Postdoctoral Colloquium

u_{xx} + \frac{1}{x} u_x = \sinh u

\]

that are real on \((0, \infty)\). They are parametrized by the monodromy parameter \( p \in \overline{\mathbb C} \), \( |p|>1 \), and an additional real parameter \( s^{\mathbb R} \) when \( p=\infty \). We describe the large \(x\) asymptotics of every solutions for fixed \(p\), as well as the transition between singular solutions (\( |p|<\infty \)) and smooth solutions (\( p=\infty \)), as \( x \to +\infty \) and \( p \to \infty \) given that \( 2\Im(p)=-s^{\mathbb R} \).

 

Abstract: An important tool for linking arithmetic data and $L$-functions is the use of $p$-adic methods, and in particular, the construction and study of $p$-adic $L$-functions. These functions are proven to be useful in connecting properties of arithmetic objects with special $L$-values, with applications to interesting problems such as the Birch and Swinnerton-Dyer conjecture.

 

For a prime $p$, $p$-adic $L$-functions can be viewed as a power series with coefficients in a local field or in the ring of integers of a local field. These functions satisfy certain growth properties and interpolate special values of complex L-functions. In this talk, I will present the construction of p-adic Asai L-functions with unbounded coefficients attached to cuspidal Bianchi modular forms at non-ordinary primes. The talk will begin with a brief review of p-adic L-functions, Bianchi modular forms, and their Asai L-functions.

 

Abstract: Accurately simulating wave propagation can require substantial computational resources. One way to reduce this cost is to build information about the governing partial differential equation directly into the numerical approximation space. Trefftz methods follow this approach by using functions that solve the equation exactly on each mesh element. For equations with variable coefficients, however, such functions are generally unavailable in closed form. Quasi-Trefftz spaces overcome this difficulty by using polynomial functions that satisfy the equation to a prescribed order in a local Taylor expansion. I will introduce the main ideas behind these spaces and their construction, then discuss applications to the time-harmonic Maxwell equations, which describe electromagnetic wave propagation at a fixed frequency. The talk will include theoretical results and numerical examples.