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1. The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems

   https://doi.org/10.1090/bproc/203  

   Belmont, Eva ; Xu, Zhouli ; Zhang, Shangjie ( January 2024 , Proceedings of the American Mathematical Society, Series B)

   We describe in terms of generators and relations the ring structure of the$RO(C_2)$-graded$C_2$-equivariant stable stems$\pi _\star ^{C_2}$modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational$C_2$-equivariant sphere$\pi _\star ^{C_2}(\mathbb {S}_\mathbb {Q})$.

    

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2. Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere

   https://doi.org/10.1090/cams/17  

   Carmeli, Shachar ; Yuan, Allen ( March 2023 , Communications of the American Mathematical Society)

   We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the$K(1)$-local sphere$\mathbb {S}_{K(1)}$at the prime$2$, in particular realizing the non-$2$-adic rational element$1+\varepsilon \in \pi _0\mathbb {S}_{K(1)}$as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in$\pi _0\mathbb {S}_{K(1)}$.

    

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3. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo ; Wang, Jinhuan ( June 2024 , Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following$L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality$\begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*}$where parameters$q$and$p$satisfy the conditions$p>d\geq 1$,$q\geq 0$. The best constant$C_{q,\infty ,p}$is given by$\begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*}$where$u_{c,\infty }$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when$u=Au_{c,\infty }(\lambda (x-x_0))$for any real numbers$A$,$\lambda >0$and$x_{0}\in \mathbb {R}^d$. In fact, the generalized Lane-Emden equation in$\mathbb {R}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For$q=0$or$d=1$,$u_{c,\infty }$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that$u_{c,m}\to u_{c,\infty }$and$C_{q,m,p}\to C_{q,\infty ,p}$as$m\to +\infty$for$d=1$, where$u_{c,m}$and$C_{q,m,p}$are the function achieving equality and the best constant of$L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively.

    

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4. On symmetric representations of 𝑆𝐿₂(ℤ)

   https://doi.org/10.1090/proc/16205  

   Ng, Siu-Hung ; Wang, Yilong ; Wilson, Samuel ( January 2023 , Proceedings of the American Mathematical Society)

   We introduce the notions of symmetric and symmetrizable representations of${\operatorname {SL}_2(\mathbb {Z})}$. The linear representations of${\operatorname {SL}_2(\mathbb {Z})}$arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of${\operatorname {SL}_2(\mathbb {Z})}$. By investigating a$\mathbb {Z}/2\mathbb {Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of${\operatorname {SL}_2(\mathbb {Z})}$are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of${\operatorname {SL}_2(\mathbb {Z})}$that are subrepresentations of a symmetric one.

    

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5. A proof of the Kahn–Kalai conjecture

   https://doi.org/10.1090/jams/1028  

   Park, Jinyoung ; Pham, Huy ( August 2023 , Journal of the American Mathematical Society)

   Proving the “expectation-threshold” conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property$\mathcal {F}$on a finite set$X$,\[$p_c(\mathcal {F})=O(q(\mathcal {F})\log \ell (\mathcal {F})),$\]where$p_c(\mathcal {F})$and$q(\mathcal {F})$are the threshold and “expectation threshold” of$\mathcal {F}$, and$\ell (\mathcal {F})$is the maximum of$2$and the maximum size of a minimal member of$\mathcal {F}$.

    

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