Vladimir Drinfeld

How to use this course

Vladimir Drinfeld’s work runs through number theory, algebraic geometry, representation theory and mathematical physics, and his definitions have a way of becoming entire subjects. Drinfeld modules and shtukas made the Langlands programme over function fields a theorem rather than a conjecture. Quantum groups gave knot theory and integrable systems a common algebra. His associators connected that algebra to the absolute Galois group of \(\mathbb Q\). With Alexander Beilinson he turned the geometric Langlands programme into a precise mathematical project.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Several theorems are stated in simplified form; the references give the full hypotheses. Drinfeld is living, and the biographical lessons keep to his career and his published work.

Timeline

Kharkiv and the Olympiad, 1954–1969

Vladimir Gershonovich Drinfeld was born in Kharkiv, then in the Ukrainian Soviet Socialist Republic, on 14 February 1954. Kharkiv had a strong mathematical tradition. He attended Kharkiv Physics and Mathematics School no. 27, one of the specialist schools that trained gifted Soviet pupils in mathematics.

In 1969, aged fifteen, he represented the Soviet Union at the International Mathematical Olympiad in Bucharest and won a gold medal with full marks, 40 out of 40. While still a schoolboy he submitted his first paper, A cyclic inequality, in November 1969; it appeared in 1971 and concerned Shapiro’s cyclic inequality for sums of fractions. The same year he entered the Faculty of Mechanics and Mathematics at Moscow State University.

Moscow and Manin, 1969–1978

Moscow State University around 1970 had one of the strongest mathematical communities in the world, with seminars run by Israel Gelfand, Yuri Manin, Igor Shafarevich and others. Drinfeld became a student of Manin, whose seminar covered algebraic geometry, number theory and, increasingly, mathematical physics. Through Manin he learned the Langlands programme, then only a few years old.

As an undergraduate he found a new way into it. Robert Langlands had conjectured a correspondence between automorphic forms and Galois representations. Over function fields of curves over finite fields, Drinfeld invented objects he called elliptic modules, now Drinfeld modules, whose moduli spaces play the role of modular curves. His 1974 paper used them to establish the correspondence for \(\mathrm{GL}_2\) in a large class of cases. He graduated in 1974, aged twenty, and stayed in Moscow as a graduate student of Manin from 1974 to 1977.

In the same period Manin drew him into gauge theory. In 1977 Drinfeld and Manin, working in parallel with Michael Atiyah and Nigel Hitchin in Oxford, found a complete description of the instantons on the four-sphere; the four published it jointly in 1978 (Lesson 06). He defended his candidate thesis at Moscow State University in 1978. That year his paper on the Langlands conjecture for \(\mathrm{GL}(2)\) over function fields, outlining the method of what are now called shtukas, appeared in the proceedings of the International Congress in Helsinki.

Ufa and Kharkiv, 1978–1990

Soviet residence rules made it hard for someone without a Moscow residence permit to obtain a post in the capital. Drinfeld went instead to Bashkir State University in Ufa, in the southern Urals, where he taught mathematics. In 1981 he returned to Kharkiv to join the Department of Mathematical Physics of the B. Verkin Institute for Low Temperature Physics and Engineering of the Ukrainian Academy of Sciences.

The 1980s in Kharkiv were his most varied decade. He completed the Langlands correspondence for \(\mathrm{GL}_2\) over function fields and in 1983 gave a geometric construction of the automorphic side that started the geometric Langlands programme. With Serge Vlăduţ he proved a sharp bound for the number of points on curves over finite fields (1983). With Vladimir Sokolov he gave the Lie-algebraic theory of KdV-type hierarchies (1985). From the quantum inverse scattering method of the Leningrad school of Ludwig Faddeev he extracted the algebraic structure he named quantum groups, set out in his 1986 address to the International Congress in Berkeley. At the end of the decade came quasi-Hopf algebras and the Grothendieck–Teichmüller group.

He received the Doctor of Sciences degree from the Steklov Institute in Moscow in 1988. On 21 August 1990, at the International Congress in Kyoto, he was awarded a Fields Medal for his work on quantum groups and on number theory. Manin gave the laudation.

