## Håkan Hedenmalm's scientific interests and long term projects

• References are to items in the list of publications.

• ONGOING INTERESTS

(Quasi)conformal mapping. An analytic function which maps any two points to two separate points is said to be univalent. For such a univalent function (also known as a conformal mapping), defined on the open unit disk, which maps onto some region of the complex plane, we have the well-known classical pointwise estimates of Koebe-Bieberbach. It is, however, natural to ask for estimates in the mean. This led Nikolai Makarov to introduce the integral means spectral function. It is considered the main remaining theme in conformal mapping to find optimal estimates of this function. In the paper [49] with Shimorin, we develop a general method to tackle this problem. Another approach based on Marcinkiewicz-Zygmund integrals was explored in the work [58] with Baranov. More recently, quasiconformal mappings with small dilatation were studied in [74] and the related paper Bloch functions, asymptotic variance, and geometric zero packing (AJM, to appear). There, an unexpected connection with the mathematical physics of superconductivity by Nobel prize laureate Abrikosov was found. Also, a well-known conjecture by Astala was shown to be incorrect, but in the positive sense that Astala's question has an even more interesting answer than expected.

Random normal matrices. The eigenvalues of a random normal nXn matrix form a system of n points in the complex plane. The joint probability density is propertional to the modulus of all the distances times a localizing potential. This point process is known to be determinantal, governed by a so-called correlation kernel. The correlation kernel is essentially the reproducing kernel for a space of of polynomials of degree at most n-1 supplied with a Fock-type inner product. The asymptotic behavior of the correlation kernel then gives the basic behavior of the random point process. This direction is represented by papers [59], [61], [65], as well as by forthcoming work with Aron Wennman.

Polyanalytic determinantal processes. We define a point process as in the case of the random normal matrices, declared to be determinantal with correlation kernel associated with a space of polynomials of both the variable and its complex conjugate. This connects with the mathematical physics of so-called higher Landau levels. This direction is represented by papers [66], [70], as well as by forthcoming work with Antti Haimi and Aron Wennman.

PAST INTERESTS (could be resumed anytime)

Wiener-type tauberian theorems. My first scientific paper, [1], concerns Wiener's famous tauberian theorem [N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1--100], in the context of a weight; the weight has the property that the Fourier transform maps the associated weighted L^1 space to a space of holomorphic functions on a strip. The result generalizes earlier work of Korenblum (1958). Not much is known about similar matters in higher dimensions, at least not when analyticity appears. The paper [10] was an attempt to alter that: it partially solves a problem of B. Ya. Levin concerning L^1 functions on the quarter-plane. However, even in the one-variable case, the situation was not fully understood. With Borichev, the main remaining difficulty -- not having a sufficiently good grasp of the regularity of Fourier transforms of functions in weighted L^p spaces -- was resolved, and the result was a Wiener tauberian theorem for the half-line even for quasi-analytic weights (in the sense of Beurling) [24]. Later, with Ben Natan, Benyamini, and Weit [25, 29], the technique was extended to obtain the non-commutative Wiener tauberian theorem asked for by Ehrenpreis and Mautner in [Some properties of the Fourier transform on semi simple Lie groups, Ann. of Math. 61 (1955), 406--439].

Transfer themes. In the study of closed ideals in commutative Banach algebras, or, more generally, invariant subspaces in Banach spaces, it is of general interest to transfer information concerning the structure in one instance to that of another. For instance, if we have a space of holomorphic functions on an annulus, and associated spaces of holomorphic functions on the two disks obtained by filling in the hole at the origin or the hole at infinity, what are the relations between the invariant subspaces, where on the annulus we have biinvariant subspaces, and on the two disks forward and backward invariant ones, respectively? These issues were resolved in [2, 3, 5, 6, 7, 34, 39]. For example, in [5], it was shown that the closed ideals of H^\infty on the unit disk can be understood in terms of the closed ideals of the bigger uniform algebra H^\infty+C.

Several complex variables. In a series of papers, [8, 9, 89-1, 11, 13], I tried to study the closed ideals of well-known algebras of holomorphic functions of several complex variables. For instance, if f is a function in the bidisk algebra, which vanishes only at the point (1,1) in the closed bidisk, then f generates the closed ideal of all functions vanishing at (1,1) if and only if the functions f(·,1) and f(1,·), which belong to the disk algebra, are both outer. This generalizes a classical theorem of Carleman (1926) to the two-dimensional setting.

Bergman spaces. In 1991, I found a factorization theorem for the Bergman space of area-summable holomorphic functions in the unit disk space [15], which is analogous to the classical inner-outer factorization of Hardy space functions. The main difference is that the inner functions have such a complicated nonlinear structure in the Bergman space setting. Further results were obtained in [16, 17, 21, 95-3, 26, 33]. Inner functions are intimately connected with the general structure of invariant subspaces also in the Bergman space, as discovered by Aleman, Richter, Sundberg [Beurling's theorem for the Bergman space, Acta Math. 177 (1996), 275--310]. Spectral notions for the shift operator on quotient Bergman spaces were introduced in [20]. The mysterious appearance of invariant subspaces in the Bergman space of index two or more was explained in concrete terms in the papers [19, 28]. With Borichev, the well-known conjecture of Shapiro, Shields, and Korenblum -- claiming that invertible elements of the Bergman space are necessarily cyclic (weakly invertible) -- was disproved with rather intricate counterexamples [26, 32]. These were later extended with Borichev and Volberg to large Bergman spaces [45].

Biharmonic operators. The structure theorem of Aleman-Richter-Sundberg for the Bergman space invariant subspaces relies on properties of the biharmonic Green function; the most important one of them is that it is positive on the bidisk. The latter property is also used for the factorization theory. Questions concerning the relationship between inner functions and corresponding invariant subspaces automatically lead to the issue whether the Green function for the operator $\Delta\omega^{-1}\Delta$ is positive, provided that $\omega$ is a weight on the unit disk with $\log\omega$ subharmonic and such that the weight times area measure is reproducing for the origin on harmonic functions. The Green function for the bilaplacian models the clamped plate, and it also occurs in the theory of creeping flow. The introduction of the weight means that we alter the geometry of the disk: $\log\omega$ subharmonic means that the geometry is hyperbolic. My first paper on the matter was [22], and later a kind of concavity property of Green functions with respect to the weight was found in [30]. The above-mentioned problem for reproducing logarithmically subharmonic weights was finally solved in a joint paper with Jakobsson and Shimorin [35, 42]. The solution involves finding the Hele-Shaw flow analogue of the classical theorem of Hadamard (1898) which says that the exponential mapping from the tangent plane to the surface is a global diffeomorphism provided that the surface is hyperbolic, complete, and simply connected.

Riemannian geometry of surfaces. The paper with Shimorin [41] on Hele-Shaw flow on curved surfaces suggests that there are connections between complex analysis and Riemannian geometry that are yet to be uncovered. For instance, with my student Yolanda Perdomo, we considered the problem of finding a smooth surface with given curvature form having minimal area under a normalizing condition [46]. This differential geometry problem leads to deep complex analysis questions regarding the possible appearance of extraneous zeros in the Bergman kernel function.

Dirichlet series. With Lindqvist and Seip [31, 36], the Hilbert space of square summable Dirichlet series is studied with respect to the pointwise multipliers. This space is analogous to the classical Hardy space $H^2$ for Taylor series. With Gordon, a complete description of the composition operators was obtained [99-3]. The analogue of Carleson's convergence theorem for Fourier series is obtained for Dirichlet series with Saksman in [03-1]. The main motivation for studying Dirichlet series is their connection with analytic number theory.