My research is in Geometric Group theory, and in particular I am concerned with studying infinite groups of homeomorphisms of spaces via properties of these groups' actions. I have a particular abiding love of the R. Thompson groups and their relatives, although my research often focuses on other groups as well. My research employs tools from various fields including algebra, geometry, theoretical computer science, and symbolic dynamical systems.
I work in symplectic and algebraic geometry. Right now, I'm interested in how we can use mirror symmetry — a proposed dictionary between these two different kinds of geometry — to bring techniques from symplectic geometry to bear on problems in algebraic geometry. Some applications that I'm working on are the realizability problem in tropical geometry and structures on toric varieties. I'm a picture-drawer at heart: most of my intuition comes from studying the geometry of Lagrangian submanifolds of symplectic manifolds.
My research is in symplectic and enumerative geometry. Most recently, I have been working on the interactions between Gromov—Witten theory, which studies holomorphic curves in a symplectic manifold, and the Fukaya category, which is a category capturing information about Lagrangian submanifolds of this symplectic manifold. I'm also interested in using this interaction to prove structural results about Gromov—Witten theory.
Yoav Len (Head of Group)
My research is in tropical geometry, a relatively recent area on the cusp between algebraic geometry and combinatorics. I use tools from combinatorics and non-archimedean geometry to explore the geometry of curves, moduli spaces, and enumerative geometry. My main focus is Brill—Noether theory, which studies special line bundles on algebraic curves. Tropical geometry reduces problems on such line bundles to a 2-player game played on the vertices of a graph, known as the chip-firing game. You can actually try your luck with the game in the following website.
I like to work on problems that feature an interaction between geometry, combinatorics, and algorithmic considerations. I also like theoretical problems that arise out of concrete, applied scenarios. I’m interested in discrete geometry generally, and the rigidity and flexibility of frameworks specifically, with connections to applications in physics, materials, and machine learning.
Thibault Poiret (Research Fellow)
My research is in algebraic geometry, more specifically the geometry of moduli spaces (spaces which parametrize some kinds of algebro-geometric objects). I am particularly interested in the geometry of families of curves, line bundles and vector bundles on them.
In recent years, my focus has been on the art of embedding non-compact moduli spaces into compact ones via the use of logarithmic geometry. Logarithmic geometry is a framework which helps describe complicated algebro-geometric objects in terms of simpler ones via the use of the combinatorial methods.
Alison La Porta (Research Fellow)
I am interested in an expanding area of mathematics known as rigidity theory, which combines geometry, combinatorics, and other areas of mathematics, to decide whether a given structure, known as framework, is ‘rigid’ or ‘flexible’. The terms ‘rigid’ and ‘flexible’ can take different meanings, depending on the setup of the problem. Most intuitively, a framework is rigid if it cannot be deformed in given ways, and it is flexible otherwise. In the past, I have shown particular interest in symmetric frameworks. I am now looking into more generic structures. Rigidity theory finds applications in different applied sciences. While at St Andrews, I am looking forward to exploring the connection between rigid frameworks and MLE problems.
Violeta Lopez
My research interests lie in algebraic geometry and tropical geometry. Particularly, Brill-Noether theory of algebraic curves and metric graphs. I am also interested in toric varieties and real algebraic geometry.