Research

My research is broadly centered around topics from algebraic topology, differential geometry, geometric topology, and applied topology. My current areas of particular interest include, but are not limited to, the following:

• Symplectic topology and geometry (in particular, symplectic asphericity, Kähler geometry, and approximations of symplectic structures).
• Positive scalar curvature (mainly on manifolds with extra structure and its influence on large-scale topology and geometry).
• Symmetric products (mainly of 2-dimensional complexes).
• Numerical homotopy invariants (in particular, Lusternik–Schnirelmann category, sequential topological complexity, cohomological dimension, and versions of these invariants).
• Čech and Vietoris–Rips complexes (mainly of spheres).

I enjoy studying interactions between topics from the above areas. Most recently, I worked with aspherical (c-)symplectic structures and their Kähler analogues, and studied their scalar curvature and algebraic topology. I have examined the curvature, macroscopic dimensions, and symplectic asphericity of symmetric products of surfaces, and used these manifolds to answer some nuanced questions arising in the study of the former topics. Some of my work has been in using the cohomology of symmetric products of finite CW complexes to study some probabilistic variants of topological complexity and Lusternik–Schnirelmann category. On the other hand, I have also explored symmetric products of surfaces from the point of view of Kähler geometry and topological robotics.

My PhD advisor is Alexander Dranishnikov. I also enjoy working with Luca F. Di Cerbo and Henry Adams, who are on my PhD committee as well. Outside UF, I have been fortunate to collaborate with John Oprea, Jesús González, Ben Knudsen, and Navnath Daundkar.