To a cobordism between links, Khovanov homology assigns a linear map that is invariant under boundary-preserving isotopy of the cobordism. In this talk, we study those maps arising from surfaces in the 4-ball and apply our findings to existence and uniqueness questions regarding slice disks bounding a given knot. This reflects joint works with Jonah Swann and Kyle Hayden.

In this talk, we present a graph-based method for revealing the low-dimensional manifold and inferring the underlying processes of time-series. This approach provides intrinsic modeling using empirical information geometry. Unlike traditional information geometry analysis, we compute a Riemannian metric between estimates of the local probability density. Then, a parameterization of the manifold is empirically attained through eigenvectors of an appropriate Laplace operator. The learned model exhibits two important properties. We show that it is invariant under different observation and instrumental modalities and is noise resilient. In addition, the learned model can be efficiently extended to newly acquired measurements in a sequential manner. Provided with such a model, we adopt the state-space formalism and present a framework for sequential processing that is applied to nonlinear and non-Gaussian filtering problems. In addition, we show applications to acoustic signal processing and biomedical signal and image processing.

We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment map. We show that these Lagrangian submanifolds are nondisplaceable by equivariant Hamiltonian diffeomorphisms.

Since their introduction in Euclidean quantum gravity in mid-70â??s, hyperkaehler Gravitational Instantons (aka tesserons) found their use in string theory and in supersymmetric quantum field theory. Their classification was recently completed and now their parameter space is being explored. We propose a systematic program of realizing each of these spaces as a moduli space of monopoles: the monopolization program. Monopolization reveals the combinatorial and geometric structure of the parameter space of all these spaces, equips each space with various natural structures (tautological bundles, Dirac-type operators, etc), and connects different types of integrable systems associated to these gravitational instantons.