Symplectic and Contact Geometry and Topology: Below are some of the most recent research paper by our group: Constructions of symplectic forms on 4-manifolds (arXiv:2403.05512) Author: Peter Lambert-Cole Given a symplectic 4-manifold (X,ω) with rational symplectic form, Auroux constructed branched coverings to (CP2,ωFS). By modifying a previous construction of Lambert-Cole--Meier--Starkston, we prove that the branch locus in CP2 can be assumed holomorphic in a neighborhood of the spine of the standard trisection of CP2. Consequently, the symplectic 4-manifold (X,ω) admits a cohomologous symplectic form that is Kähler in a neighborhood of the 2-skeleton of X. We define the Picard group of holomorphic line bundles over the holomorphic 2-skeleton. We then investigate Hodge theory and apply harmonic spinors to construct holomorphic sections over the Kähler subset. Exotic Dehn twists on sums of two contact 3-manifolds (arXiv:2211.16664) Authors: Eduardo Fernández, Juan Muñoz-Echániz Abstract: We exhibit the first examples of exotic contactomorphisms with infinite order as elements of the contact mapping class group. These are given by certain Dehn twists on the separating sphere in a connected sum of two closed contact 3-manifolds. The tools used to detect these combine an invariant for families of contact structures which generalises the Kronheimer-Mrowka contact invariant in monopole Floer homology, together with an h-principle for families of convex spheres in tight contact 3-manifolds. As a further application, we also exhibit some exotic 1-parametric phenomena in overtwisted contact 3-manifolds. Local rigidity, contact homeomorphisms, and conformal factors (arXiv:2001.08729) Author: Michael Usher In this paper, Mike shows that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a C0-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of arXiv:1306.6367. Bordered Floer homology and contact structures (arXiv:2011.08672 ) Authors: Akram Alishahi, Viktória Földvári, Kristen Hendricks, Joan Licata, Ina Petkova, Vera Vértesi In this paper, they introduce a contact invariant in the bordered sutured Heegaard Floer homology of a contact three-manifold with boundary. They use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes cD and cA in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. They also consider a natural map associated to forgetting the foliation F in favor of the dividing set, and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda-Kazez-Matić. Filtering the Heegaard Floer contact invariant (arXiv:1603.02673 ) Authors: Cagatay Kutluhan, Gordana Matic, Jeremy Van Horn-Morris, Andy Wand In this paper, they define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Z^{≥0}∪{∞}. It is zero for overtwisted contact structures, ∞ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, theyobstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class. Low Dimensional Topology: An exotic 5RP^2 in the 4-sphere (arXiv:2312.03617 ) Authors: Gordana Matić, Ferit Öztürk, Javier Reyes, András I. Stipsicz, Giancarlo Urzúa Abstract: We show an example of an embedded copy of 5RP^2 in the four-sphere which is topologically standard but smoothly knotted, i.e. smoothly not isotopic to the standard embedding. Upsilon invariant for graphs and the homology cobordism group of homology cylinders (arXiv:2202.10614) Author: Akram Alishahi Upsilon is a homomorphism on the smooth concordance group of knots defined by Ozsváth, Stipsicz and Szabó. In this paper, Akram defines a generalization of upsilon for a family of embedded graphs in rational homolog spheres. This generalized invariant induces a homomorphism on the homology cobordism group of homology cylinders. To define this invariant, she uses tangle Floer homology, lifts relative gradings on tangle Floer homology to absolute gradings (for certain tangles) and proves a concatenation formula for it. Random graph embeddings with general edge potentials (arXiv:2205.09049) Authors: Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, Erica Uehara In this paper, they study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects, Lennard-Jones potentials, bending energies, and other physically realistic models. A configuration of n monomers in R^d can be written as a collection of d coordinate vectors, each in R^n. The first main result of this paper is that entries from different coordinate vectors are uncorrelated, even when they are different coordinates of the same monomer. They predict that this property holds in realistic simulations and in actual polymer configurations (in the absence of an external field). The second main contribution is a theorem explaining when and how a probability distribution on embeddings of a complicated graph may be pushed forward to a distribution on embeddings of a simpler graph to aid in computations. This construction is based on the idea of chain maps in homology theory. They use it to give a new formula for edge covariances in phantom network theory and to compute some expectations for a freely-jointed network. Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins (arXiv:2102.12890) Author: David Gay One way to better understand the smooth mapping class group of the 4-sphere would be to give a list of generators in the form of explicit diffeomorphisms supported in neighborhoods of submanifolds, in analogy with Dehn twists on surfaces. As a step in this direction, Dave describe a surjective homomorphism from a group associated to loops of 2-spheres in S^2×S^2's onto this smooth mapping class group, discuss two natural and in some sense complementary subgroups of the domain of this homomomorphism, show that one is in the kernel, and give generators as above for the image of the other. These generators are described as twists along Montesinos twins, i.e. pairs of embedded 2-spheres in the 4-sphere intersecting transversely at two points. Trisecting 4-manifolds (arXiv:1205.1565) Authors: David T. Gay, Robion Kirby In this paper, they show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of ♮^k(S^1×B^3), intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3--manifolds. A trisection of a 4--manifold X arises from a Morse 2--function G:X→B^2 and the obvious trisection of B^2, in much the same way that a Heegaard splitting of a 3--manifold Y arises from a Morse function g:Y→B^1 and the obvious bisection of B^1.
