In this talk, we derive second-order expressions for both the one- and two-particle reduced density matrices of the Gibbs state at fixed positive temperatures. We consider a translation-invariant system of N bosons in a three-dimensional torus. These bosons interact through a repulsive two-body potential with a scattering length of order 1/N in the large N limit. This analysis provides a justification of Bogoliubov's prediction regarding the fluctuations around the condensate. The talk will primarily introduce basic concepts and settings, ensuring accessibility for all attendees. This work is a joint effort with Christian Brennecke and Phan Thành Nam.

In this talk, I will present the recent progress of understanding adversarial multiclass classification problems, motivated by the empirical observation of the sensitivity of neural networks by small adversarial attacks. Based on 'distributional robust optimization' framework, we obtain reformulations of adversarial training problem: 'generalized barycenter problem' and a family of multimarginal optimal transport problems. These new theoretical results reveal a rich geometric structure of adversarial training problems in multiclass classification and extend recent results restricted to the binary classification setting. From this optimal transport perspective understanding, we prove the existence of robust classifiers by using the duality of the reformulations without so-called 'universal sigma algebra'. Furthermore, based on these optimal transport reformulations, we provide two efficient approximate methods which provide a lower bound of the optimal adversarial risk. The basic idea is the truncation of effective interactions between classes: with small adversarial budget, high-order interactions(high-order barycenters) disappear, which helps consider only lower order tensor computations.

We investigate the global existence and optimal time decay rate of solution to the one dimensional (1D) two-phase flow described by compressible Euler equations coupled with compressible Navier-Stokes equations through the relaxation drag force on the momentum equations (Euler-Navier-Stokes system). First, we prove the global existence of strong solution and the stability of the constant equilibrium state to 1D Cauchy problem of compressible Euler-Navier-Stokes system by using the standard continuity argument for small $H^{1}$ data while its second order derivative can be large. Then we derive the optimal time decay rate to the constant equilibrium state. Compared with multi-dimensional case, it is much harder to get optimal time decay rate by direct spectrum method due to a slower convergence rate of the fundamental solution in 1D case. To overcome this main difficulty, we need to first carry out time-weighted energy estimates (not optimal) for higher order derivatives, and based on these time-weighted estimates, we can close a priori assumptions and get the optimal time decay rate by spectrum analysis method. Moreover, due to non-conserved form and insufficient decay rate of the coupled drag force terms between the two-phase flows, we essentially need to use momentum variables $(m= \rho u, M=n\omega)$, rather than velocity variables $(u, \omega)$ in the spectrum analysis, to fully cancel out those non-conserved and insufficiently time-decay drag force terms. Finally, we study the singularity formation of the two-phase flow. We consider the blow-up of Euler equations in Euler-Navier-Stokes system. For Euler equations, we use Riemann invariants to construct decoupled Riccati type ordinary differential equations for smooth solutions and provide some sufficient conditions under which the classical solutions must break down in finite time.

In this talk, I will discuss how the fundamental concepts in probability theory—the law of large numbers, the central limit theorem, and the large deviation principle—are developed in the study of real eigenvalues of asymmetric random matrices.