the burau representation of the braid group was introduced in the 1930s and has been used since the 1980s to understand the jones polynomial. a critical component of this study has been trying to understand when the burau representation is faithful. it is known to be faithful for the braid group on fewer than 4 strands and unfaithful on more than 4 strands. if the representation is unfaithful on 4 strands, it will solve the jones unknot conjecture. we will review the current state of knowledge and some new approaches suggested by the work of armond, huynh and l\^e.

we first discuss the inviscid limit of the global energy dissipation of leray solutions of incompressible navier-stokes on the torus. assuming that the solutions have besov norms bounded uniformly in viscosity, we establish an upper bound on energy dissipation. as a consequence, onsager-type ``quasi-singularities'' are required in the leray solutions, even if the total energy dissipation is vanishes in the limit v $\rightarrow$ 0. next, we discuss an extension
of onsager's conjecture for domains with solid boundaries. we give a localized regularity condition for energy conservation of weak solutions of the euler equations assuming (local) besov regularity of the velocity with exponent >1/3 and, on an arbitrary thin layer around the boundary, boundedness of velocity, pressure and continuity of the wall-normal velocity. we also prove that the global viscous dissipation vanishes in the inviscid limit for leray-hopf solutions of the navier-stokes equations under the similar assumptions, but holding uniformly in a vanishingly thin viscous boundary layer. finally, if a strong euler solution exists, we show that equicontinuity at the boundary within a o(v) strip alone suffices to conclude the absence of anomalous dissipation.
the talk concerns joint work with g. eyink and h.q. nguyen.

in this talk, space-time finite element methods will be presented as an approach to solving pdes with time-dependence. in these methods, time is treated in the same way as the spatial variables, leading to a fully unstructured finite element discretization in space-time. the fundamental properties of this approach will be discussed, especially in contrast to the method of lines. in particular, the efficiency of space-time methods on massively parallel computers will be discussed, as well as the importance of this property in modern numerical software.

we will discuss two different approaches to computing the l-functions of shimura varieties associated with gu(2,1). both approaches employ the comparison of the grothendieck-lefschetz formula with the arthur-selberg trace formula. the first approach, carried out by the author, takes as its starting point the recent work of laumon and morel. the second approach is due to flicker. in both approaches, the principal challenge is that the shimura varieties in question are non-compact, and one must use cohomology with compact supports. time permitting, we will discuss the prospects for extending these approaches to the non-compact shimura varieties associated with higher-rank unitary groups.

motion by mean curvature for networks of surfaces arises in a variety of applications, such as the dynamics of foam and the evolution of microstructure in polycrystalline materials. it is steepest descent (gradient flow) for an energy: the sum of the areas of the surfaces constituting the network.

during the evolution, surfaces may collide and junctions (where three or more surfaces meet) may merge and split off in myriad ways as the network coarsens in the process of decreasing its energy. the first idea that comes to mind for simulating this evolution -- parametrizing the surfaces and explicitly specifying rules for cutting and pasting when collisions occur -- gets hopelessly complicated. instead, one looks for algorithms that generate the correct motion, including all the necessary topological changes, indirectly but automatically via just a couple of simple operations.

a remarkably elegant such algorithm, known as threshold dynamics, was proposed by merriman, bence, and osher in 1992. extending this algorithm, while preserving its simplicity, to more general energies where each surface in the network is measured by a different, possibly anisotropic, notion of area requires new mathematical understanding of the original version, which then elucidates a systematic path to new algorithms.

i will construct a pair of calabi-yau threefolds which are deformation and derived equivalent, but not birationally equivalent. i will explain how this gives a counterexample to the birational torelli problem for calabi-yau threefolds, as well as new examples of zero divisors in the grothendieck ring of varieties. this is joint work with lev borisov and andrei caldararu.