Drinfeld modules

For the general reader. Elliptic curves are central to number theory because their points form a group with rich symmetry. Over the rational numbers they lead to modular forms. Drinfeld found the right analogue for number systems built from polynomials over a finite field. His “elliptic modules” are simpler than elliptic curves, being defined by a single polynomial operation, and they unlock the same kind of arithmetic.

Definitions

Let \(\mathbb F_q\) be a finite field, \(A=\mathbb F_q[T]\), and \(K=\mathbb F_q(T)\). (More generally \(A\) is the ring of functions on a smooth projective curve over \(\mathbb F_q\) regular away from a fixed point \(\infty\).) Let \(L\) be a field containing \(K\), and let \(\tau\) be the Frobenius \(x\mapsto x^q\). The twisted polynomial ring \(L\{\tau\}\) consists of \(\sum a_i\tau^i\) with \(\tau a=a^q\tau\); it is the ring of \(\mathbb F_q\)-linear polynomials \(\sum a_ix^{q^i}\) under composition.

For \(A=\mathbb F_q[T]\), \(\phi\) is determined by \(\phi_T=T+g_1\tau+\cdots+g_r\tau^r\) with \(g_r\ne0\). The Carlitz module \(C_T=T+\tau\), introduced by Leonard Carlitz in 1935, is the rank-1 case. Its torsion points play the role of roots of unity: for \(M\in A\) the roots of \(C_M(x)=0\) form an \(A\)-module isomorphic to \(A/M\), and \(K(C[M])/K\) is abelian with Galois group \((A/M)^\times\), the function-field analogue of cyclotomic fields. Rank-1 Drinfeld modules give explicit class field theory; rank-2 modules are the analogue of elliptic curves.

Lattices and moduli

Let \(K_\infty=\mathbb F_q((1/T))\) and let \(\mathbb C_\infty\) be the completion of its algebraic closure.

For \(r=2\) the moduli spaces are curves, the Drinfeld modular curves, and \(\Omega^2=\mathbb C_\infty\setminus K_\infty\) replaces the complex upper half-plane. Drinfeld studied the \(\ell\)-adic cohomology of these curves, as Eichler, Shimura and Deligne had done for modular curves, and found inside it the Galois representations attached to automorphic representations of \(\mathrm{GL}_2\) over \(K\) whose component at \(\infty\) is special (Steinberg). This was the first half of the Langlands correspondence for \(\mathrm{GL}_2\) over function fields.

Shtukas and Langlands for GL(2)

For the general reader. Drinfeld modules gave half of the correspondence. To get all of it, Drinfeld introduced a more flexible object, the shtuka: a vector bundle on a curve together with a way of comparing it with its own Frobenius twist. Families of shtukas are the function-field replacement for Shimura varieties, and they are still the main tool in the subject. Laurent Lafforgue used them for \(\mathrm{GL}_n\), and Vincent Lafforgue for all reductive groups.

Let \(X\) be a smooth projective geometrically connected curve over \(\mathbb F_q\), \(F\) its function field, \(\mathbb A_F\) its adèles, and \(\ell\ne p\) a prime.

The direction from \(\pi\) to \(\sigma\) is realized geometrically, in the \(\ell\)-adic cohomology of moduli spaces; the converse uses the converse theorem of Hervé Jacquet and Langlands together with Grothendieck’s functional equation for L-functions. Drinfeld modules give moduli spaces only for \(\pi\) special at a chosen place \(\infty\); to remove this condition he defined:

A Drinfeld module of rank \(r\) gives a shtuka with pole at \(\infty\). The moduli stack of shtukas maps to \(X\times X\) by \((z,p)\), and Drinfeld computed the \(\ell\)-adic cohomology of its generic fibre with the Arthur–Selberg trace formula and the Lefschetz trace formula, finding the tensor products \(\sigma_\pi\otimes\sigma_\pi^\vee\) for all cuspidal \(\pi\).

