Elasticity is a fundamental macroscopic property that emerges in any collection of interacting particles. However, at sufficiently low temperatures where thermal fluctuations are negligible, a free energetic description of such macroscopic properties is not available. Granular materials and glasses offer a paradigm where disorder in the arrangements of particles plays a fundamental role in determining the energy landscape, and thereby their stability, response and elasticity properties. Gradually introducing disorder into athermal crystalline packings can be used to build a relation between the well-established physics of crystals and that of amorphous solids. Such studies can also reveal interesting phenomena peculiar to athermal systems such as hidden order-disorder transitions. In this talk I will outline the development of exact theoretical techniques which can be used to characterize fluctuations in positions, forces and interaction energies in near-crystalline athermal systems, which offer a route towards understanding the emergent elasticity properties of ubiquitous amorphous solids.

Black hole information paradoxes are at the heart of the mysteries of quantum spacetime. Black hole interiors pose the most profound challenges of our understanding of the holographic principle which states that quantum spacetime can be encoded into degrees of freedom of an ordinary quantum system living at the boundary. Recently quantum information theory together with some simple tractable models has played a major role in elucidating how the long standing information paradoxes of black holes can be resolved, and how the black hole interiors can be decoded from the Hawking radiation via appropriate quantum channels. Other developments have pointed out that black hole dynamics can teach us the basic principles of quantum thermodynamics necessary to realize constructions of fault tolerant quantum memory and quantum gates, and efficient ways for constructing other quantum channels such as teleportation channels using strongly interacting systems, I will review some aspects of these developments, and briefly some reasons to believe why black hole microstates can also give us the key to understanding some phases of matter like strange metals.

Recently, machine learning has become a popular tool to use in fundamental science, including lattice field theory. Here I will report on recent progress, starting with (by now) basic applications (phase transitions and critical exponents), moving on to new ideas for the Inverse Renormalisation Group and ending with more speculative suggestions on quantum-field theoretical machine learning.

I summarize arguments that suggest that the phase diagram of QCD, the theory of quarks and gluons, has a critical endpoint which is analogous to the endpoint of the water-vapor transition. This point marks the onset of a first order phase transition between a quark-gluon vapor and a hadronic liquid. I will argue that the critical point can be searched for in collisions of relativistic heavy ions. The main observables are fluctuation measurements, and the expected signatures are related to critical opalescence. I summarize the ongoing theoretical and experimental efforts devoted to observing signatures of critical fluctuations. I argue that along the way, we have gained new insights into an old theory, fluid dynamics.

I briefly review the modern theory of strong interactions, Quantum ChromoDynamics, and why we believe that a qualitatively new state of matter, a Quark-Gluon Plasma, is created in the collisions of heavy ions at very high energies. I discuss, in particular, why it may be the most "ideal" liquid on earth, and the phenomenon of jet quenching. I conclude by discussing what happens when one goes down in energy, and how that may probe qualitatively new states, including a Critical End-Point and Quantum Pion Liquids.

Strongly interacting matter at temperatures more than 100.000 times larger than in the interior of our sun and at an order of magnitude larger densities than in atomic nuclei existed in the early universe and is studied today experimentally on earth in ultra-relativistic collisions of heavy-ions. The exploration of properties of such hot and dense matter also is subject to intensive theoretical research. Computer simulations of the theory of strong interactions, Quantum Chromodynamics (QCD), performed on discrete space-time lattices provide a powerful framework for the study of such matter. These simulations provide insights into the phase structure of strong interaction matter described by QCD and allow first principle calculations that can be confronted with experimental results obtained in heavy-ion collisions. We give a brief overview of the development of lattice QCD calculations at finite temperature and density and discuss computational requirements for state-of-the-art numerical calculations. We furthermore present results from studies of the chiral phase transition in QCD as well as a new, high statistics determination of the QCD equation of state. Some results on fluctuations of conserved charges and their higher order cumulants will be discussed and compared with experimental results.

