A New Discriminant for the Hardy Z-Function and the Corrected Gram's law

In this paper, we introduce a novel variational framework rooted in algebraic geometry for the analysis of the Hardy $Z$-function. Our primary contribution lies in the definition and exploration of $\Delta_n(\overline{a})$, a newly devised discriminant that measures the realness of consecutive zeros of $Z(t)$. Our investigation into $\Delta_n(\overline{a})$ and its properties yields a wealth of compelling insights into the zeros of $Z(t)$, including the corrected Gram's law, the second-order approximation of $\Delta_n(\overline{a})$, and the discovery of the G-B-G repulsion relation. Collectively, these results provide compelling evidence supporting a new plausibility argument for the Riemann hypothesis

The Symplectic Topology of Projective Manifolds with Small Dual

We study smooth projective varieties with small dual variety using methods from symplectic topology. For such varieties, we prove that the hyperplane class is an invertible element in the quantum cohomology of their hyperplane sections. We also prove that the affine part of such varieties are subcritical. We derive several topological and algebraic geometric consequences from that. The main tool in our work is the Seidel representation associated to Hamiltonian fibration