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1. Obstruction flat rigidity of the CR 3-sphere

   https://doi.org/10.1515/crelle-2021-0051  

   Curry, Sean N.; Ebenfelt, Peter (October 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable.Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite-dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable. 

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2. The moduli of sections has a canonical obstruction theory

   https://doi.org/10.1017/fms.2022.61  

   Webb, Rachel (January 2022, Forum of Mathematics, Sigma)

   We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument, we construct a dualising sheaf and trace map, in the lisse-étale topology, for families of tame twisted curves when the base stack is locally Noetherian. 

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3. Recording morphogen signals reveals mechanisms underlying gastruloid symmetry breaking

   https://doi.org/10.1038/s41556-024-01521-9  

   McNamara, Harold M; Solley, Sabrina C; Adamson, Britt; Chan, Michelle M; Toettcher, Jared E (November 2024, Nature Cell Biology)

   Aggregates of stem cells can break symmetry and self-organize into embryo-like structures with complex morphologies and gene expression patterns. Mechanisms including reaction-diffusion Turing patterns and cell sorting have been proposed to explain symmetry breaking but distinguishing between these candidate mechanisms of self-organization requires identifying which early asymmetries evolve into subsequent tissue patterns and cell fates. Here we use synthetic ‘signal-recording’ gene circuits to trace the evolution of signalling patterns in gastruloids, three-dimensional stem cell aggregates that form an anterior–posterior axis and structures resembling the mammalian primitive streak and tailbud. We find that cell sorting rearranges patchy domains of Wnt activity into a single pole that defines the gastruloid anterior–posterior axis. We also trace the emergence of Wnt domains to earlier heterogeneity in Nodal activity even before Wnt activity is detectable. Our study defines a mechanism through which aggregates of stem cells can form a patterning axis even in the absence of external spatial cues. 

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4. Exotic Mazur manifolds and knot trace invariants

   https://doi.org/10.1016/j.aim.2021.107994  

   Hayden, Kyle; Mark, Thomas E; Piccirillo, Lisa (November 2021, Advances in Mathematics)

   From a handle-theoretic perspective, the simplest contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the trace of a knot, i.e. the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along K. This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective. We also show that the concordance invariants τ and ϵ are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct surgeries to $$S^1 \times S^2$$, resolving a question from Problem 1.16 in Kirby's list. 

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5. A slicing obstruction from the 10/8+4 theorem

   Truong, Linh (January 2021, MATRIX book series)

   Wood, David R.; de Gier, Jan; Praeger, Cheryl E.; Tao, Terence (Ed.)

   Using the 10/8 + 4 theorem of Hopkins, Lin, Shi, and Xu, we derive a smooth slicing obstruction for knots in the three-sphere using a spin 4-manifold whose boundary is 0–surgery on a knot. This improves upon the slicing obstruction bound by Vafaee and Donald that relies on Furuta’s 10/8 theorem. We give an example where our obstruction is able to detect the smooth non-sliceness of a knot by using a spin 4-manifold for which the Donald-Vafaee slice obstruction fails. 

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