Research

My work focuses on developing new intrinsic results in global analysis, with emphasis on problems that connect different areas of mathematics: differential geometry, functional analysis, Sobolev theory, partial differential equations, vector bundles, and variational methods.

I am especially interested in geometric formulations that make it possible to study PDE problems on Riemannian manifolds and vector bundles while keeping both the local structure and the global phenomena visible.

Main directions

Sobolev spaces on manifolds and vector bundles

I study intrinsic definitions of Sobolev spaces, norm equivalence, density results, Sobolev embeddings, Rellich–Kondrashov compactness, and trace theory in geometric settings.

Geometric integration by parts

A central direction of my work is the formulation of higher-order integration by parts identities on Riemannian manifolds with boundary and on vector bundles, including boundary terms and formal adjoints of covariant derivatives.

PDE on vector bundles

I am currently working on nonlinear elliptic problems on vector bundles, especially quasilinear equations and ground state solutions through variational methods.

Bounded geometry and noncompact analysis

I am interested in uniform estimates on noncompact manifolds, including bounded geometry, trace theorems, local-to-global constructions, and fractional Sobolev spaces.

Topology and realcompact spaces

I also maintain an interest in general topology, especially realcompact spaces and related extensions.