Progress on a conjecture regarding the triangular distribution

Progress on Dirac's Conjecture

In 1951, Gabriel Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n 2 − c lines determined by P , for some constant c. The following weakening was proved by Beck and Szemerédi–Trotter: every set P of n non-collinear points contains a point in at least n c lines determined by P , for some large unspecified constant c. We prove that every set ...

Recent progress on the Restriction conjecture

Recent Progress on the Tate Conjecture

We survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely related with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch–Swinnerton-Dyer conjectures. We conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.

Recent progress on the Jacobian Conjecture

In this paper we describe some recent developments concerning the Jacobian Conjecture(JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form x + (Ax)∗3 with A = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings i.e. mappings of the form x − ∇f...

Progress Report on the Telescope Conjecture

The Telescope Conjecture (made public in a lecture at Northwestern University in 1977) says that the vn–periodic homotopy of a finite complex of type n has a nice algebraic description. It also gives an explicit description of certain Bousfield localizations. In this paper we outline a proof that it is false for n = 2 and p ≥ 5. A more detailed account of this work will appear in [Rav]. In view...