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We will talk about Drinfeld modules, and how they compare to elliptic curves for algorithms and computations.

Drinfeld modules can be seen as function field analogues of elliptic curves. They were introduced in the 1970's by Vladimir Drinfeld, to create an explicit class field theory of function fields. They were instrumental to prove the Langlands program for GL2 of a function field, or the function field analogue of the Riemann hypothesis.

Elliptic curves, to the surprise of many theoretical number theorists, became a fundamental computational tool, especially in the context of cryptography (elliptic curve Diffie-Hellman, isogeny-based post-quantum cryptography) and computer algebra (ECM method).

Despite a rather abstract definition, Drinfeld modules offer a lot of computational advantages over elliptic curves: one can benefit from function field arithmetics, and from objects called Ore polynomials and Anderson motives.

We will use two examples to highlight the practicality of Drinfeld modules computations, and mention some applications.

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

Let $\rm{ord}_p(a)$ be the order of $a$ in $( \mathbb{Z} / p \mathbb{Z} )^*$. In 1927, Artin conjectured that the set of primes $p$ for which an integer $a\neq -1,\square$ is a primitive root (i.e. $\rm{ord}_p(a)=p-1$) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk we will study the behaviour of $\rm{ord}_p(a)$ as $p$ varies over primes, in particular we will show, under GRH, that the set of primes $p$ for which $\rm{ord}_p(a)$ is “$k$ prime factors away” from $p-1$ has a positive asymptotic density among all primes except for particular values of $a$ and $k$. We will interpret being “$k$ prime factors away” in three different ways, namely $k=\omega(\frac{p-1}{\rm{ord}_p(a)})$, $k=\Omega(\frac{p-1} {\rm{ord}_p(a)})$ and $k=\omega(p-1)-\omega(\rm{ord}_p(a))$, and present conditional results analogous to Hooley's in all three cases and for all integer $k$. From this, we will derive conditionally the expectation for these quantities. Furthermore we will provide partial unconditional answers to some of these questions. This is joint work with Leo Goldmakher and Greg Martin.

Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. Indeed, the behaviour of Euler products inside the critical strip is very closely related to several major problems in number theory including the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip. We will end by giving applications of these results to questions related to Chebyshev's bias.

In this talk, we will discuss the question of establishing CLTs for empirical entropic optimal transport when choosing the regularisation parameter as a decreasing function of the sample size. Importantly, decreasing the regularisation parameter enables estimating the population unregularized quantities of interest. Furthermore, we will show an application to score function estimation, a central quantity in diffusion models, and will discuss parallels with recent work on the estimation of transport maps based on the linearization of the Monge—Ampère equation.

The Polya group of a number field K is a specific subgroup of the ideal class group Cl(K) of K, generated by all classes of Ostrowski ideals of K. In this talk, I will discuss the equality Po(K)=Cl(K) in two directions. First, we will see this equality happens for infinitely many "non-Galois fields'' K. Accordingly, I prove two conjectures presented by Chabert and Halberstadt concerning the Polya groups of some families of non-Galois fields. Then, I present some "finiteness theorems" for the equality Po(K)=Cl(K) for some families of "Galois" fields K obtained in joint work with Amir Akbary (University of Lethbridge).

I am going to discuss various results on moments of symmetric square L-functions and some of their applications. I will mainly focus on a recent result of R. Khan and M. Young and our improvement of it. Khan and Young proved a mean Lindelöf estimate for the second moment of Maass form symmetric-square L-functions $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the short interval of length $G >> |t_j|^{(1 + \epsilon)/t^{(2/3)}}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ as long as $|t_j|^{(6/7 + \delta)} << t < (2 - \delta)|t_j|$. We obtain a mean Lindelöf estimate for the same moment in shorter intervals, namely for $G >> |t_j|^{(1 + \epsilon)/t}$. As a corollary, we prove a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the interval $|t_j|^{(2/3 + \delta)} << t << |t_j|^{(6/7 - \delta)}$. This is joint work with Olga Balkanova.

Previously we found certain convolution sums of divisor functions arising from physics yield Fourier coefficients of modular forms. In this talk we will discuss the limitations of the current proof of these formulas. We will also explore the connection with the Petersson and Kuznetsov Trace Formulae and the possibility of extending these formulas to other cases. The work mentioned in this talk is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko, and Don Zagier.

Measures provide valuable insights into long-term and global behaviors across a broad range of dynamical systems. In this talk, we present our recent research efforts that employ measure theory and optimal transport to tackle core challenges in system identification, parameter recovery, and predictive modeling. First, we adopt a PDE-constrained optimization perspective to learn ODEs and SDEs from slowly sampled trajectories, enabling stable forward models and uncertainty quantification. We then use optimal transportation to align physical measures for parameter estimation, even when time-derivative data is unavailable. Our second result extends the celebrated Takens’ time-delay embedding, a foundational result in dynamical systems, from state space to probability distributions. It establishes a robust theoretical and computational framework for state reconstruction that remains effective under noisy and partial observations. Finally, we show that by comparing invariant measures in time-delay coordinates, one can overcome identifiability challenges and achieve unique recovery of the underlying dynamics even though it is not generally possible to uniquely reconstruct dynamics using invariant statistics alone. Collectively, these works demonstrate how measure-theoretic and transport-based methods can robustly identify, analyze, and forecast real-world dynamical systems and the great research potential of measure-theoretic approaches for dynamical systems.