Nets Hawk Katz

Finite time blow-up for a dyadic model of the Euler equations

We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.

A SLIGHT IMPROVEMENT TO GARAEV'S SUM PRODUCT ESTIMATE

max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality. It is known, via the work in [BKT], that if A is a subset of Fp, the field of p elements with p prime, and if p6 0, then one has the sum product estimate max(|? + A|,|AA|)>|,4|1+ for some e > 0. This result has found many applications in combinatorial problems and exponential sum estimates (see e.g. [BKT], [BGK], [G2]). Recently, Garaev [Gl] showed that when \A\ < pz, one has the estimate

New bounds for Kakeya problems

We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.

New bounds on cap sets

We provide an improvement over Meshulams bound on cap sets in

On Additive Doubling and Energy

F_3^N

Approximate multiplicative groups in nilpotent Lie groups

. We show that there exist universal

On the CNF-complexity of bipartite graphs containing no squares

\epsilon>0

Pseudodifferential operators onSU(2)

and

A SUM-PRODUCT ESTIMATE IN FINITE FIELDS, AND APPLICATIONS

C>0

On the Erdos distinct distance problem in the plane

so that any cap set in