The motion of a coronavirus is highly dependent on its modes of vibration. If we model a coronavirus based on a elastodynamic PDEs, then the wiggling motion of the spikes and breathing motion of the capsid can be captured, for example. Here, we give a quick review of this model and the associated modal analysis through finite element computations. We then begin to study the invasion process by a coronavirus into a healthy cell, which takes place after the virus attacked the cell and its membrane began to fuse with the membrane of the cell. The fusion causes an initial opening with diameter of about 1 nm on the cell's membrane. For the invasion process to be successful, the virus must further widen the diameter to about 100nm in order for the viral genome material to enter the healthy cell to replicate. Can we model and simulate such a process by using a coupled virus-cell elastodynamic system? We will present our most recent partial findings for this question through the showing of supercomputer simulation animations.

In this work, we describe an alternative approach to the general theory of Mean Field Control as presented in the book of P. Cardaliaguet, F. Delarue, J-M Lasry, P-L Lions: The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematical Studies, Princeton University Press, 2019. Since it uses Control Theory and not P.D.E. techniques it applies only to Mean Field Control. The general difficulty of Mean Field Control is that the state of the dynamic system is a probability. Therefore, the natural functional space for the state is the Wasserstein metric space. P.L. Lions has suggested to use the correspondence between probability measures and random variables, so that the Wasserstein metric space is replaced with the Hilbert space of square integrable random variables. This idea is called the lifting approach. Unfortunately, this brilliant idea meets some difficulties, which prevents to use it as an alternative, except in particular cases. In using a different Hilbert space, we study a Control problem with state in a Hilbert space, which solves the original Mean Field Control problem, as a particular case, and thus provides a complete alternative to the approach of Cardaliaguet, Delarue, Lasry, Lions. Based on Joint work with P. J. GRABER, P. YAM. Research supported by NSF grants DMS- 1905449 and 2204795.

Localization in the Hubbard model within Hartree-Fock theory was previously established in joint work with J. Schenker, in the regime of large disorder in arbitrary dimension and at any disorder strength in dimension one, provided the interaction strength is sufficiently small. I will present recent progress on the spectral statistics conjecture for this model. Under weak interactions and for energies in the localization regime which are also Lebesgue points of the density of states, it is shown that a suitable local eigenvalue process converges in distribution to a Poisson process with intensity given by the density of states times Lebesgue measure. If time allows, proof ideas and further research directions will be discussed, including a Minami estimate and its applications.