abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
What is called the Langlands correspondence in number theory (Langlands 67) is a conjectural correspondence (a bijection subject to various conditions) between
$n$-dimensional representations of the Galois group $Gal(\bar F/F)$ of a given number field $F$, and
certain representations – called automorphic representations – of the $n$-dimensional general linear group $GL_n(\mathbb{A}_F)$ with coefficients in the ring of adeles of $F$, arising within the representations given by functions on the double coset space $GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O})$ (where $\mathcal{O} = \prod_v \mathcal{O}_p$ is the ring of integers of all formal completions of $F$).
This conjecture is motivated from the following special case, which is fully understood:
For $n = 1$ then an $n$-dimensional representation of the Galois group factors through $GL_1$ and hence through an abelian group. Therefore, by adjunction, it is equivalently a representation of the abelianization of the Galois group. The Kronecker-Weber theorem says that for $F = \mathbb{Q}$, then the abelianized Galois group is the idele class group $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})$, and hence 1-dimensional representations of the Galois group are equivalently representations of this. Moreover, one finds that for any prime number $p$, then those representations which are “unramified at $p$” are invariant under the subring of p-adic integers, hence are representations of the double quotient group $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})/GL_1(\mathbb{Z}_p)$. More generally, the Artin reciprocity law says that for number fields there is an isomorphism between $GL_1(K) \backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O}_K)$ and the abelianized Galois group.
Various versions and refinements of this conjecture have since been considered, for some perspective see (Taylor 02, Langlands 14, Harris 14).
In particular, interpretation of the above story dually in arithmetic geometry has led to some developments. Namely under the function field analogy we have that
the Galois group is essentially the fundamental group of an algebraic curve;
hence a representation of it is a principal/vector bundle with flat connection on this curves;
while that curious double quotient $GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O}_F)$ is just the canonical Cech-realization of the moduli stack of bundles (see there for details) on that curve.
From this arithmetic geometry point of view the Langlands conjecture seems to speak of a correspondence that sends Dirac distributions on the moduli space of flat connections over an algebraic curve to certain “automorphic” functions on the moduli stack of bundles on the same curve. This suggests that the Langlands correspondence should be understood as a nonabelian version of a Fourier-Mukai-type integral transform. This version of the conjecture is known as the geometric Langlands correspondence. See there for more details.
The original conjecture is due to
Surveys of the state of the program include
Richard Taylor, Galois Representations, Proceedings of the ICM 2002 (long version pdf).
Robert Langlands, Problems in the theory of automorphic forms -- 45 years later, Oxford 2014
Michael Harris, Automorphic Galois representations and Shimura varieties, Proceedings of the ICM 2014 (pdf).
Introductions and expository surveys include
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219; Edward Frenkel, Commentary on “An elementary introduction to the Langlands Program” by Steven Gelbart, Bull. Amer. Math. Soc. 48 (2011), 513-515, abstract, pdf
Mark Goresky, Langlands’ conjectures for physicists (pdf)
Discussion with an eye towards geometric class field theory and geometric Langlands duality is in
Peter Toth, Geometric abelian class field theory, 2011 (web)
Edward Frenkel, Gauge Theory and Langlands Duality (arXiv:0906.2747)
More resources are at
the digital archive of Robert Langlands’ articles at sunsite.ubc.ca, not currently being maintained, see:
The Work of Robert Langlands at the Institute of Advanced Study.