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1. Proof of the Goldberg–Seymour conjecture on edge–colorings of multigraphs

   https://doi.org/10.1007/s10878-025-01348-6  

   Chen, Guantao; Jing, Guangming; Zang, Wenan (September 2025, Journal of Combinatorial Optimization)

   Abstract Given a multigraph$$G=(V,E)$$, the edge-coloring problem (ECP) is to color the edges ofGwith the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is referred to as the fractional edge-coloring problem (FECP). The optimal value of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index) ofG, denoted by$$\chi '(G)$$(resp.$$\chi ^*(G)$$). Let$$\Delta (G)$$be the maximum degree ofGand let$$\Gamma (G)$$be the density ofG, defined by$$\begin{aligned} \Gamma (G)=\max \left\{ \frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hspace{5.69054pt}\textrm{and} \hspace{5.69054pt}\textrm{odd} \right\} , \end{aligned}$$whereE(U) is the set of all edges ofGwith both ends inU. Clearly,$$\max \{\Delta (G), \, \lceil \Gamma (G) \rceil \}$$is a lower bound for$$\chi '(G)$$. As shown by Seymour,$$\chi ^*(G)=\max \{\Delta (G), \, \Gamma (G)\}$$. In the early 1970s Goldberg and Seymour independently conjectured that$$\chi '(G) \le \max \{\Delta (G)+1, \, \lceil \Gamma (G) \rceil \}$$. Over the past five decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated an important body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for$$\chi '(G)$$, so an analogue to Vizing’s theorem on edge-colorings of simple graphs holds for multigraphs; second, although it isNP-hard in general to determine$$\chi '(G)$$, we can approximate it within one of its true value, and find it exactly in polynomial time when$$\Gamma (G)>\Delta (G)$$; third, every multigraphGsatisfies$$\chi '(G)-\chi ^*(G) \le 1$$, and thus FECP has a fascinating integer rounding property. 

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2. 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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3. Fixed point sets and the fundamental group II: Euler characteristics

   https://doi.org/10.1017/prm.2023.75  

   Cappell, Sylvain; Weinberger, Shmuel; Yan, Min (October 2023, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   For a finite group$$G$$of not prime power order, Oliver showed that the obstruction for a finite CW-complex$$F$$to be the fixed point set of a contractible finite$$G$$-CW-complex is determined by the Euler characteristic$$\chi (F)$$. (He also has similar results for compact Lie group actions.) We show that the analogous problem for$$F$$to be the fixed point set of a finite$$G$$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on$$K_0$$[2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set. 

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4. Cyclic base change of cuspidal automorphic representations over function fields

   https://doi.org/10.1112/S0010437X24007243  

   Böckle, Gebhard; Feng, Tony; Harris, Michael; Khare, Chandrashekhar B; Thorne, Jack A (September 2024, Compositio Mathematica)

   Let$$G$$be a split semisimple group over a global function field$$K$$. Given a cuspidal automorphic representation$$\Pi$$of$$G$$satisfying a technical hypothesis, we prove that for almost all primes$$\ell$$, there is a cyclic base change lifting of$$\Pi$$along any$$\mathbb {Z}/\ell \mathbb {Z}$$-extension of$$K$$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group$$G$$over a local function field$$F$$, and almost all primes$$\ell$$, any irreducible admissible representation of$$G(F)$$admits a base change along any$$\mathbb {Z}/\ell \mathbb {Z}$$-extension of$$F$$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi. 

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5. The Ruelle invariant and convexity in higher dimensions

   https://doi.org/10.4171/jems/1671  

   Chaidez, Julian; Edtmair, Oliver (July 2025, Journal of the European Mathematical Society)

   We construct the Ruelle invariant of a volume preserving flow and a symplectic cocycle in any dimension and prove several properties. In the special case of the linearized Reeb flow on the boundary of a convex domainXin\mathbb{R}^{2n}, we prove that the Ruelle invariant\operatorname{Ru}(X), the period of the systolec(X)and the volume\operatorname{vol}(X)satisfy\operatorname{Ru}(X) \cdot c(X) \le C(n) \cdot \operatorname{vol}(X). HereC(n) > 0is an explicit constant depending onn. As an application, we construct dynamically convex contact forms onS^{2n-1}that are not convex, disproving the equivalence of convexity and dynamical convexity in every dimension. 

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