My research lies in the fields of low-dimensional geometric topology and geometric group theory, in particular mapping class groups of surfaces and Teichmuller theory. Recently, I have been exploring connections with dynamics and model theory.
My current research has four axes:
first-order logic of the curve complex and the mapping class groups;
coarse geometry and combinatorics of geometric graphs;
diameters of (modular) flip graphs: combinatorics and algorithms;
Teichmueller theory & Thurston's distance on the (classical) Teichmuller space.
I am particularly interested in studying the actions of the mapping class group on certain simplicial complexes encoding the combinatorics of curves and arcs on a surface. The most famous one is the curve complex, an important tool in the proof of Thurston’s Ending Lamination Conjecture and in the coarse geometry of the Teichmuller space and the mapping class group. Since my PhD. I have explored the combinatorics and the coarse geometry of the arc complex and the flip graph of a punctured surface, a graph whose vertices are ideal triangulations. In collaboration with Anja Randecker, Robert Tang and Huiping Pan I have recently employed these techniques in dynamics and started to investigate the affine group of (half-)translation surfaces via their actions on saddle connection complexes, which are "flat" analogues of the (topological) arc complex and the flip graph. In a collaboration with Javier de la Nuez Gonzalez and Thomas Koberda, we have started a long-term program to understand the curve complex and other similar geometric graphs from the point of view of model theory, in particular stability theory.
My research statement is available upon request.
Daniele ALESSANDRINI (Columbia University)
Mark BELL (industry)
Javier DE LA NUEZ GONZALEZ (University of Bilbao)
Thomas KOBERDA (University of Virginia)
Marta MAGNANI (Universitaet Heidelberg)
Huiping PAN (Jinan University)
Hugo PARLIER (Université du Luxembourg)
Maria Beatrice POZZETTI (Universitaet Heidelberg)
Anja RANDECKER (Universitaet Heidelberg)
Dialdeen Taha (Universitaet Heidelberg)
Robert TANG (Xi'an Jiaotong-Liverpool)