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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...

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of alge...

In this talk, the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice is revisited. A central role is played by a quantum oscillator-weighted five vertex model, which features an unusual weight conservation distinct from the conventional one. This approach clarifies the interrelations among several known results and refines their derivations, including the multiline queue construction and matrix product formulas. (Joint work with Masato Okado and Travis Scrimshaw)

The quantum K-theory of the flag variety is a ring defined by introducing a quantum product to the K-theory of the flag variety. Under appropriate localization, it is known that the following three rings (i), (ii), and (iii) are isomorphic, and this property allows for a detailed investigation of each ring: (i)the coordinate ring of the phase space of the relativistic Toda lattice, (ii) the quantum equivariant K-theory of the flag variety, and (iii) the K-equivariant homology ring of the affine Grassmannian.

The isomorphism between (i) and (ii) is derived from the Lax formalism of the relativistic Toda lattice [Ikeda-Iwao-Maeno]. The isomorphism between (ii) and (iii) is referred to as the K-Peterson isomorphism [Lam-Li-Mihalcea-Shimozono, Kato, Chow-Leung, Ikeda-Iwao-Maeno]. In this talk, I will outline how techniques from classical integrable systems, such as the construction of algebraic solutions and Bäcklund transformations, are applied to the study of geometry. This talk is ba...

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.