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1. IWASAWA THEORY FOR p -TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p

   https://doi.org/10.1017/nmj.2022.30  

   BOOHER, JEREMY; CAIS, BRYDEN (June 2023, Nagoya Mathematical Journal)

   Abstract We investigate a novel geometric Iwasawa theory for$${\mathbf Z}_p$$-extensions of function fields over a perfect fieldkof characteristic$$p>0$$by replacing the usual study ofp-torsion in class groups with the study ofp-torsion class groupschemes. That is, if$$\cdots \to X_2 \to X_1 \to X_0$$is the tower of curves overkassociated with a$${\mathbf Z}_p$$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of thep-torsion group scheme in the Jacobian of$$X_n$$as$$n\rightarrow \infty $$. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of$$X_n$$equipped with natural actions of Frobenius and of the Cartier operatorV. We formulate and test a number of conjectures which predict striking regularity in the$$k[V]$$-module structure of the space$$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$$of global regular differential forms as$$n\rightarrow \infty .$$For example, for each tower in a basic class of$${\mathbf Z}_p$$-towers, we conjecture that the dimension of the kernel of$$V^r$$on$$M_n$$is given by$$a_r p^{2n} + \lambda _r n + c_r(n)$$for allnsufficiently large, where$$a_r, \lambda _r$$are rational constants and$$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$$is a periodic function, depending onrand the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on$${\mathbf Z}_p$$-towers of curves, and we prove our conjectures in the case$$p=2$$and$$r=1$$. 

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2. The k -core in percolated dense graph sequences

   https://doi.org/10.1017/jpr.2024.66  

   Bayraktar, Erhan; Chakraborty, Suman; Zhang, Xin (March 2025, Journal of Applied Probability)

   Abstract We determine the order of thek-core in a large class of dense graph sequences. Let$$G_n$$be a sequence of undirected,n-vertex graphs with edge weights$$\{a^n_{i,j}\}_{i,j \in [n]}$$that converges to a graphon$$W\colon[0,1]^2 \to [0,+\infty)$$in the cut metric. Keeping an edge (i,j) of$$G_n$$with probability$${a^n_{i,j}}/{n}$$independently, we obtain a sequence of random graphs$$G_n({1}/{n})$$. Using a branching process and the theory of dense graph limits, under mild assumptions we obtain the order of thek-core of random graphs$$G_n({1}/{n})$$. Our result can also be used to obtain the threshold of appearance of ak-core of ordern. 

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3. Eliminating Thurston obstructions and controlling dynamics on curves

   https://doi.org/10.1017/etds.2023.114  

   BONK, MARIO; HLUSHCHANKA, MIKHAIL; ISELI, ANNINA (September 2024, Ergodic Theory and Dynamical Systems)

   Abstract Every Thurston map$$f\colon S^2\rightarrow S^2$$on a$$2$$-sphere$$S^2$$induces a pull-back operation on Jordan curves$$\alpha \subset S^2\smallsetminus {P_f}$$, where$${P_f}$$is the postcritical set off. Here the isotopy class$$[f^{-1}(\alpha )]$$(relative to$${P_f}$$) only depends on the isotopy class$$[\alpha ]$$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the mapfcan be seen as a fixed point of the pull-back operation. We show that if a Thurston mapfwith a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying$$2$$-sphere and construct a new Thurston map$$\widehat f$$for which this obstruction is eliminated. We prove that no other obstruction arises and so$$\widehat f$$is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem. 

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4. The exact consistency strength of the generic absoluteness for the universally Baire sets

   https://doi.org/10.1017/fms.2023.127  

   Sargsyan, Grigor; Trang, Nam (January 2024, Forum of Mathematics, Sigma)

   Abstract A set of reals isuniversally Baireif all of its continuous preimages in topological spaces have the Baire property.$$\mathsf {Sealing}$$is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The$$\mathsf {Largest\ Suslin\ Axiom}$$($$\mathsf {LSA}$$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let$$\mathsf {LSA-over-uB}$$be the statement that in all (set) generic extensions there is a model of$$\mathsf {LSA}$$whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory,$$\mathsf {Sealing}$$is equiconsistent with$$\mathsf {LSA-over-uB}$$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that$$\mathsf {Sealing}$$is weaker than the theory ‘$$\mathsf {ZFC} +$$there is a Woodin cardinal which is a limit of Woodin cardinals’. A variation of$$\mathsf {Sealing}$$, called$$\mathsf {Tower\ Sealing}$$, is also shown to be equiconsistent with$$\mathsf {Sealing}$$over the same large cardinal theory. The result is proven via Woodin’s$$\mathsf {Core\ Model\ Induction}$$technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of$$\mathsf {CMI}$$as explained in the paper. 

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5. On the zeroes of hypergraph independence polynomials

   https://doi.org/10.1017/S0963548323000330  

   Galvin, David; McKinley, Gwen; Perkins, Will; Sarantis, Michail; Tetali, Prasad (January 2024, Combinatorics, Probability and Computing)

   Abstract We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree$$\Delta$$have a zero-free disc almost as large as the optimal disc for graphs of maximum degree$$\Delta$$established by Shearer (of radius$$\sim 1/(e \Delta )$$). Up to logarithmic factors in$$\Delta$$this is optimal, even for hypergraphs with all edge sizes strictly greater than$$2$$. We conjecture that for$$k\ge 3$$,$$k$$-uniformlinearhypergraphs have a much larger zero-free disc of radius$$\Omega (\Delta ^{- \frac{1}{k-1}} )$$. We establish this in the case of linear hypertrees. 

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