Here, we provide a generic version of quantum Wielandt's inequality, which gives an optimal upper bound on the minimal length such that products of that length of n-dimensional matrices in a generating system span the whole matrix algebra with probability one. This length generically is of order $\Theta(\log n)$, as opposed to the general case, in which the best bound is $O(n^2)$.  This has implications for the primitivity index of random quantum channels, matrix product states and projected entangled pair states. Some results can be extended to Lie algebras. Joint work with Yifan Jia.[-]

The mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.[-]

Suspensions are ubiquitous in nature (sediments, clouds,biological fluids ... etc.) and in industry such as civil engineering (paints, polymers ... etc.) among many others. The rigorous derivation of fluid-kinetic models for suspensions has attracted a lot of attention in the last decade. This lecture aims at presenting a review of the main results that have been obtained.

The first session aims at introducing both the microscopic and the limiting equation and giving a formal derivation of the former one. The second session aims at presenting the main early results concerning the derivation of an effective model starting from the microscopic model in which particle positions and velocities are fixed or given. Such a system takes the following form for example
\begin{equation}\label{eq:Stokes}
\left \{
\begin{array}{rcl}
-\Delta u+\nabla p &=& f, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
\text{div } u&=& 0, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
u&=& V_i, \text{ on } \partial B(x_i,r)\\
u&=& 0, \text{ on } \partial \Omega
\end{array}
\right.
\end{equation}
where $\Omega$ a smooth open set of $\mathbb{R}^3$, $x_1, x_2, \cdots, x_N$ are the particles position, $r$ their radius and $V_i$ the given velocity of the $i$th particle. The aim is then to perform an asymptotic analysis when the number of particles $N$ becomes large while their radius $r$ becomes small, first results have been obtained in [1,2,3] where the limit equations depend on the scale of the holes and their typical distance; Stokes equation, Darcy equation or Stokes-Brinkman equation. After recalling the recent contributions, we will present a short argument giving insights about the derivation of the Brinkman term in a simple case.

The last session of this mini-course aims at presenting the results regarding the rigorous derivation of fluid-kinetic models when taking into account the fluid-particle interactions and particle dynamics. This means that we consider the Stokes equation [1] coupled to Newton laws where we neglect particles inertia (balance of force and torque) and the motion of the center of the particles $\dot{x}_i=V_i$.

The rigorous derivation of a fluid-kinetic model in this setting have been obtained in [6,5,7] in the case $\Omega=\mathbb{R}^3$ under some separation assumptions on the particles. The obtained equation is a Transport-Stokes equation
\begin{equation}\label{eq:TS}\tag{TS}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &=& \rho g,\\
\text{div } u&=& 0, \\
\partial_t \rho +\text{div }( ( u + \gamma^{-1} V_{\mathrm{St}})\rho) &=& 0,
\end{array}
\right.
\end{equation}
where $\gamma = \lim Nr \in (0,\infty]$.

This result is related to the mean field limit of many particles interacting through a kernel and has been extensively studied for several different problems. We present the main ideas for such a derivation using the method of reflections and stability estimates through Wasserstein distance following the approach by M. Hauray [4]. We finish by emphasizing new results based on a mean-field argument for the derivation of models of suspensions.[-]

Recent developments in quantum information led to a generalised notion of reference frames transformations, relevant when reference frames are associated to quantum systems. In this talk, I discuss whether such quantum reference frame transformations could realise a notion of deformed symmetries formalised as quantum group transformations. In particular, I show the correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at the first order in the quantum deformation parameter. This correspondence is found once the group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame.[-]

The quantisation of the spectral action for spectral triples remains a largely open problem. Even within a perturbative framework, serious challenges arise when in the presence of non-abelian gauge symmetries. This is precisely where the Batalin–Vilkovisky (BV) formalism comes into play: a powerful tool specifically designed to handle the perturbative quantisation of gauge theories. The central question I will address is whether it is possible to develop a BV formalism entirely within the framework of noncommutative geometry (NCG). After a brief introduction to the key ideas behind BV quantisation, I will report on recent progress toward this goal, showing that the BV formalism can be fully formulated within the language of NCG in the case of finite spectral triples. [-]

This talk introduces a class of Hopf algebras, called T -Poincaré, which represent, arguably, the simplest small scale/high energy quantum group deformations of the Poincaré group. Starting from some reasonable assumptions on the structure of the commutators, I am able to show that these models arise from a class of classical r-matrices on the Poincaré group. These have been known since the work of Zakrzewski and Tolstoy, and allow me to identify 16 multiparametric models. Each T -Poincaré model admits a canonical 4-dimensional quantum homogeneous spacetime, T -Minkowski, which is left invariant by the coaction of the group. A key result is the systematic unification provided by this framework, which incorporates well-established non-commutative spacetimes like Moyal, lightlike κ-Minkowski, and ρ-Minkowski as specific instances. I will then outline all the mathematical structures that are necessary in order to study field theory on these spaces: differential and integral calculus, noncommutative Fourier theory, and braided tensor products. I will then discuss how to describe (classical) Standard Model fields within this framework, and the challenges associated with quantum field theory. Particular focus is placed on the Poincar´e covariance of these models, with the goal of finding a mathematically consistent model of physics at the Planck scale that preserves the principle of Special Relativity while possessing a noncommutativity length scale.[-]

We'll discuss our joint work with Ron Donagi and Tony Pantev on the construction of the Higgs bundles associated to Hecke eigensheaves for the geometric Langlands program in the case of rank 2 local systems on a curve of genus 2 . Recall that the moduli space of bundles in this case has two connected components: $\mathbb{P}^3$ and the intersection of two quadrics in $\mathbb{P}^5$. We look for Higgs bundles on these spaces with parabolic structure and logarithmic poles along the wobbly locus. This leads to the study of the geometry of the wobbly locus and its singularities, and the use of our Dolbeault higher direct image construction for the calculation of Hecke operators.[-]

In this talk, I discuss the energy-critical half-wave maps equation (HWM). It has been known for quite some time that (HWM) is completely integrable with a Lax pair structure. However, the question about global-in-time existence of solutions has been completely open so far — even for smooth and sufficiently small initial data. I will present very recent results that prove global well-posedness for rational initial data (with no size restriction) along with a general soliton resolution result in the large-time limit. The proofs strongly exploit the Lax structure of (HWM) in combination with an explicit flow formula. This is joint work with Patrick Gérard (Paris-Saclay).[-]