Pure & Applied Mathematics Seminar

Weighted Alexandrov-Fenchel and Minkowski inequalities in hyperbolic space

Alexandrov-Fenchel and Minkowski inequalities are fundamental in differential geometry and convex geometry. In recent years, their weighted versions have attracted substantial attention, especially in the space forms setting. In this talk, I will first briefly survey weighted geometric inequalities in space forms. I will then present my recent joint work with Yong Wei on sharp weighted Minkowski and Alexandrov-Fenchel type inequalities for closed, static-convex hypersurfaces in hyperbolic space. These inequalities compare weighted curvature integrals with quermassintegrals and include weights such as $\Phi^k$ and $\cosh^k r$ for any $k\geq 1$, where $\Phi=\cosh r -1$. Our results unify and extend a number of previously known inequalities in the literature. This is joint work with Yong Wei.

Boundary-forced linear systems in surface gravity waves

This talk examines two types of boundary forcing in surface gravity waves. In the first, the forcing is prescribed: an atmospheric pressure disturbance moves across the ocean surface, and the water responds passively. In the second, the forcing is coupled: a floating elastic body interacts with the surrounding fluid, creating a two-way feedback between the wave field and the body motion.

The prescribed framework is applied to the global tsunami generated by the 2022 Hunga Tonga eruption. The atmospheric disturbance is modelled as a simple, propagating pressure wave, which captures the leading behaviour of the tsunami with accuracy comparable to more complex numerical models. The results show that tsunami amplitudes far from the source are governed primarily by the instantaneous structure of the pressure disturbance, rather than its early-time evolution.

The coupled framework is used to study interactions between ocean waves and floating icebergs. A numerical model is developed that combines fluid motion with the elastic response of the iceberg, allowing realistic geometries to be analysed. This approach is used to infer iceberg stiffness from observed vibrations, and to investigate how transient wave pulses generate flexural stresses. These stresses are found to be highly sensitive to both iceberg thickness and wave characteristics.

Together, these examples show how linear wave models can describe phenomena ranging from basin-scale tsunamis to localised hydroelastic deformation, and highlight the contrast between one-way forcing and fully coupled interactions.

Dynamics and Periodicity Conditions for the Integrable Boltzmann System

Consider a simple mechanical system proposed by Boltzmann in the 1860's: a massive particle moves in a gravitational field with a linear boundary between the particle and the center of gravity. Reflections off the boundary are absolutely elastic and obey the billiard reflection law: angles of incidence and reflection are congruent. This system was recently shown by Gallavotti and Jauslin to have a second integral of motion. We study its dynamics and prove the existence of caustics, Cayley-type periodicity conditions, and more. This is joint work with Milena Radnović (University of Sydney).

The monopolist's free boundary problem in the plane: an excursion into the economic value of private information

The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.

If the monopolists costs grow quadratically with the product type we show that a partially smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Based on works with Kelvin Shuangjian Zhang, Cale Rankin, and Lucas O'Brien in various combinations: 1) Math. Models Methods Appl. Sci. 34 (2024) 2351-2394; 2) J. Convex Anal. (Rockafellar 90 Issue), 32 (2) (2025) 579-584; 3) arXiv 2303.04937; 4) arxiv 2412.15505; 5) arXiv 2603.14100.

Boxes and Balls and RSK, oh my!

The Korteweg-de Vries (KdV) equation is a nonlinear PDE that reflects an integrable system. It has a ultradiscrete shadow called the box-ball system, which is a discrete dynamical system with a very simple (reversible) time evolution. Despite its simplicity, it has a very rich structure connected with quantum mechanics. In this talk, we will describe the box-ball system in two different ways: using its classical definition and a version of the Robinson-Shensted-Knuth (RSK) correspondence. We will describe how this relates to Kashiwara crystals and conclude with a generalization. This is based on joint work with Takashi Imamura, Matteo Mucciconi, and Tomohiro Sasamoto

Approximation of stochastic semi-linear PDEs on the unit sphere

We study the numerical approximation of a class of semi-linear stochastic partial differential equations posed on the unit sphere. The equations considered combine diffusion, nonlinear reaction terms, and random forcing, and arise naturally in models of spatially distributed phenomena evolving on spherical domains. We construct a fully discrete numerical method by combining a Galerkin approximation based on spherical radial basis functions with a suitable time-stepping scheme. Under appropriate assumptions on the linear operator, the nonlinearity, the stochastic forcing, and the initial data, we establish stability and strong convergence of the proposed approximation. The error analysis identifies the contributions arising from the spatial discretisation, the temporal discretisation, and the truncation of the driving noise. The results extend existing numerical analyses of linear stochastic equations on the sphere to a semi-linear setting and provide a framework for approximating stochastic reaction–diffusion equations on spherical geometries.

Like Squeezing Water From a Stone; Phototrophic Subaerial Biofilms Growing on Porous Substrata

Subaerial biofilms (SABs) are microbial communities that colonize exposed surfaces, both built and natural, frequently powered by photosynthesis. Their function is tightly coupled to environmental conditions which are often harsh, particularly with respect to water availability. Even without exposure to regular rainfall, it is hypothesized that overnight radiative cooling allows SABs to utilize condensation, i.e., dew, to gather high activity water, in order to be productive during the early morning before evaporation becomes significant. Further, it is proposed that SABs are able to effectively seal substratum pores to store condensed (or precipitated) water to offset evaporation for some time. A physical theory, based on the Hertz-Knudsen equation for describing evaporation/condensation at liquid-gas interfaces, is constructed. To test the theory, the derived mathematical model of water activity in SAB-stone systems is compared to prior laboratory data of water transport in limestone samples with and without SAB growth. We show that the model is able to successfully explain differences in the data between the two cases. Implications for SAB growth in the environment are discusses.