For A an abelian variety defined over a number field, a conjecture of Serre predicts that the set S of primes of ordinary reduction for A has positive density. If A is an elliptic curve without CM, Serre proved that S has density 1, and Elkies showed that, if L admits a real embedding, then the complement of S, that is the set of supersingular primes, is infinite.

In this talk, I will discuss generalizations of both Serre’s and Elkies’s theorems to abelian varieties of type IV, that is with multiplication by a CM field, which are parametrized by unitary Shimura curves.

This talk is based on joint work in progress with Victoria Cantoral-Farfan, Wanlin Li, Rachel Pries, and Yunqing Tang.

We study the dimensions of the eigenspaces for the Atkin-Lehner involutions acting on spaces of modular forms $M_k(\Gamma_0(Np))$, with the additional constraint imposed by fixing the mod $p$ Galois representation attached to the eigenforms. For this purpose we establish isomorphisms up to semisimplification between certain Hecke modules in characteristic $p$, generalizing Serre’s work relating modular forms mod $p$ to quaternion algebras. These isomorphisms are in turn obtained via a delicate study of congruences in the trace formulas for Hecke and Atkin-Lehner operators.

We discuss applications of (a part of) the Iwasawa main conjecture to the non-triviality of Kato’s Kolyvagin systems and the structure of Selmer groups of elliptic curves over the rationals without any rank restriction.

The Rapoport-Zink space for $\mathrm{GSp}(4)$ is a local counterpart of the Siegel threefold. Its $\ell$-adic etale cohomology is equipped with an action of the product of three groups: $\mathrm{GSp}_4(\mathbb{Q}_p)$, its non-trivial inner form $J(\mathbb{Q}_p)$, and the Weil group of $\mathbb{Q}_p$. This action is expected to be strongly related with the local Langlands correspondence for $\mathrm{GSp}_4$ and $J$. In this talk, I will explain how the $\mathrm{GSp}_4(\mathbb{Q}_p)$-supercuspidal part of the cohomology is described by the local Langlands correspondence. If time permits, I will also give some observations on the $\mathrm{GSp}(6)$ case.

The Ichino-Ikeda conjecture is an explicit relation between the central L-value and squares of a certain period of automorphic forms.

This conjecture has been established by Beuzart-Plessis, Yifeng Liu, Wei Zhang, Xinwen Zhu, Chaudouard and Zydor for unitary groups.
I will report on a joint work in progress with Michael Harris and Ming-Lun Hsieh on the construction of $p$-adic L-functions for $U(3)\times U(2)$ via the Ichino-Ikeda conjecture.

In this talk, we will present our recent results on the local multiplicity formula of strongly tempered spherical subgroups. In particular, we formulate the epsilon dichotomy conjecture for those spherical subgroups and prove this conjecture for the tempered representations of endoscopy type. Globally, the corresponding automorphic periods are related to the central values of certain L-functions of symplectic type. The conjectural Ikeda–Ichino type formulas are given. This is a joint project with Chen Wan at Rutgers University, Newark.

Hassett–Tschinkel and Benoist–Wittenberg recently introduced a new rationality obstruction that refines the classical the Clemens–Griffiths intermediate Jacobian obstruction to rationality, and exhibited its strength by showing that this new obstruction characterizes rationality for intersections of two quadrics. We show that this phenomenon does not extend to all geometrically rational threefolds. We construct examples of conic bundle threefolds over $\mathbb{P}^2$ that have no refined intermediate Jacobian obstruction to rationality, yet fail to be rational. This is joint work with S. Frei, L. Ji, S. Sankar, and I. Vogt.

We find new polynomial upper bounds for the size of nodal sets of eigenfunctions when the Riemannian manifold has a Gevrey or quasianalytic regularity.

The study of eigenfunctions of Laplacians lies at the interface of several areas of mathematics, including analysis, geometry, mathematical physics and number theory. These special functions arise in physics and in partial differential equations as modes of periodic vibration of drums and membranes.

A fundamental question surrounding Laplace-Beltrami eigenfunctions targets their concentration phenomena, via high-energy asymptotics or high-frequency behaviour. One popular approach to this question involves studying the growth of the $L^p$ norms of these eigenfunctions on the ambient manifold, or on submanifolds thereof. What can one say about the behaviour of eigenfunctions on rougher sets? We will discuss answers to this question, based on joint work with Suresh Eswarathasan.

One guiding principle is the idea that Laplacian eigenfunctions corresponding to larger frequencies should oscillate more. The same principle should then also be true, in some form, for linear combinations of high-frequency eigenfunctions (in one dimension, this is Sturm-Liouville theory). Recent progress on this question is based on the notion of optimal transport and a very simple idea which we formalize: if it’s easy to buy milk, then there are must be many supermarkets (and, conversely, if there are only few supermarkets at least some people have to travel a large distance to buy milk). This turns into a geometric inequality that is interesting in its own right.

An important heuristic principle in the study of eigenfunctions of the Laplace-Beltrami operator is that their properties should resemble those of polynomials. In this light, I will discuss oscillations and zeros for linear combinations of Laplace eigenfunctions on Riemannian manifolds. In particular, I will prove that zeros become dense in the manifold if not too many eigenfunctions are summed. Time permitting, I will mention related open questions on eigenfunctions sums.

In this talk, I will discuss joint work with M. Engelstein, L. Li, and S. Mayboroda, where we introduce the notion of Dahlberg-Kenig-Pipher operators in the context of domains $\Omega$ with low dimensional boundaries. We show that when the boundary of the domain is uniformly rectifiable with small constant, then elliptic measure $\omega$ associated to this domain is an $A_\infty(d\sigma)$ weight with small constant. As consequence, we show that for $C^1$ domains, $\log (d\omega/d\sigma) \in \mathrm{VMO}$. One of the main difficulties in this context is the lack of outer graphical approximations to $\Omega$, since the exterior of $\Omega$ can be empty.

Given a bounded domain $\Omega \subset \mathbb R^n$, one says that the $L^p$-regularity problem is solvable for the Laplace equation in $\Omega$ if, given any continuous function $f$ defined in $\partial \Omega$ and the harmonic extension $u$ of $f$ to $\Omega$, the non-tangential maximal function of the gradient of $u$ can be controlled in $L^p$ norm by the tangential derivative of $f$ in $\partial\Omega$. In my talk I will review a joint result with Mourgoglou where we proved the $L^p$-regularity for more general domains, and a more recent result (with the additional collaboration of Poggi) where we extend this result to elliptic PDE’s in divergence form.

In this talk I will present spectral features of the Dirac operator with infi nite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szegö type inequality as well as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given.

This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmieres-Bonafos.

In this talk, we will explore the celebrated Almgren’s monotonicity formula. This beautiful result with far-reaching consequences states that if u is harmonic in the unit ball, then a certain frequency function $N(r)$ is non-decreasing. Moreover, $N(r)=k$ for all $r<1$ if, and only if, $u$ is homogeneous of degree $k$. We will then discuss some of the many applications of this formula, and recent developments connected to it.