Time: Wednesday 4:00 pm (unless otherwise noted)
schedule for the semester (Spring 2025)
March 5: Ron Lifshitz (School of Physics & Astronomy, Tel Aviv University)
March 19: Juhun Baik (KAIST)
April 16: Ethan Ackelsberg (École Polytechnique Fédérale de Lausanne)
May 7: Jiyoung Han (Pusan National University)
May 21: Minsung Kim (KTH Royal Institute of Technology)
3.5: Ron Lifshitz
Title: What is a Crystal? A new paradigm for an old question
Abstract: The discovery of quasicrystals signalled the beginning of a remarkable scientific revolution, in which some of the most basic notions of condensed matter physics and material science have undergone a thorough re-examination. Over four decades later, the field continues to intrigue us with scientific puzzles, surprising discoveries, and new possibilities for applications. I will focus on some current issues from my own research – such as soft matter quasicrystals and photonic applications based on metamaterials – but only after giving a concise overview for non-specialists of what quasicrystals are, and why their discovery was so important.
March 19: Juhun Baik
Title: Quadratic polynomials and big mapping classes
Abstract: The Cantor set arises naturally in various dynamical systems. In particular, the iteration of a complex polynomial yields an invariant set (called Julia set), which is often homeomorphic to a Cantor set. In this talk, I will focus on the iteration of quadratic polynomials and how the Julia set changes when the parameter varies. The behavior of the Julia set produce a mapping class of a plane-Cantor set as a result. I will also discuss how to construct the mapping class of the plane - Cantor set which reflects the polynomial dynamics. This is a joint work with Prof. Hyungryul Baik.
April 16: Ethan Ackelsberg
Title: Equidistribution of orbits in 2-step nilpotent translational systems and applications
Abstract: Solutions to several major problems in additive combinatorics rely on a dichotomy between “structure” and “randomness,” characterized in many cases by Gowers uniformity norms. In the ergodic theory context, there is a corresponding family of seminorms (the Host—Kra seminorms) producing Host—Kra factors as the “structure” behind combinatorial phenomena. A recent result of Jamneshan, Shalom, and Tao (2024) describes order 2 Host—Kra factors for actions of abelian groups as inverse limits of 2-step nilpotent translational systems defined on homogeneous spaces of 2-nilpotent locally compact Polish groups.
In this talk, we will discuss a new Ratner-type equidistribution theorem for orbits in 2-step nilpotent translational systems, generalizing previous equidistribution results for (2-step) nilsystems. We will also discuss a combinatorial application of the equidistribution theorem to a problem about infinite triple sumsets in sets of positive density in abelian groups.
Based on joint work with Asgar Jamneshan.
May 7: Jiyoung Han
Title: On the distribution of values of bilinear forms at tuples of integer vectors II: random quantification.
Abstract: The famous Margulis theorem (Oppenheim conjecture) says that one can determine when the values of the given non-degenerate indefinite quadratic form at integer vectors are dense in the real line. The theorem can be extendible using Ratner’s theorem to the case of symmetric bilinear forms with $(d-1)$-tuples of integer vectors, where $d$ is the rank of the symmetric matrix associated with the given bilinear form, which is the work of Dani and Margulis.
In this talk, we will see how one can quantify their theorem randomly using Rogers’ higher moment formulas and generalize these problems to the case of symplectic forms.
May 21: Minsung Kim
Title: Anisotropic spaces and nil-automorphisms.
Abstract: Interconnections between parabolic and hyperbolic dynamics have been recently studied regarding renormalization techniques. In particular, they were observed in the deviation of ergodic integrals and solving cohomological equations for some parabolic flows (cf. Liverani-Giulietti, Adam-Baladi, Faure-Gouëzel-Lanneau, and Butterley-Simonelli, etc.).
In this talk, we introduce the geometric anisotropic Banach spaces on Heisenberg nilmanifolds. This construction shows how the Ruelle resonances for the transfer operator associated with the renormalization map (partially hyperbolic nil-automorphism) are related to the invariant distributions for cohomological equations of Heisenberg nilflows. This is joint work with Oliver Butterley.