Khare and Jean-Pierre Wintenberger (2009).

Use by Wiles

Let \(E\) be a semistable elliptic curve over \(\mathbb Q\) and \(\bar\rho_{E,3}:\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)\to\mathrm{GL}_2(\mathbb F_3)\) the action on its 3-torsion points. The group \(\mathrm{GL}_2(\mathbb F_3)\) is solvable, and it embeds in \(\mathrm{GL}_2(\mathbb Z[\sqrt{-2}])\) compatibly with reduction modulo the prime \((1+\sqrt{-2})\) above 3. This lifts \(\bar\rho_{E,3}\) to an odd complex representation with solvable image. Langlands–Tunnell gives a weight 1 form; multiplying by a weight 1 Eisenstein series congruent to 1 modulo 3 and applying the Deligne–Serre lemma gives a weight 2 eigenform congruent to \(E\) modulo 3. So \(\bar\rho_{E,3}\) is modular, and Wiles’s modularity lifting theorem then shows \(E\) itself is modular when \(\bar\rho_{E,3}\) is irreducible. The 3–5 switch handles the remaining case.

Percolation and conformal invariance

For the general reader. Colour each cell of a fine grid black or white by a fair coin toss. Does a black path cross a rectangle from left to right? In the 1980s Langlands turned from number theory to this question from statistical physics. With colleagues in Montreal he showed by careful computation that the crossing probability does not depend on the grid, and that it changes in a precise way when the rectangle is deformed by an angle-preserving map. These experiments helped launch one of the most active areas of modern probability.

Crossing probabilities

In bond percolation on \(\mathbb Z^2\) each edge is open with probability \(p\); the critical value is \(p_c=\tfrac12\) (Harry Kesten, 1980). For a rectangle of width \(a\) and height \(b\), scaled by a mesh \(\delta\to0\), let \(\pi_h(r)\) be the limiting probability at criticality that open edges connect the left side to the right side, with \(r=a/b\). Langlands, Philippe Pouliot and Yvan Saint-Aubin, with Claude Pichet, computed such probabilities numerically for many lattices and models with high precision.

Their 1994 survey in the Bulletin of the AMS tested conformal invariance, an idea credited there to Michael Aizenman, against extensive simulations. John Cardy used boundary conformal field theory to predict the exact formula. For a conformal rectangle mapped to the upper half-plane with marked points \(0,\eta,1,\infty\) (so that \(\eta\) is a cross-ratio),

Lennart Carleson observed that the formula becomes linear on an equilateral triangle \(ABC\): for \(X\) on side \(BC\), the probability of an open crossing from \(AC\) to the segment \(BX\) is \(|BX|/|BC|\).

Stanislav Smirnov’s proof, together with Oded Schramm’s SLE processes and the work of Gregory Lawler and Wendelin Werner, put the physics predictions on rigorous ground. Universality for other lattices, such as bond percolation on \(\mathbb Z^2\), remains open.

The geometric programme

For the general reader. André Weil liked to compare number fields with fields of functions on curves over finite fields, and these with Riemann surfaces. In the geometric Langlands programme the automorphic side becomes geometry on spaces of bundles over a curve. Others, beginning with Vladimir Drinfeld, built that theory. In his eighties Langlands returned to it and proposed an analytic version for complex curves, closer to his original harmonic analysis.

Function fields

Let \(X\) be a smooth projective curve over \(\mathbb F_q\), with function field \(K\). Automorphic forms for \(\mathrm{GL}_n\) over \(K\) unramified everywhere are functions on the set of isomorphism classes of rank \(n\) vector bundles on \(X\); Hecke operators modify a bundle at one point. Drinfeld proved the Langlands correspondence for \(\mathrm{GL}_2\) over function fields (1974–1983), Laurent Lafforgue for \(\mathrm{GL}_n\) (2002), and Vincent Lafforgue constructed the automorphic-to-Galois direction for general \(G\) (2018).

