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In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology TX cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+, there exists a unique class of cubics maximizing κX, not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [ABSTRACT FROM AUTHOR]

Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk and a denotes one of them, then the closed disk of center a and radius 1 contains a critical point of P (i.e. a zero of its derivative P'). The main result of this paper is to prove that, for each a, there exists an integer N such that the disk |ζ -a| ≤ 1 contains a critical point of P when the degree of P is larger than N. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture. [ABSTRACT FROM AUTHOR]