Starting with the Gauss-Bonnet formula : rigidity phenomena on bounded symmetric domains Ngaiming MOK (The University of Hong Kong, Hong Kong) Let $E$ be a compact Riemann surface of genus 1, and $Z$ be a compact Riemann surface of genus $≥ 2$. Then, every holomorphic map $f : E → Z$ is constant, as can be proven by contradiction by pulling back a nontrivial holomorphic differential on $Z$ which necessarily vanishes at some point. A metric version of the proof using the Gauss-Bonnet formula is more flexible, and a variation of the proof based on a Chern integral gives a Hermitian metric rigidity theorem, first established by the author in 1987 in the case of compact quotients $X\left\lceil := \Omega/\right\lceil$ of irreducible bounded symmetric domains $\mathrm{X}_{Γ} := \Omega/Γ$ of rank $≥ 2$ and then extended in the finite-volume case by To in 1989, which gives rigidity results on holomorphic maps from $X\lceil$ to Kähler manifolds of nonpositive holomorphic bisectional curvature, and geometric superrigidity results in the special cases of $Γ\G/K$ for $G/K$ of Hermitian type and of rank $≥ 2$ and for cocompact lattices $Γ ⊂ G$ via the use of harmonic maps and the $∂∂$-Bochner-Kodaira formula of Siu's in 1980. The Hermitian metric rigidity theorem was the starting point of the author's investigation on rigidityphenomena mostly on bounded symmetric domains $\Omega$ irreducible of rank $≥ 2$, but also, in the presence of irreducible lattices Γ ⊂ G := Aut0(Ω), on reducible $Omega$, and, for certain problems also on the rank-1 cases of n-dimensional complex unit balls Bn. The proof of Hermitian metric rigidity serves both (I)as a prototype for metric rigidity theorems and (II) as a source for proving rigidity results or making conjectures on rigidity phenomena for holomorphic maps. For type-I results the author will explain (1) the finiteness theorem on Mordell-Weil groups of universal polarized Abelian varieties over functionfields of Shimura varieties, established by Mok (1991) and by Mok-To (1993), (2) a Finsler metric rigidity theorem of the author's (2004) for quotients $XΓ := Ω/Γ$ of bounded symmetric domains Ω of rank $\ge2$ by irreducible lattices and a recent application by He-Liu-Mok (2024) proving the triviality of the spectral base when $XΓ$ is compact, (3) a rigidity result of Clozel-Ullmo (2003) characterizing commutants of certain Hecke correspondences on irreducible bounded symmetric domains Ω of rank $\ge 2$ via a reduction to a characterization of holomorphic isometries and the proof of Hermitian metric rigidity. For type-II results the author will focus on irreducible bounded symmetric domains Ω of rank $\ge2$ and explain (4) the rigidity results of Mok-Tsai (1992) on the characterization of realizations of Ω as convex domains in Euclidean spaces, (5) its ramification to a rigidity result of Tsai's (1994) on proper holomorphic maps in the equal rank case, (6) a theorem of Mok-Wong (2023) characterizing Γ-equivariant holomorphic maps into arbitrary bounded domains inducing isomorphisms on fundamental groups, and (7) a semi-rigidity theorem of Kim-Mok-Seo (2025) on proper holomorphic maps between irreducible bounded symmetric domains of rank $\ge2$ in the non-equirank case. Through Hermitian metric rigidity the author wishes to highlight the fact that complex differential geometry links up with many research areas of mathematics, as illustrated for instance by the aforementioned results (6) of Mok-Wong in which harmonic analysis meets ergodic theory and Kähler geometry, and (7) of Kim-Mok-Seo on proper holomorphic maps in which techniques of several complex variables cross-fertilize with those in $CR$ geometry and the geometric theory of varieties of minimal rational tangents ($VMRTs$).[-]

In this talk, I will present a work in progress with Matthieu Astorg and Lorena Lopez-Hernanz. We are interested in studying holomorphic endomorphisms of $C2$ which are tangent to the identity at the origin, and our goal is to understand how the dynamics changes when we perturb such maps. In particular, we generalize a result obtained by Bianchi and show a statement à la Lavaurs when the unperturbed map admits a basin parabolic centered in a characteristic direction, but it does not fix a complex line. I will recall the motivation and results in the one-dimensional case before moving to dimension 2.[-]

The delta symbol developed by Duke-Friedlander-Iwaniec and Heath-Brown has played an important role in studying rational points on hypersurfaces of low degrees. We present a two dimensional delta symbol and apply it to establish a quantitative Hasse principle for a smooth intersection of two quadratic forms defined over $Q$ in at least ten variables. The goal of these delta symbols is to carry out a (double) Kloosterman refinement of the circle method. This is based on a joint work with Simon Rydin Myerson and Pankaj Vishe.[-]

In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k\geq2$, consider the set of $k$-tuples of reduced fractions $\frac{a1}{q1} , . . . , \frac{ak}{qk} \in I$, where $I$ is an interval around 0. How many $k$-tuples are there with $\sum_{i} \frac{ai}{qi} \in \mathbb{Z} $? When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from “diagonal” terms, where the fractions $\frac{ai}{qi}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In joint work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.[-]

Let $\lambda$ be the Liouville function, defined by $\lambda(n) = (-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ (with multiplicity). This completely multiplicative Let $\lambda$ be the Liouville function, defined by $\lambda(n)=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ (with multiplicity). This completely multiplicative function is believed to exhibit pseudo-random statistical properties. For example, its partial sums are conjectured to obey the square-root cancellation estimate $\sum_{n \leq x} \lambda(n)=O\left(x^{1 / 2+\varepsilon}\right)$; this is equivalent to the Riemann Hypothesis.

The Fourier uniformity conjecture (a close cousin of the Chowla and Sarnak conjectures) concerns the pseudo-random behaviour of the Liouville function in short intervals. In 2023, Walsh proved that, for $\exp \left((\log X)^{1 / 2+\varepsilon}\right) \leq H \leq X$,

$
\sum_{X \lt x \lt 2X} \sup _{\alpha \in \mathbb{R}}\left|\sum_{x\lt n \lt x+H} \lambda(n) e(n \alpha)\right|=o(H X)
$

as $X \rightarrow \infty$. This non-correlation estimate is expected to hold for any $H=H(X)$ tending arbitrarily slowly to infinity with $X$ : this is the Fourier uniformity conjecture.

We improve on Walsh's range, proving that the Fourier uniformity conjecture holds for intervals of length $H \geq \exp \left((\log X)^{2 / 5+\varepsilon}\right)$.[-]

Following Zhang's breakthrough on bounded gaps between primes, much work has gone into improving upper bounds on the smallest integer which appears infinitely often as the gap between a given number of primes. Equidistribution estimates for primes in certain arithmetic progressions are a key ingredient of Zhang's proof and later work of Polymath. In this talk, I will highlight how bounded gaps and primes in arithmetic progressions are linked, and I will discuss obstacles and recent successes in using various types of old and new equistribution estimates to improve on the results of Polymath for bounded gaps between primes.[-]