The geometric correspondence

Following Alexander Grothendieck’s dictionary between functions and sheaves, Drinfeld and Gérard Laumon reformulated the problem geometrically. For a curve \(X\) over \(\mathbb C\) and a reductive group \(G\):

Alexander Beilinson and Drinfeld constructed eigensheaves for opers using quantization of Hitchin’s integrable system. Edward Witten and Anton Kapustin connected the conjecture to electric–magnetic duality in four-dimensional gauge theory (2007). A proof of the categorical conjecture in the de Rham setting was announced in 2024 by Dennis Gaitsgory, Sam Raskin and their collaborators in a series of long papers.

Langlands’ analytic proposal

Langlands found the sheaf-theoretic version far from the analysis he cared about. His Abel lecture of 2018 was titled On the geometric theory. He then wrote a long paper in Russian, later translated by him into English as On the analytic form of the geometric theory of automorphic forms (posted 2020). It proposes studying genuine functions on \(\mathrm{Bun}_G\) for a complex curve, as eigenfunctions of Hecke operators defined by integration, with spectrum to be described by data from the dual group. Edward Frenkel showed the Hecke operators are well defined in the abelian case and identified difficulties in the non-abelian case. Pavel Etingof, Frenkel and David Kazhdan developed an analytic Langlands correspondence for complex curves based on these ideas, using differential operators on half-densities on \(\mathrm{Bun}_G\).

Ankara, Yale and the Institute

In 1967–68 Langlands spent a year at the Middle East Technical University, with an office next to the Turkish number theorist Cahit Arf. He learned Turkish, and decades later wrote a paper in it. From Ankara he sent letters to Jean-Pierre Serre on representations of \(p\)-adic groups. Ideas in them became the Deligne–Langlands conjecture on representations of Hecke algebras, proved by David Kazhdan and George Lusztig in 1987.

Back at Yale (1967–72), he wrote Problems in the theory of automorphic forms (1970), the first published account of functoriality, and with Hervé Jacquet the monograph Automorphic Forms on GL(2) (1970). His first doctoral students were James Arthur (1970) and Diana Shelstad (1974). In July 1972 he returned to Princeton as a permanent professor at the Institute for Advanced Study, later as Hermann Weyl Professor.

The 1970s and 1980s were years of building tools. Langlands studied Shimura varieties and their zeta functions, proposing in the 1979 Corvallis proceedings a description of their points modulo primes, later made precise with Michael Rapoport (1987). With Jean-Pierre Labesse he found L-indistinguishability for \(\mathrm{SL}_2\) (1979): automorphic representations of \(\mathrm{SL}_2\) come in packets with the same L-functions. This phenomenon, endoscopy, led to his work with Shelstad on transfer factors (1987) and to the fundamental lemma they formulated, proved by Ngô Bao Châu in 2008.

From the mid-1980s he also spent long periods on percolation and conformal invariance (Lesson 09), working with physicists and mathematicians at the Université de Montréal. Langlands became emeritus in 2007 and continued to work at the Institute, writing on the trace formula and, in Beyond endoscopy (2004), proposing a route to functoriality through the analysis of poles of L-functions in the trace formula.

He has spent much effort on making his work accessible. The Institute maintains an online archive of his papers, letters and lectures, many with his later comments. He reads Russian and wrote his late geometric paper in that language.

Honours, students and legacy

Langlands was elected a Fellow of the Royal Society of Canada in 1972 and of the Royal Society of London in 1981, and later to the National Academy of Sciences (1993) and the American Philosophical Society (2004). He gave an invited address at the International Congress of Mathematicians in Helsinki in 1978. His prizes include the Jeffery–Williams Prize (1980), the Cole Prize in Number Theory (1982), the first NAS Award in Mathematics (1988), the Wolf Prize shared with Andrew Wiles (1995–96), the Grande Médaille d’Or of the French Academy of Sciences (2000), the Steele Prize (2005), the Nemmers Prize (2006) and the Shaw Prize shared with Richard Taylor (2007).

In 2018 the Norwegian Academy of Science and Letters awarded him the Abel Prize for his visionary program connecting representation theory to number theory. In 2019 he was made a Companion of the Order of Canada. He holds more than thirty honorary doctorates.

