The University of Georgia maintains a strong research presence in the geometry and topology of symplectic and contact structures on smooth manifolds, and their relations with algebraic geometry. Symplectic structures live on even dimensional manifolds while contact structures are their odd dimensional analogs. Below are some highlights of our history and current activity:
- Gordana Matic pioneered much of our modern understanding of gauge theoretic contact invariants in her seminal work with Paolo Lisca on tight contact structures and Seiberg-Witten invariants.
- Will Kazez has continued his work on contact topology in collaboration with Rachel Roberts by studying the existence of tight contact structures as perturbations of taut foliations. Matic is studying refinements of the Ozsvath-Szabo contact invariant that can detect an infinite number of possible degrees of tightness of contact structure.
Will and Gordana collaborated with then-UGA postdoc Ko Honda to study 3-dimensional contact topology from the point of view of the embedded surfaces in the ambient manifold. This lead to a decomposition theory of contact manifolds and an interpretation of the Ozsvath-Szabo Heegaard-Floer invariants from the point of view of open book decompositions. This has lead to insights into monoids of surface automorphisms related to the tightness of contact structures and generalizations to bordered 3-manifolds.
- Jason Cantarella has been using the symplectic structure on space polygons to prove new results in the theory of random walks. These include a new algorithm for directly sampling closed random walks with equal length edges which were selected by the Journal of Physics as one of the highlights of the year in mathematical physics in 2016. (http://iopscience.iop.org/journal/1751-8121/page/Highlights-of-2016). At UGA, he is working with Erik Schreyer, Philipp Reiter, and Kyle Chapman.
- Mike Usher established relationships between questions about pseudo-holomorphic curves and aspects of smooth four-manifold topology such as Lefschetz fibrations and minimality. More recently he has focused largely on filtered Floer theory on general symplectic manifolds, both developing foundational tools as in his joint work with then-UGA grad student Jun Zhang on generalized persistence barcodes, and applying such tools to prove results concerning, for example, the geometry of the Hamiltonian diffeomorphism group and the existence and uniqueness of symplectic embeddings between domains in R^4.
- David Gay began his career working on handle decompositions and surgeries in the low-dimensional symplectic and contact settings, and has recently been studying 4-manifolds quite broadly using the new theory of trisections, developed in collaboration with Rob Kirby. David is working closely with Gordana, postdoc Adam Saltz and grad student Bill Olsen on expanding the Floer theoretic techniques of Oszvath and Szabo to this setting, and is trying to understand the connections between algebraic geometry, symplectic geometry and the smooth topology of trisections. We currently host a blog on Floer theory and trisections, have a look!
- Weiwei Wu is our most recent hire in the field and has especially strong connections with algebraic geometry through his work on mirror symmetry and algebraic surfaces from a symplectic point of view. His recently studies autoequivalences on Fukaya categories induced by symplectomorphisms. He is also working on various aspects of automorphism groups of symplectic rational surfaces, which are infinite dimensional Lie groups that exhibit interesting rigidities that mimic the Cremona group in classical algebraic geometry.