Talks

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 modern astronomy, artificial intelligence (AI) is increasingly utilized to analyze large volumes of data, significantly reducing the need for human computational resources and time. Machine learning (ML) techniques are at the forefront of revealing astronomical mysteries by analyzing observed data. Here, we will introduce the application of machine learning to Intensity Interferometry (II) data for high-resolution optical astronomy, aiming to overcome the limitations of traditional image reconstruction methods. In this presentation, we demonstrate successful image reconstruction of a fast-rotating star using conditional Generative Adversarial Networks (cGANs), a supervised machine learning approach. Simulations of II are based on an assembly of four telescopes similar to existing arrays. However, the sensitivity of the signal and high resolution are expected to improve with additional baselines. It makes the current and future Cherenkov Telescope Array Observatory (CTAO) an ideal candidate for II applications. Our approach is highly relevant and innovative, addressing key challenges in phase reconstruction and proposing novel solutions that could revolutionize high-resolution imaging in astronomy.

Even with dense sampling of the uv plane intensity correlations only contain half the information required to reconstruct an image. Intensity correlations do contain the full information of the image power spectrum and therefore of the image 2-point correlation function. With some practice one can gain intuitive understanding in interpreting 2-point correlation function "images". This is illustrated with both toy examples and modeling of real astronomical images. In some assumptions one can even interpret these 2-point correlation function "images" with only a few baselines.