Students

His doctoral students include James Arthur, who built the general trace formula and classified automorphic representations of classical groups; Diana Shelstad, whose work with him on endoscopy and transfer factors underlies the stabilization of the trace formula; and Thomas Hales, who worked on the fundamental lemma before proving the Kepler conjecture on sphere packing.

Legacy

• Number theory: the modularity theorem, Fermat’s Last Theorem, Sato–Tate and the proof of Serre’s modularity conjecture are all cases of reciprocity.
• Representation theory: the local Langlands correspondence for \(\mathrm{GL}_n\) (Michael Harris and Taylor; Guy Henniart, 2000–01), and the classification of representations by L-packets.
• Geometry and physics: the geometric Langlands programme, its links to gauge theory, and the conformal invariance of percolation.
• Prizes for the programme: Fields Medals to Drinfeld (1990), Laurent Lafforgue (2002), Ngô (2010) and Peter Scholze (2018) recognised work on its central problems.

Connections

People linked to Robert Langlands across this collection. A name without a link has no page here yet. See Robert Langlands in the network of all 337 lives →

Teachers and advisors

• Cassius Ionescu Tulcea 1960

Students

• Diana Shelstad 1974
• James Arthur 1970
• Thomas Hales

Influenced by

• André Weil 1967
• Atle Selberg
• Harish-Chandra

Influenced

• Andrew Wiles
• Vladimir Drinfeld
• Edward Frenkel
• Jerrold Tunnell 1981
• Ngô Bao Châu

Colleagues and collaborators

• Atle Selberg
• Jean-Pierre Serre 1967-1968
• Cahit Arf 1967-1968
• Hervé Jacquet 1970
• Jean-Pierre Labesse 1979
• Michael Rapoport 1987
• Philippe Pouliot 1992-1994
• Salomon Bochner 1960-1967
• Yvan Saint-Aubin 1992-1994

Also mentioned in this life

• Emil Artin
• Stefan Banach
• Pierre Deligne
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Pierre de Fermat
• Georg Frobenius
• Évariste Galois
• Alexander Grothendieck
• Erich Hecke
• Johannes Kepler
• Felix Klein
• Leopold Kronecker
• Sophus Lie
• Isaac Newton
• Siméon Denis Poisson
• Bernhard Riemann
• Hermann Weyl
• Edward Witten

Fields

• number theory 45 lives
• automorphic forms 4 lives
• representation theory 8 lives
• mathematical physics 45 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Robert Phelan Langlands, MacTutor History of Mathematics.
2. Robert Langlands, Wikipedia.
3. The Abel Prize, Robert P. Langlands, laureate 2018, with lectures and interview.
4. Institute for Advanced Study, Robert P. Langlands, faculty page.
5. The Work of Robert Langlands, IAS digital archive of papers, letters and comments.

Mathematics

1. R. P. Langlands, Letter to André Weil, January 1967, in the IAS archive above.
2. R. P. Langlands, Euler Products, Yale Mathematical Monographs, Yale University Press.
3. R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications III, Lecture Notes in Math. 170, Springer, 1970.
4. H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math. 114, Springer, 1970.
5. R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer, 1976.
6. R. P. Langlands, Base Change for GL(2), Annals of Mathematics Studies 96, Princeton University Press, 1980.
7. J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. AMS 5 (1981), 173–175.
8. R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987).
9. R. Langlands, P. Pouliot and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. AMS 30 (1994), 1–61.
10. R. P. Langlands, On the analytic form of the geometric theory of automorphic forms, English translation, 2020.
11. E. Frenkel, Is there an analytic theory of automorphic functions for complex algebraic curves?, arXiv:1812.08160.
12. S. Gelbart, An elementary introduction to the Langlands program, Bull. AMS 10 (1984), 177–219.
13. J. Arthur, The principle of functoriality, Bull. AMS 40 (2003), 39–53.
14. H. Duminil-Copin, Sixty years of percolation, arXiv:1712.04651.
15. E. Frenkel, Lectures on the Langlands program and conformal field theory, arXiv:hep-th/0512172.