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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. Parametrized Motion Planning and Topological Complexity

   https://doi.org/10.1007/978-3-031-21090-7_1  

   Farber, M; Weinberger, S (October 2023, Algorithmic Foundations of Robotics XV)

   LaValle, S.; O'Kane, J.; Otte, M.; Sadigh, D.; Tokekar, P. (Ed.)

   In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing the recent paper [2], we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an explicit parametrized motion planning algorithm solving the motion planning problem for any number of robots and obstacles in Rd. This algorithm is optimal, it has minimal possible topological complexity for any d ≥ 3 odd. Besides, we describe a modification of this algorithm which is optimal for d ≥ 2 even. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes. 

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3. Algorithmic Aspects of Immersibility and Embeddability

   https://doi.org/10.1093/imrn/rnae170  

   Manin, Fedor; Weinberger, Shmuel (August 2024, International Mathematics Research Notices)

   Abstract We analyze an algorithmic question about immersion theory: for which $$m$$, $$n$$, and $$CAT=\textbf{Diff}$$ or $$\textbf{PL}$$ is the question of whether an $$m$$-dimensional $CAT$-manifold is immersible in $$\mathbb{R}^{n}$$ decidable? We show that PL immersibility is decidable in all cases except for codimension 2, whereas smooth immersibility is decidable in all odd codimensions and undecidable in many even codimensions. As a corollary, we show that the smooth embeddability of an $$m$$-manifold with boundary in $$\mathbb{R}^{n}$$ is undecidable when $n-m$ is even and $$11m \geq 10n+1$$. 

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4. Bourgeois contact structures: Tightness, fillability and applications

   https://doi.org/10.1007/s00222-022-01131-y  

   Bowden, Jonathan; Gironella, Fabio; Moreno, Agustin (July 2022, Inventiones mathematicae)

   Abstract Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $$V \times {\mathbb {T}}^2$$ V × T 2 . We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $${\mathbb {S}}^1$$ S 1 -invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n -torus has a unique symplectically aspherical strong filling up to diffeomorphism. 

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5. Odd-Dimensional GKM-Manifolds of Non-Negative Curvature

   https://doi.org/10.1093/imrn/rnab137  

   Escher, Christine; Goertsches, Oliver; Searle, Catherine (October 2021, International Mathematics Research Notices)

   Abstract Let $$M$$ be a closed, odd GKM$$_3$$ manifold of non-negative sectional curvature. We show that in this situation one can associate an ordinary abstract GKM$$_3$$ graph to $$M$$ and prove that if this graph is orientable, then both the equivariant and the ordinary rational cohomology of $$M$$ split off the cohomology of an odd-dimensional sphere. 

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