Abstract: Consider a random polynomial whose coefficients are independent Rademacher random variables (taking the values ±1 with equal probabilities). A central conjecture in probabilistic Galois theory predicts that such polynomials are irreducible asymptotically almost surely as their degree approaches infinity. Here irreducibility is considered over the field of rational numbers. In the first part of the talk I will discuss the recent progress that has shown that this conjecture follows from the Generalized Riemann Hypothesis and that the limiting infimum of the irreducibility probability is positive. In the second part of the talk, we will explore ideas from the proof of the following result: the limiting supremum of the irreducibility probability is 1, unconditionally. Specifically, we demonstrate that along special sequences of degrees, the polynomial is irreducible asymptotically almost surely. This result is based on joint work with Hokken, Kozma, and Poonen.

Future Talks:

Past Talks: Fall 2024:

Abstract: We shall review some, mostly classical, results pertaining to approximation of continuous functions on closed sets in the complex plane by polynomials. The talk will also serve as an introduction to a later talk, in which we shall, hopefully, discuss some results obtained in a joint work in progress with Gady Kozma

Abstract: We will begin with defining the concepts of harmonic measure and different notions of dimensions. We will then connect those notions with what is called multi-fractal spectrum. Next, we will discuss finer features of the relationship between those dimensions. Lastly, we will define the universal counterparts and discuss an approximation theorem, showing the importance of domains arising from multifractal formalism.
This talk is based on a joint work with I. Binder.