Research

My research is at the intersection of differential geometry, complex geometry, geometric topology, and algebraic topology. Broadly, I am interested in geometric structures on manifolds, in the global constraints they impose on the topology and large-scale geometry of manifolds, and in identifying and describing the precise form of those restrictions through both general results and carefully chosen examples.

My research statement (describing my previous research projects and directions for future research) is available upon request.

My current areas of particular interest include, but are not limited to, the following:

• Positive scalar curvature (mainly on manifolds with extra structure and its influence on large-scale topology and geometry).
• Complex and birational geometry (in particular, complex algebraic geometry, Kähler geometry, and the minimal model theory).
• Symplectic topology and geometry (in particular, symplectic asphericity and approximations of symplectic structures).
• Symmetric products of surfaces (mainly of closed Riemann surfaces).
• Numerical homotopy invariants (in particular, Lusternik–Schnirelmann category, sequential topological complexity, cohomological dimension, and versions of these invariants).

I enjoy studying interactions between topics from the above areas. A recurring question in my research is how additional structures force global and large-scale restrictions on a manifold. I approach this question through the isometry-invariant scalar curvature and geometric obstructions to its positivity, through the condition of symplectic asphericity as a mechanism for producing new curvature obstructions and for understanding topological constraints,
and through symmetric products of surfaces as explicit spaces in which these interactions can be detected and examined. I have also studied numerical homotopy invariants of symmetric products, and have used the latter to develop the theory of probabilistic variants of the former.

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

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