Hecke operators as modifications

By André Weil’s adelic description of vector bundles, unramified automorphic forms for \(\mathrm{GL}_n\) over \(F\) are functions on the set \(\mathrm{Bun}_n(\mathbb F_q)\) of isomorphism classes of rank-\(n\) bundles on \(X\). For a closed point \(x\), the Hecke operator is \((T_xf)(\mathcal E)=\sum f(\mathcal E')\) over subsheaves \(\mathcal E'\subset\mathcal E\) with \(\mathcal E/\mathcal E'\) isomorphic to the skyscraper \(k_x\). Drinfeld’s 1983 idea was to replace functions by sheaves on the stack \(\mathrm{Bun}_n\), the starting point of Lesson 09.

Instantons: the ADHM construction

For the general reader. Instantons are solutions of the equations of Yang–Mills gauge theory, the theory behind the strong and weak nuclear forces, in four-dimensional Euclidean space. Physicists wanted all of them. In 1977 two teams, Atiyah and Hitchin in Oxford and Drinfeld and Manin in Moscow, reduced the problem to linear algebra: every instanton comes from a few matrices satisfying quadratic equations.

An \(\mathrm{SU}(n)\) connection \(A\) on \(\mathbb R^4\) is an instanton if its curvature is anti-self-dual, \(F_A=-{\star}F_A\), and has finite action \(\int|F_A|^2\). By Karen Uhlenbeck’s removable singularities theorem it extends to \(S^4\), and its charge \(k=\frac1{8\pi^2}\int\operatorname{tr}F_A\wedge F_A\) is a non-negative integer.

The proof goes through Roger Penrose’s twistor space \(\mathbb{CP}^3\to S^4\). By Richard Ward’s correspondence, as developed by Atiyah and Ward, instantons are holomorphic vector bundles on \(\mathbb{CP}^3\) that are trivial on the real twistor lines and carry a real structure. Results of Wolf Barth and Geoffrey Horrocks show that such bundles are the cohomology of monads, complexes \(\mathcal O(-1)^k\to\mathcal O^{2k+n}\to\mathcal O(1)^k\) of linear maps. Writing out the monad and the real structure yields the ADHM data. The construction is now part of the hyperkähler quotient picture: the two equations are the three components of a hyperkähler moment map.

Quantum groups

For the general reader. In the early 1980s physicists in Leningrad solved many one-dimensional quantum models with a method built on the Yang–Baxter equation. Drinfeld recognized the algebra underneath: deformations of the symmetry algebras of Lie groups, with a twisted way of combining representations. He named them quantum groups. They explain why the solutions exist, produce invariants of knots, and have become a field of their own.

Hopf algebras and R-matrices

A Hopf algebra \(H\) has a multiplication and a coproduct \(\Delta:H\to H\otimes H\), with a counit and an antipode, satisfying compatibility axioms. The coproduct lets \(H\) act on tensor products of representations. For the enveloping algebra \(U(\mathfrak g)\), \(\Delta(x)=x\otimes1+1\otimes x\) is symmetric, so \(V\otimes W\cong W\otimes V\) by the flip. Drinfeld’s key definition relaxes this.

For \(\mathfrak{sl}_2\), with \(q=e^{h}\), \(U_q(\mathfrak{sl}_2)\) has generators \(E,F,K^{\pm1}\) and relations

with \(\Delta(K)=K\otimes K\), \(\Delta(E)=E\otimes K+1\otimes E\), \(\Delta(F)=F\otimes1+K^{-1}\otimes F\). On the two-dimensional representation \(V=\mathbb C^2\) with basis \(v_1,v_2\), the R-matrix is

where \(E_{ij}\) are matrix units. It satisfies the Yang–Baxter equation, and \(\check R=PR\) satisfies the braid relation on \(V^{\otimes3}\). Nicolai Reshetikhin and Vladimir Turaev (1990) used such R-matrices to build invariants of knots and 3-manifolds; for \(\mathfrak{sl}_2\) and \(V=\mathbb C^2\) one recovers the Jones polynomial.

Associators and Grothendieck–Teichmüller

For the general reader. Tensor products of representations can be bracketed in different ways, \((U\otimes V)\otimes W\) or \(U\otimes(V\otimes W)\). Drinfeld allowed the identification between them to be non-trivial and found that a single universal series, built from the values of Riemann’s zeta function, does the job for every Lie algebra. The symmetries of such series form a group that contains the Galois group of all algebraic numbers. The discovery joined quantum algebra to arithmetic.

