Research

In Hamiltonian mechanics, the natural space to study is "phase space", whose coordinates correspond to the position and momentum of particles. Geometry on this space is governed by the equations of motion and one of the most important questions in the area is determining if there are orbits in these equations of motion. For example, the motion of two planets is determined by a twelve-dimensional phase space (three space and momentum coordinates for each planet) and orbits of these equations correspond to actual orbits. Symplectic geometry is the study of spaces locally described by phase space. In the 1990s, Andreas Floer proved that the existence of orbits associated with the equations of motion on these spaces could be determined by the topology of the space. This breakthrough led to the development of a toolkit (now known as Floer theory) that has become a powerful tool for probing structures not only in symplectic geometry but topology as well. Hicks researches the geometry of Lagrangian submanifolds and its connections to tropical and algebraic geometry.

Tropical geometry explores the combinatorial core underpinning geometric objects, mostly in algebraic geometry. It has deep connections to classical geometry, particularly in the study of moduli spaces, where tropical techniques provide insights into families of geometric objects and their degenerations. In Brill–Noether theory, tropical methods lead to the study of linear series on curves via chip-firing game on graphs.   This perspective has led to breakthroughs in areas such as counting curves, understanding limit linear series, and modular compactifications of moduli spaces.

Rigidity theory is the study of the structural rigidity and flexibility of geometric structures, typically modelled as graphs with vertices representing joints and edges representing rigid bars. It explores conditions under which a framework maintains its shape under continuous deformations, with applications in fields such as engineering, crystallography, and robotics. Key concepts include infinitesimal rigidity, which examines deformations through linearized constraints, and global rigidity, which ensures uniqueness of a structure up to rigid motions. Connections to combinatorics and algebraic geometry arise in the study of rigidity matroids and moduli spaces of frameworks, influencing areas like structural design and motion planning

Geometric group theory is a field of mathematics that explores the interplay between algebra and geometry by studying groups through their actions on geometric spaces. It employs tools from topology, combinatorics, and hyperbolic geometry to understand infinite groups and their properties. Key topics include quasi-isometries, growth of groups, and the study of hyperbolic and CAT(0) groups. Thompson groups, which arise in this field, are particularly intriguing due to their rich algebraic structure and unusual geometric properties, making them a central topic of ongoing research. Geometric group theory has broad applications, influencing topology, dynamics, and theoretical computer science.