A very natural question is whether the periodic table of the elements is indeed periodic and whether this can be proved mathematically.

From a mathematical point of view this is more interesting if we allow ourselves to extrapolate beyond the physical atomic numbers bounded by 92 to arbitrarily large atomic numbers. This relates to the famous ionization conjecture in mathematical physics. It states that quantities such as the radius, maximal ionization, and ionization energies of atoms remain bounded as the atomic number tends to infinity. This conjecture is still open for the full non-relativistic many-body Schrödinger description of atoms. Several years ago, I proved the ionization conjecture in the approximate Hartree-Fock model.

A generalization of the ionization conjecture asks whether there is even a limiting behavior as the atomic number tends to infinity. In this talk I will describe another approximate model, the Thomas-Fermi mean field model, in which there indeed is a limiting behavior of large atoms. It leads to an exactly periodic limiting “periodic table”.

The infinite atoms are described by a periodic family of self-adjoint realizations of a very singular Schrödinger operator. It corresponds to what in the theory of self-adjoint extensions is referred to as a Weyl limit circle case.

What are the removable singularities of harmonic functions with bounded gradient?  This problem, that takes its origins in certain problems of complex analysis, which are 140 years old, was solved relatively recently. Its solution is based on extension to a new territory of classical theory of singular integrals.

Singular integrals are ubiquitous objects and play an important part in Geometric Measure Theory. The simplest ones are called Calderon–Zygmund operators. Their theory was completed in the 50′s by Zygmund and Calderon. Or it seemed like that. The last 25 years saw the need to consider CZ operators in very bad environment, so kernels are still very good, but the ambient set/measure has no regularity whatsoever.

Initially such situations appeared from the wish to solve some outstanding problems in complex analysis: such as problems of Painleve, Ahlfors, Denjoy and Vitushkin.

The analysis of CZ operators on very bad sets  is very fruitful in the part of Geometric Measure Theory that deals with removability mentioned above and rectifiability. It can be viewed as the study of very low regularity free boundary problems.  We will explain the genesis of ideas that led to several  long and difficult proves that culminated in the solutions of problems of Denjoy, Vitushkin, David-Semmes, and Bishop, and brought also the solution by Tolsa of Painleve’s problem.

Computing the derivative of long-time-averaged observables with respect to system parameters is a central problem for many numerical applications. Conventionally, there are three straight-forward formulas for this derivative: the pathwise perturbation formula (including the backpropagation method used by the machine learning community), the divergence formula, and the kernel differentiation formula. We shall explain why none works for the general case, which is typically chaotic (also known as the gradient explosion phenomenon), high-dimensional, and small-noise.

We present the fast response formula, which is a 'Monte-Carlo' type formula for the parameter-derivative of hyperbolic chaos. It is the average of some function of u-many vectors over an orbit, where u is the unstable dimension, and those vectors can be computed recursively. The fast response overcomes all three difficulties under hyperbolicity assumptions. Then we discuss how to further incorporate the kernel differentiation trick to overcome non-hyperbolicity.

The modified diagonal on the triple product of a curve was first introduced by Gross and Schoen in the 90’s. This simply defined object holds fundamental importance in the study of the geometry and arithmetic of curves. One basic question is whether the modified diagonal vanishes under “deformation”. I will introduce the origin of this type of question and provide a brief history of the study of the modified diagonal. Subsequently, I will discuss my collaborative works with W. Zhang, where we demonstrated that such vanishing can be dictated by the symmetry of the curve. As an application in number theory, we proved a case of the notorious Beilinson—Bloch conjecture, a generalization of the Millennium Birch—Swinnerton-Dyer conjecture. Finally, I want to propose some new questions.

Ricci flow is an important tool in geometric analysis. There have been remarkable topology applications of Ricci flow on closed manifolds, such as the Poincaré Conjecture resolved by Perelman, and the recent Generalized Smale Conjecture resolved by Bamler-Kleiner. In contrast, much less is known about the Ricci flow on open manifolds. Solitons produce self-similar Ricci flows, and they often arise as singularity models. Collapsed singularities and solitons create additional difficulties for open manifolds. In this talk, I will survey some recent developments in Ricci flow on open manifolds. In particular, I will talk about the resolution of Hamilton's Flying Wing Conjecture, and the resulting collapsed steady solitons.