Wednesday, February 10, 2021
02/10/2021 - 10:15am
On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures how "highly connected" the surface is. We study the spectral gap of a random covering space of a fixed surface, and show that for every ε>0 , with high probability as the degree of the cover tends to ∞, the smallest new eigenvalue is at least 3/16-ε.
Our main tool is a new method to analyze random permutations "sampled by surface groups". I intend to give some background to the result and discuss some ideas from the proof.
This is based on joint works with Michael Magee and Frederic Naud.
02/10/2021 - 4:15pm
Motivated by the centuries old conjecture that there are infinitely many integers n for which n^2+1 is a prime number, in a joint work with Will Sawin we show that over some finite field there are infinitely many polynomials f(x) for which the polynomial f(x)^2 + x is irreducible. Our proof combines analytic techniques previously used in the study of the aforementioned conjecture, with Grothendieck’s function-sheaf dictionary, and Deligne’s Riemann Hypothesis. We are then led to study the cohomology of local systems on the complement of an arrangement of hyperplanes in positive characteristic affine space, borrowing ideas from the analogous topological problem over the complex numbers.