tbranzov

In this talk, the emphasis will be on Brill–Noether theorems using the language of linear systems on algebraic curves developed in the previous talk. We will focus on existence and dimension results for linear series, illustrating the theory through concrete examples. Finally, we will briefly discuss recent developments and related research directions, if time permits.

The notion of visibility from the origin along straight lines has been generalised by considering lattice points viewed through nonlinear trajectories. Inspired by this, we define polynomial Farey sequences, reducing to the classical Farey sequences for linear polynomials. In this talk, we discuss the distributional properties of polynomial Farey fractions and their subsequences using discrepancy and pair correlation measures

This will be a two-talk mini-series. In the first talk I will give a very gentle introduction to Brill–Noether theory for complex algebraic curves. I will start from the basic question: what kinds of functions can exist on a curve, and how many independent ones can we hope for? After introducing some necessary concepts, I will explain the classical results of Brill–Noether theory, which give a simple criterion for when such functions should exist on a “typical” curve. The emphasis will be on ideas and examples rather than technical details.