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1. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

   https://doi.org/10.1090/jams/1045  

   Annala, Toni; Hoyois, Marc; Iwasa, Ryomei (March 2024, Journal of the American Mathematical Society)

   We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-category satisfies ${\mathbb{P}}^1$-homotopy invariance and weighted ${\mathbb{A}}^1$-homotopy invariance, which we use in place of ${\mathbb{A}}^1$-homotopy invariance to obtain analogues of several key results from ${\mathbb{A}}^1$-homotopy theory. These allow us in particular to define a universal oriented motivic ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$. 

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2. 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 $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. 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 $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$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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3. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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4. Dens, nests and the Loehr-Warrington conjecture

   https://doi.org/10.1090/jams/1057  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G (June 2025, Journal of the American Mathematical Society)

   We prove and extend the longest-standing conjecture in ‘ $q,t$-Catalan combinatorics,’ namely, the combinatorial formula for ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$conjectured by Loehr and Warrington, where $s_{\mathrm{\mu <\#comment/>}}$is a Schur function and $\mathrm{\nabla <\#comment/>}$is an eigenoperator on Macdonald polynomials. Our approach is to establish a stronger identity of infinite series of $G{L}_l$characters involvingSchur Catalanimals; these were recently shown by the authors to represent Schur functions $s_{\mathrm{\mu <\#comment/>}}[-<\#comment/>M{X}^{m,n}]$in subalgebras $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)\subset <\#comment/>\mathcal{E}$isomorphic to the algebra of symmetric functions $\mathrm{\Lambda <\#comment/>}$over $\mathbb{Q}(q,t)$, where $\mathcal{E}$is the elliptic Hall algebra of Burban and Schiffmann. We establish a combinatorial formula for Schur Catalanimals as weighted sums of LLT polynomials, with terms indexed by configurations of nested lattice paths callednests, having endpoints and bounding constraints controlled by data called aden. The special case for $\mathrm{\Lambda <\#comment/>}\left(X^{m,1}\right)$proves the Loehr-Warrington conjecture, giving ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. In general, for $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)$our formula implies a new $(m,n)$version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduce to our previous shuffle theorem for paths under any line. Both this and the $(m,n)$Loehr-Warrington formula generalize the $(km,kn)$shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our formula here unifies these two generalizations. 

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5. The Vlasov–Poisson–Landau system in the weakly collisional regime

   https://doi.org/10.1090/jams/1014  

   Chaturvedi, Sanchit; Luk, Jonathan; Nguyen, Toan (January 2023, Journal of the American Mathematical Society)

   Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $\begin{array}{c}{\mathrm{\partial <\#comment/>}}_{t}F(t,x,v)+v_i{\mathrm{\partial <\#comment/>}}_{x_i}F(t,x,v)+E_i(t,x){\mathrm{\partial <\#comment/>}}_{v_i}F(t,x,v)=\mathrm{\nu <\#comment/>}Q(F,F)(t,x,v),\\ E(t,x)=\mathrm{\nabla <\#comment/>}{\mathrm{\Delta <\#comment/>}}^{-<\#comment/>1}\left({\int <\#comment/>}_{{\mathbb{R}}^3}F\right(t,x,v)\mathrm{d}v-<\#comment/>{\int <\#comment/>-<\#comment/>}_{{\mathbb{T}}^3}{\int <\#comment/>}_{{\mathbb{R}}^3}F(t,x,v\left)\mathrm{d}v\mathrm{d}x\right),\end{array}$with $\mathrm{\nu <\#comment/>}\ll <\#comment/>1$. We prove that for $\mathrm{ϵ<\#comment/>}>0$sufficiently small (but independent of $\mathrm{\nu <\#comment/>}$), initial data which are $O\left(\mathrm{ϵ<\#comment/>}{\mathrm{\nu <\#comment/>}}^{1/3}\right)$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to <\#comment/>\mathrm{\infty <\#comment/>}$. The solutions exhibit uniform-in- $\mathrm{\nu <\#comment/>}$Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. 

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