The APDE seminar on Monday, 10/3, will be given by our own Mengxuan Yang in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Wave trace and resonances of the Aharonov–Bohm Hamiltonian

Abstract: I want to discuss propagation of singularities of the
magnetic Hamiltonian with singular vector potentials, which is related
to the so-called Aharonov–Bohm effect. In addition, I shall discuss a
Duistermaat–Guillemin type trace formula, as well as some
applications to scattering resonances in this setting.

The APDE seminar on Monday, 9/26, will be given by Mihai Tohaneanu (U Kentucky) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: The weak null condition on Kerr backgrounds

Abstract: Understanding global existence for systems satisfying the weak null condition plays a crucial role in the proof of stability of Minkowski in harmonic coordinates. In this talk I will present a proof of global existence for a semilinear system of equations on Kerr spacetimes satisfying the weak null condition. This is joint work with Hans Lindblad.

The APDE seminar on Monday, 9/19, will be given by Thibault Lefeuvre (Sorbonne U) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: On isospectral connections

Abstract: Kac’s celebrated inverse spectral question “Can one hear the shape of a drum?” consists in recovering a metric from the knowledge of the
spectrum of its Laplacian. I will discuss a very similar question on negatively-curved manifolds, where the word “metric” is now replaced by “connection” on a vector bundle. This problem turns out to be very rich and connects unexpectedly to two other a priori unrelated fields of
mathematics:
1) in dynamical systems: the study of the ergodic behaviour of partially hyperbolic flows obtained as isometric extensions of the geodesic flow (over negatively-curved Riemannian manifolds);
2) in algebraic geometry: the classification of non-trivial algebraic maps between spheres.

Using this relation, I will explain a positive answer to Kac’s inverse spectral problem for connections under a low rank assumption. Joint work with Mihajlo Cekić.

The APDE seminar on Monday, 9/12, will be given by Nets Katz (Caltech) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: A proto-inverse Szemer\’edi Trotter theorem

Abstract: The symmetric case of the Szemer\’edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of $N$ point and $N$ lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\’edi Trotter is densely related to a successful instance of the recipe. We discuss the relation of this statement to the inverse Szemer\’edi Trotter problem. (joint work in progress with Olivine Sillier.)