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1. Log Symplectic Manifolds and [ Q,R ]=0

   https://doi.org/10.1093/imrn/rnab140  

   Lin, Yi; Loizides, Yiannis; Sjamaar, Reyer; Song, Yanli (June 2021, International Mathematics Research Notices)

   Abstract We show, under an orientation hypothesis, that a log symplectic manifold with simple normal crossing singularities has a stable almost complex structure, and hence is Spin$$_c$$. In the compact Hamiltonian case we prove that the index of the Spin$$_c$$ Dirac operator twisted by a prequantum line bundle satisfies a $[Q,R]=0$ theorem. 

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2. Ambient Prime Geodesic Theorems on Hyperbolic 3-Manifolds

   https://doi.org/10.1093/imrn/rnab048  

   Dever, Lindsay; Milićević, Djordje (October 2021, International Mathematics Research Notices)

   Abstract We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy. 

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3. An almost flat manifold with a cyclic or quaternionic holonomy group bounds

   Davis, James F; Fang, Fuquan (January 2016, Journal of differential geometry)

   A long-standing conjecture of Farrell and Zdravkovska and independently S.T. Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions. 

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4. Topological rigidity in twisted, elastic ribbons

   Moguel-Lehmer, Carlos E; Santangelo, Christian (February 2025, arXiv)

   Topology is an important determinant of the behavior of a great number of condensed-matter systems, but until recently has played a minor role in elasticity. We develop a theory for the deformations of a class of twisted non-Euclidean sheets which have a symmetry based on the celebrated Bonnet isometry. We show that non-orientability is an obstruction to realizing the symmetry globally, and induces a geometric phase that captures a memory analogous to a previously identified one in 2D metamaterials. However, we show that orientable ribbons can also obstruct realizing the symmetry globally. This new obstruction is mediated by how the unit normal vector winds around the centerline of the ribbon, and provides conditions for constructing soft modes of deformation compatible with the topology of multiply-twisted connected ribbons. 

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5. On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

   https://doi.org/10.4171/JEMS/1485  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (July 2024, Journal of the European Mathematical Society)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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