1. Principal Eigenvalues, Geometry, and Shape Optimization

A major part of my current research concerns principal eigenvalue problems for fully nonlinear elliptic operators, especially Pucci extremal operators. I am interested in understanding how the geometry of a domain influences the corresponding principal eigenvalue.

This direction includes shape optimization problems, Pólya-type spectral questions, symmetry, eigenvalue minimization, and geometric inequalities for fully nonlinear operators. The long-term goal is to understand how classical spectral optimization ideas change when the Laplacian is replaced by a fully nonlinear non-divergence operator.

2. Fully Nonlinear and Semilinear Elliptic PDEs

This direction contains much of my past and ongoing work on nonlinear elliptic equations and systems. I study existence, uniqueness, multiplicity, bifurcation, regularity, and boundary behavior of positive solutions.

A substantial part of this work concerns fully nonlinear elliptic equations involving Pucci extremal operators, singular nonlinearities, natural gradient growth, weakly coupled systems, Lane–Emden type systems, and semipositone problems. The methods include viscosity solution theory, comparison principles, sub- and supersolution techniques, fixed point methods, a priori estimates, Liouville-type theorems, and bifurcation analysis.

3. Optimal Control and Mathematical Ecology

A developing direction of my research concerns optimal control problems arising in mathematical ecology and population dynamics. I study spatial models for optimal harvesting, sustainable resource use, grazing effects, density-dependent dispersal, and nonlinear boundary migration.

The typical models involve elliptic or parabolic reaction-diffusion equations where the control represents harvesting effort and the state variable represents population density. A key theme is the interaction between harvesting, population distribution, and movement across ecological or policy boundaries.

In this direction, I am particularly interested in the feedback mechanism: harvesting changes density; density changes boundary migration; and boundary migration changes the optimal harvesting strategy.

4. HJB Equations, Pucci Operators, and Reinforcement Learning

A new long-term direction of my research is to understand the mathematical connection between fully nonlinear PDEs, Hamilton–Jacobi–Bellman equations, optimal control, and reinforcement learning.

The guiding bridge is dynamic programming, Bellman equations, HJB equations, viscosity solutions, and reinforcement learning. This direction is naturally connected to Pucci operators and fully nonlinear PDEs. In stochastic control, robust control, and differential games, HJB, and Isaacs equations often lead to fully nonlinear second-order equations.

My long-term goal is to study HJB equations, viscosity solutions, monotone numerical schemes, and learning-based approximation methods, with an eye toward the mathematical foundations of reinforcement learning and artificial intelligence.