Program

Complex Analysis and Dynamical Systems,

a workshop in honor of the retirement of Professor

David Shoikhet

November 10, 2022

11:30–16:00

VIP room, Braude College of Engineering

•11:30-11:45: Gathering (Coﬀee in VIP)

•11:45-12:00: Opening

Prof. Arie Maharshak, President

Prof. Sarit Sivan, Vice President

Prof. Aviv Gibali, Head of the Mathematics Department

•12:00-12:45: Prof. S. Reich, Technion – Israel Institute of

Technology

•12:45-13:30: Prof. M. Agranovsky, Bar–Ilan University

•13:30-14:15: Lunch

•14:45-15:00: Prof. B. Solomyak, Bar–Ilan University

•15:00-15:45: Prof. M. Elin, Braude College of Engineering

•15:45-16:00: Closing

Abstracts

On some problems of geometric tomography - after Newton’s

Lemma about Ovals

Mark Agranovsky

Bar–Ilan University

e-mail: agranovs@math.biu.ac.il

In 1687, I. Newton proved, while exploring Kepler’s law in celestial me-

chanics, that no convex smoothly bounded domain in the plane is alge-

braically integrable. This means that there is no algebraic equation relating

the areas of the two portions of the domain on both sides of a chord and the

parameters determining the chord . In 1987, V. Arnold suggested studying

the phenomenon of algebraic (or, the opposite, transcendental) integrability

of geometric bodies in higher dimensions. In my talk, I will touch brieﬂy on

the history of the subject and discuss some recent related results.

Filtration of semigroup generators

Mark Elin

Braude College of Engineering

e-mail: mark elin@braude.ac.il

In this talk we present results based on joint works [1, 2, 3] and de-

voted to parametric embedding (ﬁltration) of various classes of semigroup

generators. Originally, the idea of ‘ﬁltration’ was intended to establish a

veriﬁable condition for a function to be a generator. When studying various

ﬁltrations, it turned out they describe well the dynamic properties of the

generated semigroups. Moreover, certain non-linear ﬁltrations make it possi-

ble to solve problems in geometric function theory, in particular, the inverse

Fekete–Szeg¨o problem.

References

[1] F. Bracci, M. D. Contreras, S. D´ıaz-Madrigal, M. Elin and D. Shoikhet,

Filtrations of inﬁnitesimal generators, Funct. Approx. Comment. Math.

59 (2018).

[2] M. Elin, D. Shoikhet, and T. Sugawa, Filtration of semi-complete vector

ﬁelds revisited, in: Trends in Math., Birkh¨auser/Springer, Cham, 2018.

[3] M. Elin, F. Jacobzon and N. Tuneski, The Fekete–Szeg¨o problem and

ﬁltration of generators, Rendiconti del Circolo Matematico di Palermo

Series 2, DOI 10.1007/s12215-022-00824-w.

Polynomial estimates for the method of cyclic projections in

Hilbert spaces

Simeon Reich and Rafa l Zalas

Technion – Israel Institute of Technology

e-mail: sreich@technion.ac.il, rafalz@technion.ac.il

We study the method of cyclic projections when applied to closed and

linear subspaces Mi,i= 1, . . . , m, of a real Hilbert space H. We show

that the average distance to individual sets enjoys a polynomial behaviour

o(k−1/2) along the trajectory of the generated iterates. Surprisingly, when

the starting points are chosen from the subspace Pm

i=1 M⊥

i, our result yields a

polynomial rate of convergence O(k−1/2) for the method of cyclic projections

itself. Moreover, if Pm

i=1 M⊥

iis not closed, then both of the aforementioned

rates are best possible in the sense that the corresponding polynomial k1/2

cannot be replaced by k1/2+εfor any ε > 0.

“Mandelbrot set” for pairs of linear maps and complex Bernoulli

convolutions

Boris Solomyak

Bar–Ilan University

e-mail: bsolom3@gmail.com

This survey-type talk is devoted to self-similar iterated function systems

in the complex plane and related questions. I will mention some parallels,

and even direct connections, with the classical complex dynamics of quadratic

polynomials.