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We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This would allow for example asymptotic formulas for the density of solutions to nonsingular systems of Diophantine inequalities in sufficiently many variables.

We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This would allow for example asymptotic formulas for the density of solutions to nonsingular systems of Diophantine inequalities in sufficiently many variables.

I will discuss the problem of obtaining an asymptotic formula for counting integral solutions to an equation of the form f(x, y, z, w)=N in an expanding box, where N is a non-zero integer and f is an indefinite quadratic form over the integers. For forms of the shape axy-bzw, I will then explain how we can obtain a significantly strong error term by applying deep methods from the spectral theory of automorphic forms. This is a joint work with Rachita Guria.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon L. Rydin Myerson (warwick) and Junxian Li (UC Davis).

n this talk, I will discuss the problem of obtaining lower bounds for the number of rational points of bounded height on cubic hypersufaces. Our main tools will be the circle method, the Ekedahl sieve and the geometry of numbers.

In practice, L-functions appear as generating functions encapsulating information about various objects, such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We discuss a connection between low-lying zeros and central values of L-functions, in particular showing that results about the distribution of low-lying zeros (towards the density conjecture of Katz-Sarnak) implies results about the distribution of the central values (towards the normal distribution conjecture of Keating-Snaith). Even though we discuss this principle in general, we instanciate it in the case of modular forms in the level aspect to give a statement an...

For a $SL(2,\mathbb{Z})$ form $f$, we obtain a sub-Weyl bound: \begin{equation*}L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}\end{equation*} for some explicit $\delta>0$, crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.

I will present a new zero-free region for all $\mathrm{GL}(1)$-twists of $\mathrm{GL}(m)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions. The proof is inspired by Siegel's celebrated lower bound for Dirichlet $L$-functions at $s=1$. I will also discuss two applications briefly. Joint work with Jesse Thorner.

We use the Petersson trace formula over totally real number fields as a delta symbol to prove a t-aspect subconvexity bound for L-functions of Hilbert modular forms. This seems to be the first instance of using a delta symbol approach over number fields for proving a subconvexity result. This is an ongoing work, joint with Naomi Tanabe.

In this talk, I'll describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make these connections explicit. Recently, a new tableau formula was found for the modified Macdonald polynomial $\widetilde{H}_{\lambda}$ in terms of a queue inversion statistic that is naturally related to the dynamics of the TAZRP. We give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ using the same queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle, and the queue inversion statistic encodes to the dynamics of the ASEP.  Our tableaux are in bijection with Martin's multiline queues, from which we obtain an alternative multiline queue formula for $P_{\la...