Yuri I. Manin

Automorphic Pseudodifferential Operators

The theme of this paper is the correspondence between classical modular forms and pseudodifferential operators (ΨDO’s) which have some kind of automorphic behaviour. In the simplest case, this correspondence is as follows. Let Γ be a discrete subgroup of PSL 2(ℝ) acting on the complex upper half-plane H in the usual way, and f(z) a modular form of even weight k on Γ. Then there is a unique lifting from f to a Γ-invariant ΨDO with leading term f(z)∂-k/2, where ∂ is the differential operator \( \frac{d}{{dz}} \). This lifting and the fact that the product of two invariant ΨDO’s is again an invariant ΨDO imply a non-commutative multiplicative structure on the space of all modular forms whose components are scalar multiples of the so-called Rankin-Cohen brackets (canonical bilinear maps on the space of modular forms on Γ defined by certain bilinear combinations of derivatives; the definition will be recalled later). This was already discussed briefly in the earlier paper [Z], where it was given as one of several “raisons d’etre” for the Rankin-Cohen brackets.

The notion of dimension in geometry and algebra

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared. Glenn Gould disapproved of his own recording of Goldberg variations. “There is a lot of piano playing going on there, and I mean that as the most disparaging comment possible.” NYRB, Oct. 7, 2004, p. 10

Iterated integrals of modular forms and noncommutative modular symbols

The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper half-plane, possibly multiplied by z s−1, along geodesics connecting two cusps. This setting generalizes simultaneously the theory of modular symbols and that of multiple zeta values.

Theta Functions, Quantum Tori and Heisenberg Groups

A linear algebraic group G is represented by the linear space of its algebraic functions F(G) endowed with multiplication and comultiplication which turn it into a Hopf algebra. Supplying G with a Poisson structure, we get a quantized version Fq(G) which has the same linear structure and comultiplication, but deformed multiplication. This paper develops a similar theory for Abelian varieties. A description of Abelian varieties A in terms of linear algebra data was given by Mumford: F(G) is replaced by the graded ring of theta functions with symmetric automorphy factors, and comultiplication is replaced by the Mumford morphism M* acting on pairs of points as M(x, y) = (x + y, x − y). After supplementing this with a Poisson structure and replacing the classical theta functions by the quantized theta functions, introduced earlier by the author, we obtain a structure which essentially coincides with the classical one so far as comultiplication is concerned, but has a deformed multiplication which, moreover, becomes only partial. The classical graded ring is thus replaced by a linear category. Another important difference from the linear case is that Abelian varieties with different period groups (for multiplication) and different quantization parameters (for comultiplication) become interconnected after quantization.

Unfoldings of meromorphic connections and a construction of Frobenius manifolds

Let M be a complex manifold. A structure of a Frobenius manifold on M defined by B. Dubrovin consists of several pieces of data of which the most important are: (a) the choice of a flat structure on M represented by a subsheaf of flat vector fields T M f of the sheaf of holomorphic vector fields T M; (b) a commutative and associative O M-bilinear multiplication o on T M.

Renormalisation and computation ii: Time cut-off and the halting problem

This is the second instalment in the project initiated in Manin (2012). In the first Part, we argued that both the philosophy and technique of perturbative renormalisation in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, we address some of the issues raised in Manin (2012) and develop them further in three contexts: a categorification of the algorithmic computations; time cut-off and anytime algorithms; and, finally, a Hopf algebra renormalisation of the Halting Problem.

Quantum Theta Functions and Gabor Frames for Modulation Spaces

Representations of the celebrated Heisenberg commutation relations in quantum mechanics (and their exponentiated versions) form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we will try to bridge the two communities, represented by the two co-authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show, e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.

An update on semisimple quantum cohomology and F-manifolds

AbstractIn the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative

Moduli, Motives, Mirrors

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