Georgia Institute of Technology College of Sciences

Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. C. De Lellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. This approach involves correcting “approximate solutions" by adding rapid oscillations, which are designed to reduce the error term in solving the equation. In this talk, I will discuss an improved convex integration framework, which yields solutions with Holder regularity 1/5- as well as other recent results.

We will start by counting lattice points in a polytope and showhow this produces many familiar objects in mathematics.For example if one scales the polytope, the number of lattice points givesrise to the Ehrhart polynomials, including binomals and other well knownfunctions.Things get more interesting once we take a weighted sum over the latticepoints instead of just counting them. I will explain how toextend Ehrhart's theory in this case and discuss an application to knottheory. We will derive a new state sum for the colored HOMFLYpolynomial using q-Ehrhart polynomials, following my recent preprint Arxiv1501.00123.

The existence, stability, and bifurcation structure of localized radially symmetric solutions to the Swift--Hohenberg equation is explored both numerically through continuation and analytically through the use of geometric blow-up techniques. The bifurcation structure for these solutions is elucidated by formally treating the dimension as a continuous parameter in the equations. This reveals a family of solutions with an anomalous amplitude scaling that is far larger than expected from a formal scaling in the far field. One key advantage of the geometric blow-up techniques is that a priori knowledge of this scaling is unnecessary as it naturally emerges from the construction. The stability of these patterned states will also be discussed.