The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. 

A closely related question due to Borwein, Erd\'elyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.

A number of results on classical problems in analytic number theory rely on bounds for multilinear forms of Kloosterman sums, which in turn use deep inputs from the spectral theory of automorphic forms. We’ll discuss our recent work available at arxiv.org/abs/2404.04239, which uses this interplay between counting problems, exponential sums, and automorphic forms to improve results on the greatest prime factor of $n^2+1$, and on the exponents of distribution of primes and smooth numbers in arithmetic progressions.
The key ingredient in this work are certain “large sieve inequalities” for exceptional Maass forms, which improve classical results of Deshouillers-Iwaniec in special settings. These act as on-average substitutes for Selberg’s eigenvalue conjecture, narrowing (and sometimes completely closing) the gap between previous conditional and unconditional results.

The Junior Algebra and Representation Theory Seminar will kick-off the start of the academic year with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Let F be a p-adic field. In this talk I'll study the Om(F)-distinction of some specific principal series representations  of Glm(F). The main goal is to give a computing method to see if those representations are distinguished or not so we can also explicitly find a non zero  Om(F)-equivariant linear form. This linear form will be given by the integral of the representation's matrix coefficient over Om(F).
 

After explaining on what specific principal series representations I'm working and why I need those specificities, I'll explain the different steps to compute the integral of my representation's matrix coefficient over Om(F). I'll explicitly give the obtained result for the case m=3. After that I'll explain an asymptotic result we can obtain when we can't compute the integral explicitly.

I will report on work with Andrew Graham in which we prove new results towards the Bloch--Kato conjecture for automorphic forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$.