We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. A curious object is Laplace eigenfunctions on two-dimensional sphere, which are restrictions of homogeneous harmonic polynomials of three variables onto 2-dimensional sphere. They are called spherical harmonics. Zero sets of such functions are unions of smooth curves with equiangular intersections. Topology of zero set could be quite complicated, but the total length of the zero set of any spherical harmonic of degree n is comparable to n. Though the Laplace eigenfunctions are known for ages, we still don't understand them well enough (even the spherical harmonics). 

Employing standard, as well as novel techniques of asymptotics, three different problems will be discussed: (i) The computation of the large time asymptotics of initial-boundary value problems via the unified transform (also known as the Fokas Method, www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large t-asymptotics to all orders of the Riemann zeta function [2], and the introduction of a new approach to the Lindelöf Hypothesis [3]. (iii) The proof that the ultra-relativistic limit of the Minkowskian approximation of general relativity [4] yields a force with characteristics of the strong force, including confinement and asymptotic freedom [5].

[1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925 (2015).
J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with t-Periodic Data: II. Perturbative Results, Proc. R. Soc. A 471, 20140926 (2015).
[2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear).
[3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis, Transactions of Mathematics and its Applications (to appear).
[4] L. Blanchet and A.S. Fokas, Equations of Motion of Self-Gravitating N-Body Systems in the First Post-Minkowskian
Approximation, Phys. Rev. D 98, 084005 (2018).
[5] A.S. Fokas, Super Relativistic Gravity has Properties Associated with the Strong Force, Eur. Phys. J. C (to appear).

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.