A quasi-Hopf algebra (Drinfeld, 1989) is an algebra with a coproduct that is coassociative only up to conjugation by an invertible \(\Phi\in H^{\otimes3}\), the associator, which satisfies the pentagon equation. With an R-matrix satisfying two hexagon equations, its representations form a braided monoidal category.

The KZ associator

Let \(A,B\) be non-commuting variables. Consider the Knizhnik–Zamolodchikov equation on \((0,1)\),

with values in formal power series in \(A,B\). Let \(G_0\) and \(G_1\) be the solutions with \(G_0(z)\sim z^{A}\) as \(z\to0\) and \(G_1(z)\sim(1-z)^{B}\) as \(z\to1\). Then \(\Phi_{\mathrm{KZ}}(A,B)=G_1^{-1}G_0\) is a constant series.

The homomorphism comes from the action of the Galois group on the profinite fundamental group of \(\mathbb P^1\setminus\{0,1,\infty\}\). It is injective by Gennadii Belyi’s theorem that every curve over \(\overline{\mathbb Q}\) is a covering of \(\mathbb P^1\) ramified only over \(0,1,\infty\). This realizes a programme sketched by Alexander Grothendieck in his Esquisse d’un programme. Whether the map is an isomorphism is still open. Associators were later used by Dror Bar-Natan for knot invariants and by Pavel Etingof and David Kazhdan to quantize every Lie bialgebra (1996).

Opers and geometric Langlands

For the general reader. In 1983 Drinfeld showed that the functions in the Langlands correspondence over function fields come from geometric objects, sheaves on the space of all vector bundles on a curve. The idea makes sense over the complex numbers as well, where it becomes a statement about differential equations. With Alexander Beilinson he found that the differential equations on one side are special connections called opers, and constructed the corresponding sheaves using quantum integrable systems.

Automorphic sheaves

Grothendieck’s function–sheaf dictionary turns an \(\ell\)-adic sheaf on a variety over \(\mathbb F_q\) into a function on its rational points, by taking traces of Frobenius. Drinfeld’s 1983 paper started from a 2-dimensional \(\ell\)-adic local system \(E\) on \(X\) and built, through a geometric version of the Whittaker model, a perverse sheaf \(\mathcal A_E\) on the moduli stack of rank-2 bundles whose function is the automorphic form attached to \(E\), and which is an eigensheaf for geometric Hecke operators. Gérard Laumon extended the construction conjecturally to \(\mathrm{GL}_n\), and Edward Frenkel, Dennis Gaitsgory and Kari Vilonen proved it for \(\mathrm{GL}_n\) (2002).

Opers

Over \(\mathbb C\), local systems become flat connections. Let \(X\) be a compact Riemann surface. An \(\mathfrak{sl}_2\)-oper on \(X\) is a second-order operator \(\partial_z^2-q(z)\) acting from \(K^{-1/2}\) to \(K^{3/2}\), with \(K\) the canonical bundle; equivalently a projective structure on \(X\). Under a change of coordinate \(z=f(w)\), the potential transforms as

where \(\{f,w\}\) is the Schwarzian derivative. Opers for a general group \(G\) are \(G\)-connections with a reduction to a Borel subgroup in special position. Drinfeld and Vladimir Sokolov (1985) found them in the theory of integrable equations: Hamiltonian reduction from a loop algebra produces the space of operators \(\partial^n+u_{n-2}\partial^{n-2}+\cdots+u_0\) and the generalized KdV hierarchies.

The proof quantizes Nigel Hitchin’s integrable system on \(T^*\mathrm{Bun}_G\) and uses the theorem of Boris Feigin and Frenkel identifying the centre of the affine Kac–Moody algebra at the critical level with functions on opers on the disc. This gave the first construction of Hecke eigensheaves for general \(G\), on the oper locus. A proof of the full categorical conjecture in the de Rham setting was announced in 2024 by Gaitsgory, Sam Raskin and their collaborators in a series of long papers.

Points on curves and chiral algebras

For the general reader. How many solutions can a polynomial equation in two variables have over a finite field? André Weil’s bound depends on the genus of the curve. Drinfeld and Vlăduţ showed that for curves of large genus the true maximum is much smaller, and their bound is exactly attained. The result matters for error-correcting codes. The lesson ends with Drinfeld’s long collaboration with Beilinson on chiral algebras, the algebra of two-dimensional conformal field theory.

