학과 세미나 및 콜로퀴엄
We study stochastic motion of objects in micrometer-scale living systems: tracer particles in living cells, pathogens in mucus, and single cells foraging for food. We use stochastic models and state space models to track objects through time and infer properties of objects and their surroundings. For example, we can calculate the distribution of first passage times for a pathogen to cross a mucus barrier, or we can spatially resolve the fluid properties of the cytoplasm in a living cell. Recently developed computational tools, particularly in the area of Markov Chain Monte Carlo, are creating new opportunities to improve multiple object tracking. The primary remaining challenge, called the data association problem, involves mapping measurement data (e.g., positions of objects in a video) to objects through time. I will discuss new developments in the field and ongoing efforts in my lab to implement them. I will motivate these techniques with specific examples that include tracking salmonella in GI mucus, genetically expressed proteins in the cell cytoplasm, active transport of nuclei in multinucleate fungal cells, and raphid diatoms in seawater surface interfaces.
Abstract:
We consider the initial-boundary value problem (IBVP) for the 1D isentropic Navier-Stokes equation (NS) in the half space. Unlike the whole space problem, a boundary layer may appear due to the influence of viscosity.
In this talk, we first briefly study the asymptotic behavior for the initial value problem of NS in the whole space. Afterwards, we will present the characterization of the expected asymptotics for the IBVP of NS in the half space. Here, we focus only on the inflow problem, where the fluid velocity is positive on the boundary.
Reference:
Matsumura, Akitaka. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl. Anal. 8 (2001), no. 4, 645–666.
In this talk, we will introduce vector field method for the wave equation. The key step is to establish the Klainerman-Sobolev inequality developed in [1]. Using this inequality, we will provide dispersive estimates of the linear wave equation, and prove small-data global existence for some nonlinear wave equations. The main reference will be Chapter II in [2].
참고문헌:
[1]. Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477
[2]. Christopher D. Sogge, Lectures on Nonlinear Wave Equations, Second Edition
In this note, we investigate threshold conditions for global well-posedness and finite-time blow-up of solutions to the focusing cubic nonlinear Klein–Gordon equation (NLKG) on $\bbR^{1+3}$ and the focusing cubic nonlinear Schrödinger equation (NLS) on $\bbR$. Our approach is based on the Payne–Sattinger theory, which identifies invariant sets through energy functionals and conserved quantities. For NLKG, we review the Payne–Sattinger theory to establish a sharp dichotomy between global existence and blow-up. For NLS, we apply this theory with a scaling argument to construct scale-invariant thresholds, replacing the standard mass-energy conditions with a $\dot{H}^{\frac12}$-critical functional. This unified framework provides a natural derivation of global behavior thresholds for both equations.