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1. On flat manifold bundles and the connectivity of Haefliger’s classifying spaces

   https://doi.org/10.1090/proc/16941  

   Nariman, Sam (November 2024, Proceedings of the American Mathematical Society)

   We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston’s conjecture predicts that every $M$-bundle over a manifold $B$where $\dim <\#comment/>\left(B\right)\le <\#comment/>\dim <\#comment/>\left(M\right)$is cobordant to a flat $M$-bundle. In particular, we study the bordism class of flat $M$-bundles over low dimensional manifolds, comparing a finite dimensional Lie group $G$with ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0\left(G\right)$. 

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2. 𝐺-cohomologically rigid local systems are integral

   https://doi.org/10.1090/tran/8610  

   Klevdal, Christian; Patrikis, Stefan (February 2022, Transactions of the American Mathematical Society)

   Let G G be a reductive group, and let X X be a smooth quasi-projective complex variety. We prove that any G G -irreducible, G G -cohomologically rigid local system on X X with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig [Selecta Math. (N. S. ) 24 (2018), pp. 4279–4292; Acta Math. 225 (2020), pp. 103–158] when G = G L n G= \mathrm {GL}_n , and it answers positively a conjecture of Simpson [Inst. Hautes Études Sci. Publ. Math. 75 (1992), pp. 5–95; Inst. Hautes Études Sci. Publ. Math. 80 (1994), pp. 5–79] for G G -cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple; this moreover holds when we relax cohomological rigidity to rigidity. 

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3. Odoni’s conjecture on arboreal Galois representations is false

   https://doi.org/10.1090/proc/15920  

   Dittmann, Philip; Kadets, Borys (August 2022, Proceedings of the American Mathematical Society)

   Suppose f ∈ K [ x ] f \in K[x] is a polynomial. The absolute Galois group of K K acts on the preimage tree T \mathrm {T} of 0 0 under f f . The resulting homomorphism ϕ f : Gal K → Aut ⁡ T \phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T} is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K K there exists a polynomial f f for which ϕ f \phi _f is surjective. We show that this conjecture is false. 

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4. Derived 𝑝-adic heights and the leading coefficient of the Bertolini–Darmon–Prasanna 𝑝-adic 𝐿-function

   https://doi.org/10.1090/btran/227  

   Castella, Francesc; Hsu, Chi-Yun; Kundu, Debanjana; Lee, Yu-Sheng; Liu, Zheng (May 2025, Transactions of the American Mathematical Society, Series B)

   Let $E/\mathbf{Q}$be an elliptic curve and let $p$be an odd prime of good reduction for $E$. Let $K$be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which $p$splits. The goal of this paper is two-fold: (1) we formulate a $p$-adic BSD conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue $F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$, we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected “maximal non-degeneracy” of an anticyclotomic $p$-adic height. In particular, when the Iwasawa–Greenberg Main Conjecture $\left(F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)=\left(L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)$is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$at $T=0$up to a $p$-adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes $p$under mild hypotheses. In the $p$-ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Théor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes. 

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5. Dimension drop for diagonalizable flows on homogeneous spaces

   https://doi.org/10.3934/jmd.2024012  

   Kleinbock, Dmitry; Mirzadeh, Shahriar (January 2024, Journal of Modern Dynamics)

   Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms. 

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