Title: The 8th Holon Workshop in Complex Analysis Place: H.I.T. - Holon Institute of Technology, Auditorium (room 400), Science Faculty, Bldg. 8 Date: March 25, 2010 09:40 Registration 10:00 ^Ö 10:10 Welcome 10:10 ^Ö 10:50 Alexander Olevskii (Tel Aviv University) Wiener's "Closer of Translates" Problem and Piatetskii-Shapiro Uniqueness Phenomenon Abstract. Wiener characterized cyclic vectors (with respect to translations) in $l^p(Z)$ and $L^p(R)$ ($p=1,2$) in terms of zero sets of Fourier transform. He conjectured that a similar characterization should be true for $1 < p < 2$. I will discuss this conjecture. Joint work with Nir Lev. 11:00 ^Ö 11:40 Yosef Yomdin (Weizmann Institute) Invisible Sets for Integral Measurements and Moment Vanishing Problem Abstract. The problem of reconstruction of semi-algebraic sets and functions from integral measurements, like moments or Fourier transform, naturally arises in Signal Processing. For certain (incomplete) measurements there are "invisible" signals. Their description leads to the moment vanishing problem: give conditions for identical vanishing of the moments $m_k = \int P^k(x)q(x)dx$, for various classes of $P$ and $q$, and various integration domains. Recently a serious progress has been achieved in some special cases of this problem, and relations have been found with the Mathieu conjecture in representations of compact Lie groups, and (through the recent work of Wenhua Zhao) with certain questions around the Jacobian conjecture. **Coffee Break** 12:20 ^Ö 13:00 Dan Mangoubi (Hebrew University) Geometry of Eigenfunctions and Harmonic Functions Abstract. Let $(M, g)$ be a compact Riemannian manifold of dimension $n$. Let $L$ be the Laplace-Beltrami operator on $M$, and let $u$ be an eigenfunction: $Lu=\lambda u$. We study the geometry of nodal sets and the nodal domains as $\lambda$ becomes arbitrary large. In particular, we address the following problem: Faber-Krahn inequality shows that one has a lower bound on the volume of each nodal domain $A_\lambda$: $$ {\rm Vol}(A_\lambda) > C/\lambda^{n/2}.$$ Can one find a ball inside $A_\lambda$ of radius $C_\varepsilon/\lambda^{1/2+\varepsilon}$ inside $A_\lambda$? The answer is positive in dimension two and open in dimension $n\ge 3$. We will show that in order to obtain bounds in dimension $n\ge 3$, one is naturally lead to consider harmonic functions f in the unit ball of $\mathbb R^n$, with $f(0)=0$, and to look for lower bounds on the volume of $f>0$. I will describe what is known about these bounds. 13:10 ^Ö 13:50 Mark Agranovsky (Bar-Ilan University) Boundary Forelli Theorem and Meromorphic Extension from Circles Abstract. Characterization of holomorphic functions, of one and several complex variables, and their boundary values, in integral-geometric spirit (via moment conditions on varieties of curves) attract attention of complex analysts since '70s. However, many natural and simply formulated questions are still not answered. I will present a recent result on characterization of boundary values of holomorphic functions in the complex unit ball, in terms of one-dimensional holomorphic extension along bundles of complex lines. This result is a boundary analog of celebrated Forelli theorem about holomorphicity on slices, which in turn is a variation of the clasical Hartogs' theorem about separate analyticity. The proof of boundary Forelli theorem is based on reduction to characterization of polyanalytic functions in planar domains in terms of meromorphic extendibility from chains of circles. 13:50 Closing Organizing Committee: Anatoly Golberg (H.I.T.), Eduard Yakubov (H.I.T.), Lawrence Zalcman (BIU) --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Gershon Wolansky Announcement from: Anatoly Golberg