2026 Graduate Student Mathematical Modeling Camp

Title: Dynamic Optimal Fire Containment 
Abstract: Our fictional fire‑analytics startup aims to improve the responsiveness and cost‑effectiveness of wildfire containment. We have implemented several quantitative heuristics to advise fire authorities on containment strategies, but we now wish to strengthen our portfolio with methods based on first‑principles modeling that can incorporate multiple data modalities.

Fire containment and fire modeling are weakly coupled processes. Containment strategies naturally rely on accurate fire modeling, as the goal is to surround a spreading process with a bounded continuous curve such that the burn is bounded in infinite time. Fire modeling alone is difficult in real scenarios because it depends sensitively on geography, vegetation, weather, and climate across multiple timescales. Therefore, we are amenable to minimal PDE toy models that yield provably optimal containment paths , as well as “one‑step” algorithms based on more sophisticated toy models that incorporate two or more significant factors such as vegetation, wind, temperature, or humidity, etc. We also seek a strategy and mathematical treatment for handling a "kitchen sink" model which can adapt to an evolving situation with uncertainty incorporated—for example, Kalman or other filtering based methods. A full implementation or information of a historical case study is welcome but not expected.

In summary, a successful group will:

• Implement a diffusive "fuel only" fire spread model (e.g. an SIR or Fisher-KPP like model) and outline a mathematical model for optimal containment, possibly with an analytical solution, assuming a point source and zero initial containment.
• Propose a PDE model for fire spread which incorporates at least one additional modeling aspect with numerical simulation, and a numerical scheme (heuristic or analytic) for updating a containment path in time with numerical experiments.
• Outline method(s) for modeling and parameter estimation in time with uncertainty, e.g. Kalman filters or other filtering methods and how they would be applied in this context.

Optionally, the group could incorporate real fire spread data and possibly a case study of fire spread and containment and compare their spectrum of their models to the historical event. Any partial work in this direction is welcomed. We also welcome participants revisiting our implicit assumptions or exploring tangential directions.

Title: Modeling and Stabilization of Transport-Dominated Flows
Abstract: Many physical systems are governed by transport-dominated partial differential equations (PDEs), where advection significantly outweighs diffusion. Examples include atmospheric flows on the rotating sphere (our planet Earth), pollutant transport, and tracer dynamics in oceans. Classical numerical discretizations of these systems often produce nonphysical spurious oscillations and/or loss of positivity of some quantities of interest. To mitigate these, typical ad-hoc numerical fixes are introduced, such as artificial diffusion.

In this project, we consider the problem of transporting a scalar quantity (e.g., a tracer concentration) on a curved domain such as the sphere. The goal is to design and analyze numerical models that balance three competing objectives: stability (suppression of oscillations), accuracy (with minimal numerical diffusion), and physical consistency (e.g., conservation or positivity).

Participants will explore and compare different modeling approaches, including:

diffusion or hyperdiffusion terms,
residual-based stabilization methods (e.g., streamline-aligned corrections),
flux limiting or positivity-preserving techniques.

Key questions include:

How can we quantify and reduce numerical oscillations in transport-dominated regimes?
What trade-offs arise between stability, accuracy, and performance?
Which methods are better at preserving important physical properties such as positivity or invariants?

This project involves formulating mathematical models, implementing numerical methods (most likely in high-level languages, such as Julia or Python), and testing them on benchmark problems such as solid-body rotation or the shallow-water equations on the sphere.

Title: A Kinetic Monte Carlo (kMC) Model for the Dynamics of Infectious Diseases
Abstract: We will explore a Monte Carlo simulation approach for modeling the spread of infectious diseases. We will start with a model built for COVID-19. Our goal is to abstract the model to be applicable to a wider range of infectious diseases and time-permitting apply it to the hantavirus.