• Speaker: Pablo Groisman, Universidad de Buenos Aires
• Title: Learning distances to learn topologies, to learn dynamical systems, to learn from chaos.
• Date: 05/12/2022
• Time: 2:30 PM - 3:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova (turanova@msu.edu)

Consider a finite sample of points on a manifold embedded in Euclidean space. We'll address the following issues. 1. How we can infer, from the sample, intrinsic distances that are meaningful to understand the data. 2. How we can use these estimated distances to infer the geometry and topology of the manifold (manifold learning). 3. How we can use this knowledge to validate dynamical systems models for chaotic phenomena. 4. Time permitting, we will show applications of this machinery to understand data from the production of songs in canaries. We will prove that if the sample is given by iid points with density f supported on the manifold, the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges in the sense of Gromov–Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. Then we'll apply this result to estimate the topology of the manifold by constructing intrinsic persistence diagrams (an estimator of the homology of the manifold), which are less sensitive to the particular embedding of the manifold in the Euclidean space and to outliers. We'll also discuss how to use these tools to validate (or to refute) models for chaotic dynamical systems, with applications to the study of birdsongs.

• Speaker: Zhenqi Wang, Michigan State University
• Title: Smooth local rigidity for hyperbolic toral automorphisms
• Date: 05/24/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

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

• Speaker: Selin Aviyente, Michigan State University
• Title: Multiview Graph Learning
• Date: 05/26/2022
• Time: 2:30 PM - 3:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova (turanova@msu.edu)

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.

• Speaker: Zhenqi Wang, Michigan State University
• Title: Smooth local rigidity for hyperbolic toral automorphisms
• Date: 06/07/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

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.

• Speaker: Huyi Hu, Michigan State University
• Title:  The SRB measures for pointwise hyperbolic systems on open regions
• Date: 06/14/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

A pointwise partially hyperbolic diffeomorphism is different from a partially hyperbolic one if the expansion and contraction depend on points. If the system is defined on an open set, then the hyperbolicity may not be uniform. We show that under certain conditions such a suystem has unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure. If the system is pointwise hyperbolic, then the u-Gibbs measure $\mu$ is an Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok's map have the properties.

• Speaker: Fan Yang, Michigan State University
• Title: A dichotomy for generic star flows
• Date: 07/12/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

In this talk we will present some recent progress on the topological structure of the so-called star flows (flows whose critical elements, i.e., singularities and periodic orbits, are hyperbolic in a robust way). In particular, we will prove that for $C^1$ generic star flows, every chain recurrent class with positive topological entropy must be isolated. On the other hand, zero entropy chain recurrent classes only support trivial ergodic measures.

• Speaker: Yun Yang, Virginia Tech
• Title: Entropy rigidity for 3D Anosov flows
• Date: 07/25/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

Anosov systems are among the most well-understood dynamical systems. Special among them are the {\bf algebraic systems}. In the diffeomorphism case, these are automorphisms of tori and nilmanifolds. In the flow case, the algebraic models are suspensions of such diffeomorphisms and geodesic flows on negatively curved rank one symmetric spaces. In this talk, we will show that given an integer $k \ge 5$, and a $C^k$ Anosov flow $\Phi$ on some compact connected $3$-manifold preserving a smooth volume, the measure of maximal entropy is the volume measure if and only if $\Phi$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow for $\varepsilon > 0$ arbitrarily small. This is a joint work with Jacopo De Simoi, Martin Leguil and Kurt Vinhage.

• Speaker: Ross Akhmechet, University of Virginia
• Title: Lattice cohomology and q-series invariants of 3-manifolds
• Date: 07/29/2022
• Time: 10:30 AM - 11:30 AM
• Place: C304 Wells Hall
• Contact: Teena Meredith Gerhardt (gerhar18@msu.edu)

I will discuss an invariant of negative definite plumbed 3-manifolds which unifies and extends two theories with quite different origins and structures. The first is lattice cohomology, due to Némethi, which is motivated by normal surface singularities and is isomorphic to Heegaard Floer homology for a large class of plumbings. The second theory is the $\widehat{Z}$ series of Gukov-Pei-Putrov-Vafa, a power series which conjecturally recovers SU(2) quantum invariants at roots of unity and satisfies remarkable modularity properties. I will explain lattice cohomology, $\widehat{Z}$, and our unification of these theories. I will also discuss some key features of our new invariant: it leads to a 2-variable refinement of $\widehat{Z}$, and, unlike both lattice cohomology and $\widehat{Z}$, it is sensitive to $spinc^c$-conjugation. This is joint work with Peter Johnson and Slava Krushkal.