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1. Recent Developments in Line Bundle Cohomology and Applications to String Phenomenology

   Brodie, Callum; Constantin, Andrei; Gray, James; Lukas, Andre; Ruehle, Fabian (September 2022, Nankai Symposium on Mathematical Dialogues)

   He, Yang-Hui; Ge, Mo-Lin; Bai, Cheng-Ming; Bao Jiakang; Hirst, Edward (Ed.)

   Vector bundle cohomology represents a key ingredient for string phenomenology, being associated with the massless spectrum arising in string compactifications on smooth compact manifolds. Although standard algorithmic techniques exist for performing cohomology calculations, they are laborious and ill-suited for scanning over large sets of string compactifications to find those most relevant to particle physics. In this article we review some recent progress in deriving closed-form expressions for line bundle cohomology and discuss some applications to string phenomenology. 

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2. Motivic Action on Coherent Cohomology of Hilbert Modular Varieties

   https://doi.org/10.1093/imrn/rnac126  

   Horawa, Aleksander (May 2022, International Mathematics Research Notices)

   Abstract We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $$p$$ and over $${\mathbb {C}}$$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark’s conjecture on special values of Artin $$L$$-functions and by numerical evidence in base change cases. 

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3. Modularity of d$$d$$‐elliptic loci with level structure

   https://doi.org/10.1112/jlms.70212  

   Greer, François; Lian, Carl (June 2025, Journal of the London Mathematical Society)

   Abstract We consider the generating series of special cycles on , with full level structure, valued in the cohomology of degree . The modularity theorem of Kudla–Millson for locally symmetric spaces implies that these series are modular. When , the images of these loci in are the ‐elliptic Noether–Lefschetz loci, which are conjectured to be modular. In the Appendix, it is shown that the resulting modular forms are nonzero for when and . 

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4. The cohomology of Torelli groups is algebraic

   https://doi.org/10.1017/fms.2020.41  

   Kupers, Alexander; Randal-Williams, Oscar (January 2020, Forum of Mathematics, Sigma)

   Abstract The Torelli group of $$W_g = \#^g S^n \times S^n$$ is the group of diffeomorphisms of $$W_g$$ fixing a disc that act trivially on $$H_n(W_g;\mathbb{Z} )$$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $$\text{Sp}_{2g}(\mathbb{Z} )$$ or $$\text{O}_{g,g}(\mathbb{Z} )$$ . In this article we prove that for $$2n \geq 6$$ and $$g \geq 2$$ , they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent. 

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5. Efficient Algorithms for Complexes of Persistence Modules with Applications

   https://doi.org/10.4230/LIPICS.SOCG.2024.51  

   Dey, Tamal K; Russold, Florian; Samaga, Shreyas N (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes. 

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