Emergent Behavior in Complex Systems of Interacting Agents 

In this talk we address global existence of weak solutions, their regularization, and global relaxation to Maxwellian for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite energy, $(1+ |v|^2) f_0 \in L^1, f_0 \in L^\infty$, and finite higher moment $|v|^q f \in L^2$, $q \gg 2$ , gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast.

The results are achieved through the use of a new thickness-based renormalization, which circumvents the problem of degenerate diffusion in non-perturbative regime.

The linear interpolation convexity (LIC) has served as the essential condition to guarantee the uniqueness of minimizer for pairwise interaction energies. In particular, for power-law potentials $W(x) = \frac{|x|^a}{a} - \frac{|x|^b}{b}$, it is known that LIC holds for $-d \lt b \leq 2, 2 \leq a \leq 4, b \lt a, (a, b) \neq (4, 2)$. We extend the notion of LIC by requiring the energy convexity only for linear interpolation between probability measures supported on a prescribed ball. This allows us to prove the uniqueness of minimizer for power-law potentials with $a$ slightly smaller than 2 or larger than 4.

 In this talk, I will present some results on the kinetic Vicsek model. The mixing and enhanced dissipation phenomena stabilize the dynamics within specific parameter regimes and ensure effective communication among agents. Consequently, the solution exhibits features similar to those of a spatially-homogeneous system. As a result, we confirm the phase transition observed in the agent-based Vicsek model at the kinetic level. This is a joint work with  Mengyang Gu.

This talk reviews fundamental principles of contraction theory in dynamical
systems, including an analysis of the contractivity properties exhibited by
neural networks. Then I will discuss the network contraction theorem and
explain why it proves excessively restrictive for several networked
systems. Finally, I will present some elements of an emerging theory of
semicontracting systems. I will formulate a duality theorem that explains
why the Markov-Dobrushin coefficient is the rate of contraction for both
averaging and conservation flows in discrete time. If time allows, I will
show applications to the Kuramoto model of coupled oscillators.