For a smooth projective curve \(X\) of genus \(g\) over \(\mathbb F_q\), Weil’s theorem gives \(\#X(\mathbb F_{q^m})=q^m+1-\sum_{j=1}^{2g}\alpha_j^m\) with \(|\alpha_j|=\sqrt q\), hence \(|\#X(\mathbb F_q)-q-1|\le2g\sqrt q\). Let \(N_q(g)\) be the maximum of \(\#X(\mathbb F_q)\) over curves of genus \(g\), and \(A(q)=\limsup_{g\to\infty}N_q(g)/g\). Weil’s bound gives \(A(q)\le2\sqrt q\); Yasutaka Ihara (1981) improved this to \(A(q)\le\tfrac12(\sqrt{8q+1}-1)\).

Proof sketch. Write \(\alpha_j=\sqrt q\,e^{i\theta_j}\) and \(N_m=\#X(\mathbb F_{q^m})\), so \(N_m\ge N_1\). Since the \(\alpha_j\) come in conjugate pairs, \(\sum_{j=1}^{2g}\cos m\theta_j=q^{-m/2}(q^m+1-N_m)\). Take \(c_m\ge0\) such that \(f(\theta)=1+2\sum_{m=1}^{M}c_m\cos m\theta\ge0\). Summing \(f(\theta_j)\ge0\) over \(j\) and using \(N_m\ge N_1\) gives

Divide by \(g\) and let \(g\to\infty\): \(A(q)\le\big(\sum_mc_mq^{-m/2}\big)^{-1}\). The Fejér choice \(c_m=1-m/M\) makes \(f\ge0\), and as \(M\to\infty\), \(\sum_mc_mq^{-m/2}\to\sum_{m\ge1}q^{-m/2}=\frac1{\sqrt q-1}\).

Chiral algebras

Vertex algebras, introduced by Richard Borcherds in 1986, encode the operator product expansions of two-dimensional conformal field theory, but their definition depends on a coordinate. Beilinson and Drinfeld gave a coordinate-free version: a chiral algebra on a curve \(X\) is a D-module \(\mathcal A\) with a chiral bracket \(j_*j^*(\mathcal A\boxtimes\mathcal A)\to\Delta_*\mathcal A\), where \(j\) is the inclusion of the complement of the diagonal in \(X\times X\). Their book Chiral Algebras (2004) develops the theory together with factorization algebras and the chiral homology of curves. The same circle of ideas produced the Beilinson–Drinfeld Grassmannian, a family of affine Grassmannians over powers of \(X\) that is basic to the geometric Satake equivalence and to Vincent Lafforgue’s work.

Chicago, from 1999

After the dissolution of the Soviet Union, Drinfeld remained at the Verkin Institute in Kharkiv for most of the 1990s. In 1992 he was elected to the National Academy of Sciences of Ukraine. In 1998 he moved to the United States. The University of Chicago appointed him in December 1998, and he has taught there since 1999. In 2001 he became Harry Pratt Judson Distinguished Service Professor of Mathematics. At Chicago he and Alexander Beilinson completed Chiral Algebras.

His later papers range widely: DG quotients of DG categories (2004), infinite-dimensional vector bundles in algebraic geometry, algebraic spaces with an action of \(\mathbb G_m\), finiteness questions for algebraic stacks and compact generation of D-modules on \(\mathrm{Bun}_G\) with Dennis Gaitsgory, and, from 2020, the prismatization of \(p\)-adic formal schemes, a stack-theoretic approach to the prismatic cohomology of Bhargav Bhatt and Peter Scholze developed in parallel with Bhatt and Jacob Lurie.

At Chicago he has supervised many doctoral students, among them Mitya Boyarchenko, Masoud Kamgarpour, Tanmay Deshpande and Jonathan Wang. Earlier students include Alexander Stolin, Eugene Karolinsky, Sergey Lysenko and Dmitry Arinkin.

