we revisit the preservation of analyticity and gevrey regularity for the euler equation. we provide a result on preservation of gevrey norm and analyticity in lagrangian formulation of the euler equation and discuss the validity of the result in the eulerian variables. next, we consider the navier-stokes equations posed on the half space, with dirichlet boundary conditions. we give a direct energy based proof for the instantaneous space-time analyticity and gevrey class regularity of the solutions, uniformly up to the boundary of the half space.

this is a brief introduction to mathematical programming with complementarity constraints.
this type of optimization problem is ill-posed in the usual sense. stationarities and
constraint qualifications for this problem are introduced, together with some existing methods.

free entropy theory is an analogue of information theory in a non-commutative setting, which has had great applications to the examination of structural properties of von neumann algebras. i will discuss some ongoing joint work with paul skoufranis to extend this approach to the setting of bi-free probability which attempts to study simultaneously ``left'' and ``right'' non-commutative variables. i will speak in particular of an approach to a bi-free fisher information and bi-free conjugate variables -- analogues of fisher's information measure and the score function of information theory. the focus will be on constructing these tools in the non-commutative setting, and time permitting, i will also mention some results such as bi-free cramer-rao and stam inequalities, and some quirks of the bi-free setting which are not present in the free setting.

trying to understand the mechanisms of chemotaxis,during which biological cells move in the presence of chemical gradients, is a challenging and fascinating problem. chemotaxis and cell migration are key components in a multitude of biological processes, including neuronal patterning, wound healing, embryogenesis, and cancer metastasis. even though many of the key components, cell migration remains a poorly understood process. in this talk, i will present our modeling and experimental efforts aimed towards a better quantitative and mechanistic understanding of chemotaxis and cell motility.

given an n-tuple of non-commutative random variables, its free stein discrepancy relative to the semicircle law measures how ​``close '' the distribution is to the semicircle law. by considering free stein discrepancies relative to a broader class of laws, one can define a quantity called the free stein information. in this talk, we will discuss this and its relation to other free probabilistic quantities such as the free fisher information and the non-microstates free entropy dimension. this is based on joint work in progress with ian charlesworth.