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1. On the construction of tame supercuspidal representations

   https://doi.org/10.1112/S0010437X21007636  

   Fintzen, Jessica (December 2021, Compositio Mathematica)

   Let $$F$$ be a non-archimedean local field of residual characteristic $$p \neq 2$$ . Let $$G$$ be a (connected) reductive group over $$F$$ that splits over a tamely ramified field extension of $$F$$ . We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [ Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference. 

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2. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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3. The factorisation property of l ^∞ ( X_k )

   https://doi.org/10.1017/s0305004120000304  

   LECHNER, RICHARD; MOTAKIS, PAVLOS; MÜLLER, PAUL F.X.; SCHLUMPRECHT, THOMAS (September 2021, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by $${(e_{(k,j)}^*)_j}$$ , for $$k\in\N$$ , let $$Z=\ell^\infty(X_k:k\kin\N)$$ be their l ∞ -sum, and let $$T:Z\to Z$$ be a bounded linear operator with a large diagonal, i.e. , $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive. 

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4. Sharp Finiteness Principles For Lipschitz Selections

   Charles Fefferman, Pavel Shvartsman (September 2018, Geometric and functional analysis)

   Let (M,ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness constant. 

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5. Multitrees in Random Graphs

   https://doi.org/10.37236/10804  

   Frieze, Alan; Pegden, Wesley (January 2023, The Electronic Journal of Combinatorics)

   Let $$N=\binom{n}{2}$$ and $$s\geq 2$$. Let $$e_{i,j},\,i=1,2,\ldots,N,\,j=1,2,\ldots,s$$ be $$s$$ independent permutations of the edges $$E(K_n)$$ of the complete graph $$K_n$$. A MultiTree is a set $$I\subseteq [N]$$ such that the edge sets $$E_{I,j}$$ induce spanning trees for $$j=1,2,\ldots,s$$. In this paper we study the following question: what is the smallest $m=m(n)$ such that w.h.p. $[m]$ contains a MultiTree. We prove a hitting time result for $s=2$ and an $$O(n\log n)$$ bound for $$s\geq 3$$. 

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