I shall give a pedagogical narration of how a simple, but intriguing relation about colors in classical optics impacted the most modern developments in physics, even to the point of anticipating String Theory. The strong thread that held these pearls of scientific creativity was the powerful mathematical idea of analyticity, which in this physical context turned out, rather surprisingly, to be a consequence of the deeply cherished physical principle of causality. The talk will be at a level accessible to a wide audience.

Emergence of low energy degrees of freedom is a recurrent theme in condensed matter systems. In more familiar systems such as crystalline solids the emergence of phonons as collective modes of vibration are associated mainly with broken translational symmetry that governs the physics at low temperatures. However, in a wide class of systems, local energetic constraints may take the form of Gauss's law giving rise to an emergent electromagnetism at low energies. In this talk, I shall start with a brief review of such emergent electromagnetism in discussing their basic features as well as the general settings in which they appear. Then I shall apply it to understand the mechanical response of naturally abundant granular solids such as sand grains where local conditions of mechanical equilibrium, i.e., force and torque balance on each grain, as I shall show, have the mathematical structure of a generalized Gauss's law for a rank-2 U(1) electromagnetism. The electrostatic limit of this tensor electromagnetism successfully captures the anisotropic "elasticity" of granular solids and provides a new framework to understand a large class of such systems.

An early wonder of our mathematical life happens when we come across the identity : $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{π^2}{6}$. Even better, in Euler product form, $\prod_p \frac{1}{(1-1/p^2)} = \frac{π^2}{6}$, where p runs through the prime numbers. In the course of the ninetieth century, it appeared that the (zeta) function of a complex variable $ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$, and its variants, is key to understand some of the subtle laws that govern prime numbers. Thus have the hearts of analytic number theorists been set beating. Meanwhile, algebraic number theorists have attempted to understand the meaning of "$\frac{π^2}{6}$". And, out of arithmetic geometry, they have defined a class of variants of ζ: the L-functions, series of the form $\sum_{k=1}^\infty \frac{a_k}{k^s}$ that can also be expressed as Euler products. Their valuations at integers produce the mysterious L-values, that we seek to understand. Well established conjectures now predict what the correct replacement for $\frac{π^2}{6}$ should be. But is the prediction complete? Contrary to what is often believed, not quite. Explanation for all this, including why elaborate conjectures still fall short, will not rely on general explanation of what L-functions are, but on illustrative examples based on elliptic curves and Dirichlet characters. An intriguing formula involving L-values will be offered to help reflect on the "not quite".

We review the current state of precision predictions for the Large Hadron Collider LHC covering the aspects of perturbative corrections to hard scattering processes in quantum chromodynamics at higher orders as well as our knowledge on fundamental parameters of the Standard Model such as the strong coupling constant and heavy quark masses. We illustrate how the precision of available experimental data challenges current theoretical predictions and discuss the mathematical requirements needed to advance the latter. We present an outlook and outline areas for future improvements.

Until the twentieth century it was assumed that knowledge means control. Automatic control came in the sixties for electronics with Bellman's dynamic programming and Kalman's filter and received a boost in the eighties with robust and H∞ control. Will artificial intelligence algorithms change the practice of control drastically? Parallel Optimal Control, which dates from the calculus of variations of Hadamard and the Pontryagin principle is a more functional approach to the optimization of systems. It is heavily used for the design of mechanical devices like airplanes (optimal shape design) and the topological optimisation of materials. Stochastic control remained up to now a mathematical field except for the rare semi-analytical solutions as in the case of linear quadratic control. It is now computationally feasible and its applications to finance for instance, though challenged by deep neural networks, are in daily use for risk assessment of bank's portfolios. Finally, perhaps the most mathematically demanding is the mean-field type control and its application to the Monge-Ampere problem. As this is a colloquium talk, the problems and the main results will be stated only, without assuming any prior knowledge of these sometimes difficult fields. Yet the talk is for a mathematically trained audience.