Honours and influence

Drinfeld received the Fields Medal in 1990 for his work on quantum groups and on number theory. He was elected to the National Academy of Sciences of Ukraine in 1992, the American Academy of Arts and Sciences in 2008 and the US National Academy of Sciences in 2016. In 2018 he shared the Wolf Prize in Mathematics with Alexander Beilinson for their work in algebraic geometry, mathematical physics and representation theory. In 2022 he became an Honorary Member of the London Mathematical Society. In 2023 he shared the Shaw Prize in Mathematical Sciences with Shing-Tung Yau, for contributions to mathematical physics and arithmetic geometry.

Influence

• Function-field Langlands: shtukas led to Laurent Lafforgue’s proof for \(\mathrm{GL}_n\) (Fields Medal 2002) and to Vincent Lafforgue’s construction of Langlands parameters for every reductive group (2018). Drinfeld modules are a standard tool in function-field arithmetic.
• Geometric Langlands: automorphic sheaves, opers and the Beilinson–Drinfeld Grassmannian are its basic objects; they are the setting of the proof of the categorical conjecture announced in 2024.
• Quantum algebra: quantum groups, Yangians, the quantum double and quasi-Hopf algebras underlie knot invariants, tensor categories, integrable systems and conformal field theory. Drinfeld’s name is attached to the Drinfeld double, the Drinfeld twist, the Drinfeld centre of a monoidal category and the Drinfeld–Jimbo quantum groups.
• Arithmetic and topology: associators and the Grothendieck–Teichmüller group connect multiple zeta values, the Galois group of \(\mathbb Q\), Kontsevich’s formality and the theory of finite-type knot invariants.
• Gauge theory and coding: the ADHM construction is the model for moduli of instantons on many four-manifolds, and the Drinfeld–Vlăduţ bound sets the limit for algebraic-geometry codes from curves.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Vladimir Gershonovich Drinfeld, MacTutor History of Mathematics.
2. Vladimir Drinfeld, Wikipedia.
3. B. Verkin Institute for Low Temperature Physics and Engineering, Vladimir G. Drinfeld, staff page.
4. University of Chicago News, UChicago mathematician Vladimir Drinfeld wins prestigious Shaw Prize, 2023.
5. Wolf Foundation, Vladimir Drinfeld, Wolf Prize in Mathematics 2018.
6. Mathematics Genealogy Project, Vladimir Gershonovich Drinfeld.
7. Yu. I. Manin, On the mathematical work of Vladimir Drinfeld, Proceedings of the ICM, Kyoto 1990.

Mathematics

1. V. G. Drinfeld, Elliptic modules, Mat. Sbornik 94 (1974); English translation in Math. USSR Sbornik 23 (1974).
2. V. G. Drinfeld, Langlands’ conjecture for GL(2) over functional fields, Proceedings of the ICM, Helsinki 1978.
3. M. F. Atiyah, V. G. Drinfeld, N. J. Hitchin and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978).
4. V. G. Drinfeld, Two-dimensional \(\ell\)-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2), Amer. J. Math. 105 (1983).
5. V. G. Drinfeld and S. G. Vlăduţ, Number of points of an algebraic curve, Funct. Anal. Appl. 17 (1983).
6. V. G. Drinfeld and V. V. Sokolov, Lie algebras and equations of Korteweg–de Vries type, J. Soviet Math. 30 (1985).
7. V. G. Drinfeld, Hopf algebras and the quantum Yang–Baxter equation, Soviet Math. Dokl. 32 (1985).
8. V. G. Drinfeld, Quantum groups, Proceedings of the ICM, Berkeley 1986, vol. 1, 798–820.
9. V. G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990).
10. V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(\(\overline{\mathbb Q}/\mathbb Q\)), Leningrad Math. J. 2 (1991).
11. A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint, 1991–1997.
12. A. Beilinson and V. Drinfeld, Chiral Algebras, AMS Colloquium Publications 51, 2004.
13. M. Jimbo, A q-difference analogue of U(g) and the Yang–Baxter equation, Lett. Math. Phys. 10 (1985).
14. C. Kassel, Quantum Groups, Springer GTM 155, 1995.
15. D. Goss, Basic Structures of Function Field Arithmetic, Springer, 1996.
16. E. Frenkel, Lectures on the Langlands program and conformal field theory, arXiv:hep-th/0512172.