• Speaker: Huyi Hu, Michigan State University
• Title:  The SRB measures for pointwise hyperbolic systems on open regions
• Date: 07/06/2022
• 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 μ 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: Jing Zhou, Penn State University
• Title:  Uniqueness of equilibrium state for Lorenz attractors
• Date: 07/19/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

In this talk we show the existence of an infinite measure set of exponentially escaping orbits for a resonant Fermi accelerator, which is realized as a square billiard with a periodically oscillating platform. We use normal forms to describe the energy change in a period and employ techniques from the theory of hyperbolic systems with singularities to show the exponential drift given by these normal forms on a divided time-energy phase.

• 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: Wenbo Sun, Virginia Tech
• Title:  Equidistribution for nilsequences along spheres
• Date: 07/25/2022
• Time: 3:30 PM - 4:30 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

The nilsequence is a generalization for exponential sums which has a wide range of applications in analysis and combinatorics. A well know result of Green and Tao in 2010 states that the equidistribution property of a nilsequence is determined by its projection on the horizontal torus.

• 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.

• Speaker: Michael Willis, Stanford University
• Title: Annular Khovanov stable homotopy and sl_2
• Date: 07/29/2022
• Time: 1:00 PM - 2:00 PM
• Place: C304 Wells Hall
• Contact: Teena Meredith Gerhardt (gerhar18@msu.edu)

The Khovanov complex of a link $L$ in a thickened annulus carries a filtration; the associated graded complex gives rise to the annular Khovanov homology of $L$. Grigsby-Licata-Wehrli show that this annular homology admits an action by the Lie algebra $\mathfrak{sl}_2$. Using the techniques of Lipshitz-Sarkar, one can define a stable homotopy lift of the annular Khovanov homology of $L$. In this talk I will describe (in part) how to lift the $\mathfrak{sl}_2$-action to the stable homotopy category as well, illustrating some features of how one might hope to lift signed maps with cancellations via framed flow categories. This is joint work with Ross Akhmechet and Slava Krushkal.