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1. PL-Genus of surfaces in homology balls

   https://doi.org/10.1017/fms.2023.126  

   Hom, Jennifer; Stoffregen, Matthew; Zhou, Hugo (January 2024, Forum of Mathematics, Sigma)

   Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology. 

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2. 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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3. KSB stability is automatic in codimension $$\boldsymbol{\geq 3}$$

   https://doi.org/10.1112/mod.2024.8  

   Kollár, János; Kovács, Sándor J (January 2024, Moduli)

   Abstract KSB stability holds at codimension$$1$$points trivially, and it is quite well understood at codimension$$2$$points because we have a complete classification of$$2$$-dimensional slc singularities. We show that it is automatic in codimension$$3$$. 

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4. Transversal C_k -factors in subgraphs of the balanced blow-up of C_k

   https://doi.org/10.1017/S0963548322000086  

   Ergemlidze, Beka; Molla, Theodore (November 2022, Combinatorics, Probability and Computing)

   Abstract For a subgraph$$G$$of the blow-up of a graph$$F$$, we let$$\delta ^*(G)$$be the smallest minimum degree over all of the bipartite subgraphs of$$G$$induced by pairs of parts that correspond to edges of$$F$$. Johansson proved that if$$G$$is a spanning subgraph of the blow-up of$$C_3$$with parts of size$$n$$and$$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$$, then$$G$$contains$$n$$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$$G$$is a spanning subgraph of the blow-up of$$C_k$$with parts of size$$n$$and$$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$$, then$$G$$contains$$n$$vertex disjoint copies of$$C_k$$such that each$$C_k$$intersects each of the$$k$$parts exactly once. A similar conjecture was also made by Fischer and the case$$k=3$$was proved for large$$n$$by Magyar and Martin. In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of$$G$$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically. 

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5. Small subsets with large sumset: Beyond the Cauchy–Davenport bound

   https://doi.org/10.1017/S0963548324000014  

   Fox, Jacob; Luo, Sammy; Pham, Huy Tuan; Zhou, Yunkun (July 2024, Combinatorics, Probability and Computing)

   Abstract For a subset$$A$$of an abelian group$$G$$, given its size$$|A|$$, its doubling$$\kappa =|A+A|/|A|$$, and a parameter$$s$$which is small compared to$$|A|$$, we study the size of the largest sumset$$A+A'$$that can be guaranteed for a subset$$A'$$of$$A$$of size at most$$s$$. We show that a subset$$A'\subseteq A$$of size at most$$s$$can be found so that$$|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling$$\kappa$$is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets$$A,B$$of$$\mathbb{F}_p$$of size at most$$\alpha p$$for an appropriate constant$$\alpha \gt 0$$, one only needs three elements$$b_1,b_2,b_3\in B$$to guarantee$$|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$$. Allowing the use of larger subsets$$A'$$, we show that for sets$$A$$of bounded doubling, one only needs a subset$$A'$$with$$o(|A|)$$elements to guarantee that$$A+A'=A+A$$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset. 

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