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1. Khovanov homology and exotic surfaces in the 4-ball

   https://doi.org/10.1515/crelle-2024-0001  

   Hayden, Kyle; Sundberg, Isaac (March 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms).We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations.We also show that Plamenevskaya’s transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms. 

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2. Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

   https://doi.org/10.1017/etds.2020.113  

   BARTHELMÉ, THOMAS; FENLEY, SERGIO R.; FRANKEL, STEVEN; POTRIE, RAFAEL (November 2021, Ergodic Theory and Dynamical Systems)

   Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol. , to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint , 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint , 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover. 

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3. Geometric analysis of the generalized surface quasi-geostrophic equations

   https://doi.org/10.1007/s00208-024-02867-z  

   Bauer, Martin; Heslin, Patrick; Misiołek, Gerard; Preston, Stephen_C (April 2024, Mathematische Annalen)

   Abstract We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach 

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4. Gompf’s Cork and Heegaard Floer Homology

   https://doi.org/10.1093/imrn/rnae180  

   Dai, Irving; Mallick, Abhishek; Zemke, Ian (August 2024, International Mathematics Research Notices)

   Abstract Gompf showed that for $$K$$ in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $$K \# -K$$ into a cork boundary. We derive a general Floer-theoretic condition on $$K$$ under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf’s method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms. 

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5. Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

   https://doi.org/10.1017/etds.2021.42  

   AVILA, A.; VIANA, MARCELO; WILKINSON, A. (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact. 

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