Past mathematics seminar

2025 Semester 1
WeekDateTimeSpeaker
2 Monday 3 March 2025 12:00-13:00 Dmitry Pelinovsky
Traveling waves and breathers in the nonlocal NLS models

A nonlocal derivative nonlinear Schrodinger equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero-Moser-Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove linear and nonlinear stability of the nonzero constant background under some restrictions. For both versions, we characterize the traveling periodic wave solutions by using Hirota's bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the stable background.

4 Friday 21 March 2025 11:00-12:00 Chien-Hua Chen
Function field arithmetic and irreducibility of division polynomials associated to Drinfeld modules.

In 1993, Masser and W\"ustholz proved a famous result on existence of isogeny, with degree bounded by an explicit formula,  between two isogenous Elliptic curves. Building upon this achievement, they subsequently employed the isogeny estimation to establish an explicit bound on the irreducibility of division polynomials associated to elliptic curves over a number field without complex multiplication (CM).  This bound is then used to deduce a bound on the surjectivity of mod l Galois representation associated to elliptic curves over a number field without CM. Inspired by elliptic curve theory, David and Denis introduced an isogeny estimate applicable to Anderson t-modules, specifically deriving an isogeny estimate for Drinfeld modules over global function fields. This raises the question of whether the Masser-Wüstholz strategy can be adapted to obtain a similar bound on the irreducibility of division polynomials associated with Drinfeld modules without CM. However, the Masser-W\"ustholz strategy can not be applied directly to the context of Drinfeld modules. Thus we develop an alternative strategy, which involves heights of isogenous Drinfeld modules studied by Breuer, Pazuki, and Razafinjatovo, to deduce a function field analogue of Masser-W\"ustholz irreducibility bound. In this talk, I will begin by reviewing the theory of elliptic curves and the Masser-Wüstholz theorem. Then, I will compare classical elliptic curve theory with function field arithmetic and Drinfeld modules. Finally, I will explain how we adapt the Masser-Wüstholz strategy to the context of Drinfeld modules.

Break 1 Monday 14 April 2025 11:00-12:00 Yong Wei
Curvature measures and volume preserving curvature flows

Volume preserving mean curvature flow was introduced by Huisken in 1987 and it was proved that the flow deforms convex initial hypersurface smoothly to a round sphere. This was generalized later by McCoy in 2005 and 2017 to volume preserving flows driven by a large class of 1-homogeneous curvature functions. In this talk, we discuss the flows with higher homogeneity and describe the convergence result for volume preserving curvature flows in Euclidean space by arbitrary positive powers of k-th mean curvature for all k=1,...,n. As a key step, the curvature measure theory will be used to prove the Hausdorff convergence to a sphere. We also discuss some generalizations including the flows in the hyperbolic setting and the flows in the anisotropic setting.

8 Friday 9 May 2025 11:00-12:00 Florian Breuer
Adventures in the Parallel Worlds of Number Theory

I will explain the parallel worlds of number fields and function fields, and my recent work on modular polynomials, modular forms as well as parity of quadratic units and the Cohen-Lenstra-Martinet heuristics.

9 Friday 9 May 2025 11:00-12:00 Ji Li
Flag Hardy space theory—an answer to a question by E.M. Stein.

The theory of multi-parameter flag singular integral originates from the study of ¯∂-problem on the Heisenberg group by D. Phong and E.M. Stein. In 1999, E. M. Stein asked “What is the Hardy space theory in the flag setting?” in the conference at Washington University in Saint Louis to celebrate the 70th birthday of G. Weiss. In our recent work, we established a complete flag Hardy space theory on the Heisenberg group, including characterisations via Littlewood–Paley area function, square function, non-tangential and radial maximal functions, atoms, and the flag Riesz transforms. It provided a unified approach for proving the Lp boundedness of different types of singular integrals, and led to the endpoint L log(L) → L^{1,∞} estimates. The representations of flag BMO functions are also provided.

10 Friday 16 May 2025 10:30-11:30 Josef Dick
Quasi-Monte Carlo methods for PDEs with random coefficients

Mathematical models often contain uncertainty in parameters and measurements. In this talk we focus on partial differential equations where some parameters are modelled by random variables. The main example comes for the diffusion equation where the diffusion parameters is modelled as a random field which randomly fluctuates around a given mean. To sample the random fluctuations we use quasi-Monte Carlo methods. We provide an elementary introduction to PDEs using the diffusion equation with random coefficients and discuss some of the newer methods towards the end.

11 Friday 23 May 2025 11:00-12:00 Erik Neefjes
A neural-network surrogate Bayesian algorithm for the Helmholtz inverse-shape problem

We present a novel approach to the classical inverse problem of reconstructing the shape of scatterers from noisy far-field data. The far-field data in our model is generated by multiple incident waves striking an impenetrable scatterer. This data can be mathematically modeled using the Helmholtz equation in the unbounded region outside the scatterer. For reconstruction, our method employs a Bayesian framework that incorporates data and utilizes Markov Chain Monte Carlo (MCMC) sampling. To address the computational challenges posed by the high-dimensional prior space, we introduce a physics-property informed (PPI) neural network (NN) surrogate model for the forward problem. The PPI component of the algorithm facilitates the training of the NN using far-field data from just one incident wave. Subsequently, the PPINN-surrogate model takes advantage of rotational symmetries in the prior space, allowing for efficient evaluation of the forward model across several incident wave directions. We showcase our method’s effectiveness by demonstrating excellent reconstruction for a range of test scatterer shapes.

12 Friday 30 May 2025 11:00-12:00 Chris Lustri
Stability of breather solutions to the discrete Nonlinear Schrodinger equation

In this talk, I will show how ideas from exponential asymptotics, resurgence, and complex analysis can be used to explain the stability properties of breathers (or stationary waves that are periodic in time) in discrete systems. I first introduce the history and ideas of exponential asymptotics and Stokes' phenomenon. I will then introduce the discrete analogue of the famous nonlinear Schrodinger equation explain why the only breather solutions that can exist are site-centered (on-site) or midpoint-centered (inter-site) solutions, before determining the asymptotic behaviour of the (exponentially) small eigenvalues to determine the stability of these breathers. Finally, I will explain why these methods break down once long-range interactions are introduced to the system, and how this can be resolved by using Borel transform theory and conformal mapping.