Events of the Week

This dissertation presents mathematical modeling and numerical analysis of the dynamics of antichemotactic droplets in one and two spatial dimensions. Motivated by experimental observations of self-avoiding droplets, the diffusion of chemicals secreted by the droplets is modeled by a diffusion equation with a smooth regularized delta function as the source term. The motion of each droplet is described by an evolution equation in which the droplet velocity is proportional to the local gradient of the chemical field evaluated at its center. The resulting reaction-diffusion system is solved numerically using the immersed boundary framework. For the model in which interactions between the droplets and the surrounding fluid are not explicitly considered, a single dimensionless parameter governing the droplet dynamics in a periodic domain is derived in both one and two dimensions. In one dimension the stationary solution is explicitly derived for a single droplet, and a linear stability analysis of the droplet motion is carried out. It is shown mathematically that the droplet dynamics undergo a transition from stable to unstable behavior as this dimensionless parameter varies. In addition, we analytically derive traveling wave solutions for the one droplet case that exist for values of the governing parameter large enough. Numerical simulations further demonstrate the multiple droplet dynamics that there is a transition from damped to oscillatory dynamics as the parameter increases. In two dimensions, equilibrium configurations of the droplets are characterized geometrically using Voronoi tessellations, and the collective dynamics arising from droplet-droplet and chemical-droplet interactions are quantitatively analyzed through the mean squared displacement. In addition, comparisons with experimental observations calibrate the dimensionless parameter of the proposed model with the experimental systems. Finally, to account for the physical nature of droplets as objects suspended in a fluid, a new computational framework incorporating chemical advection and inertia induced by the surrounding fluid is proposed. The target-point Immersed Boundary method is employed to describe the coupled dynamics of the chemical field, droplets, and fluid. This framework provides a foundation for investigating more general active matter systems in which chemical interactions and fluid motion are simultaneously coupled.

This thesis investigates the symbolic powers of ideals through two complementary lenses. One part addresses Demailly's Conjecture. It improves existing thresholds on the number of points required for the conjecture to hold, for both general and very general points in Projective Spaces. It also proves new cases for very general points in Projective Spaces, and establishes a weaker form valid for all dimensions. Another part introduces and develops the theory of rational symbolic powers of ideals, unifying the classical notions of symbolic powers and rational powers into a single framework. This thesis demonstrates key properties of the new algebraic object and specializes to monomial ideals—providing valuations and polyhedral characterizations.

TBA