We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its
smooth perturbation $f$. We show that if $H$ is weakly differentiable then it is $C^{1+\text{Holder}}$ and, if $A$ is also weakly irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish results of independent interest on Holder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve
regularity of the conjugacy to $C^\infty$ in prior local rigidity results. This is a joint work with B. Kalinin an V. Sadovskaya

Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. We can ask: what Legendrian knots can be both concordant to and from the unstabilized Legendrian unknot? In this talk I'll explain a strategy that can be used to obstruct many knots from this double concordance.

Modern data analysis involves large sets of structured data, where the structure carries critical information about the nature of the data. These relationships between entities, such as features or data samples, are usually described by a graph structure. While many real-world data are intrinsically graph-structured, e.g. social and traffic networks, there is still a large number of applications, where the graph topology is not readily available. For instance, gene regulations in biological applications or neuronal connections in the brain are not known. In these applications, the graphs need to be learned since they reveal the relational structure and may assist in a variety of learning tasks. Graph learning (GL) deals with the inference of a topological structure among entities from a set of observations on these entities, i.e., graph signals. Most of the existing work on graph learning focuses on learning a single graph structure, assuming that the relations between the observed data samples are homogeneous. However, in many real-world applications, there are different forms of interactions between data samples, such as single-cell RNA sequencing (scRNA-seq) across multiple cell types. This talk will present a new framework for multiview graph learning in two settings: i) multiple views of the same data and ii) heterogeneous data with unknown cluster information. In the first case, a joint learning approach where both individual graphs and a consensus graph are learned will be developed. In the second case, a unified framework that merges classical spectral clustering with graph signal smoothness will be developed for joint clustering and multiview graph learning.
This is joint work with Abdullah Karaaslanli, Satabdi Saha and Taps Maiti.