Analysis Seminar

The Analysis Seminar @ The Hebrew University of Jerusalem


Organizers: Adi Glücksam and Sasha Sodin
Time and Place: Weekly on Thursdays 12:00-14:00 at Ross 70.

What is it?

Meant for faculty as well as advanced students (younger students may contact the organizers above), the seminar aims to expose its participants to various new branches of analysis. The uniqueness of the seminar is that is aspires to combine proofs in every lecture to deepen our understanding of the topic and the main tools used.


January 23rd: Oleg Ivrii

Shapes of trees. Oleg Ivrii
Abstract: A finite tree in the plane is *conformally balanced* if every edge has the same harmonic measure as seen from infinity, and harmonic measures on the two sides of every edge are identical. It is well known that a finite tree has a conformally balanced shape, which is unique up to scale. In this talk, we study shapes of infinite trees, focusing on the case of an infinite trivalent tree. To conformally balance the infinite trivalent tree, we truncate it at level n, form the true tree T_n and take n to infinity. We show that the Hausdorff limit of the T_n contains the boundary of the developed deltoid, the domain obtained by repeatedly reflecting the deltoid in its sides. This is joint work with P. Lin, S. Rohde and E. Sygal.

Future Talks:

  • January 30th: Eyal Seelig


    Past Talks: Fall 2024:


  • January 16th: Distribution of powers of random unitary matrices through singularities of hyperplane arrangements. Itay Glazer
    Abstract: Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. In their seminal work, Diaconis and Shahshahani have shown that for any fixed m, the sequence (tr(X),tr(X^2),...,tr(X^m)) converges, as n goes to infinity, to m independent complex normal random variables (suitably normalized). This can be seen as a statement about the low-dimensional Fourier coefficients of \tau_m. In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: (a) Can one give sharp estimates on the rate of decay of its Fourier coefficients? (b) For which values of p, is the density of \tau_m L^p-integrable? In the first part of the talk, I will answer (a) and (b). In addition, using works of Rains about the distribution of X^m, I will show how Items (a) and (b) are closely related (and that (b) is in fact equivalent) to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements. In the second part of the talk, I will sketch the proof of this geometric problem, building on a known algorithm for resolution of singularities of hyperplane arrangements. Based on joint works with Julia Gordon and Yotam Hendel ([GGH]), and with Nir Avni and Michael Larsen ([AGL]).


  • January 9th:The number variance of dilations of integer sequences. Nadav Yesha
    Abstract: Let (x_n) be a sequence of positive integers. We will discuss the fluctuation of the number of elements modulo 1 of dilations (\alpha x_n) in short intervals, for generic values of alpha. The main motivation is to compare statistics such as the number variance with the random model, thereby observing pseudo-random behaviour for some interesting examples, such as x_n = n^2. Based on joint works with Zonglin Li and with Christoph Aistleitner (in progress).


  • December 19th- Riesz bases of exponentials. Gady Kozma

    Abstract: A Riesz basis is the second best thing after an orthonormal basis. We will survey the question: for a subset S of the real line, when does the space of functions on S have a Riesz basis of functions of the form exp(itx)? Some results presented are joint with Shahaf Nitzan and/or Alexander Olevskii.


  • December 12th- Creating periodic points using holomorphic curves Shira Tanny

    Abstract: An old question of Poincaré concerns creating periodic points via perturbations of Hamiltonian diffeomorphism. Hamiltonian diffeomorphisms are solutions of a certain ODE and arise from classical mechanics. In the first part of the talk, I will provide an overview of results related to this problem. In the second part, I will discuss how holomorphic curves—complex 1-dimensional submanifolds—can be used to study the existence and properties of periodic points of Hamiltonian diffeomorphisms. The talk is based on a joint work with Julian Chaidez.


  • December 5th- An introduction to weighted polynomial approximation in the complex domain – part 2 Sasha Sodin

    Abstract: This is the second part of the talk, but it should be mostly independent of the first part. We shall discuss the conditions under which a measure on a union of rays is uniquely determined by its Fourier transform. The result (based on joint work with Gady Kozma) is a generalisation of a theorem of Vul from 1959. The proof is based on a new Phragmén-Lindelöf principle, which may be of independent interest.


  • November 28st- Irreducibility of Random Polynomials with Rademacher Coefficients Lior Bary-Soroker

    Abstract: Consider a random polynomial whose coefficients are independent Rademacher random variables (taking the values ±1 with equal probabilities). A central conjecture in probabilistic Galois theory predicts that such polynomials are irreducible asymptotically almost surely as their degree approaches infinity. Here irreducibility is considered over the field of rational numbers. In the first part of the talk I will discuss the recent progress that has shown that this conjecture follows from the Generalized Riemann Hypothesis and that the limiting infimum of the irreducibility probability is positive. In the second part of the talk, we will explore ideas from the proof of the following result: the limiting supremum of the irreducibility probability is 1, unconditionally. Specifically, we demonstrate that along special sequences of degrees, the polynomial is irreducible asymptotically almost surely. This result is based on joint work with Hokken, Kozma, and Poonen.


  • November 21st- An introduction to weighted polynomial approximation in the complex domain Sasha Sodin

    Abstract: We shall review some, mostly classical, results pertaining to approximation of continuous functions on closed sets in the complex plane by polynomials. The talk will also serve as an introduction to a later talk, in which we shall, hopefully, discuss some results obtained in a joint work in progress with Gady Kozma.


  • October 31st, November 7th, November 14th- Multi-fractal spectrum of planar harmonic measures Adi Glücksam

    Abstract: We will begin with defining the concepts of harmonic measure and different notions of dimensions. We will then connect those notions with what is called multi-fractal spectrum. Next, we will discuss finer features of the relationship between those dimensions. Lastly, we will define the universal counterparts and discuss an approximation theorem, showing the importance of domains arising from multifractal formalism.
    This talk is based on a joint work with I. Binder.