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1. Integral Points on Varieties With Infinite Étale Fundamental Group

   https://doi.org/10.1093/imrn/rnad147  

   Achenjang, Niven T ; Morrow, Jackson S ( July 2023 , International Mathematics Research Notices)

   Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi \colon Y \to X$ of degree at least $2\dim (X)^{2}$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable. 

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2. A PDE hierarchy for directed polymers in random environments

   https://doi.org/10.1088/1361-6544/ac23b7  

   Gu, Yu ; Henderson, Christopher ( September 2021 , Nonlinearity)

   Abstract For a Brownian directed polymer in a Gaussian random environment, with q ( t , ⋅) denoting the quenched endpoint density and Q n ( t , x 1 , … , x n ) = E [ q ( t , x 1 ) … q ( t , x n ) ] , we derive a hierarchical PDE system satisfied by { Q n } n ⩾ 1 . We present two applications of the system: (i) we compute the generator of { μ t ( d x ) = q ( t , x ) d x } t ⩾ 0 for some special functionals, where { μ t ( d x ) } t ⩾ 0 is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d ⩾ 3, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive O ( t 2/3 ) scaling. 

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3. DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

   https://doi.org/10.1017/S0017089518000022  

   OSTROVSKA, S. ; OSTROVSKII, M. I. ( January 2019 , Glasgow Mathematical Journal)

   Abstract Given a Banach space X and a real number α ≥ 1, we write: (1) D ( X ) ≤ α if, for any locally finite metric space A , all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C , the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C ; (2) D ( X ) = α + if, for every ϵ > 0, the condition D ( X ) ≤ α + ϵ holds, while D ( X ) ≤ α does not; (3) D ( X ) ≤ α + if D ( X ) = α + or D ( X ) ≤ α. It is known that D ( X ) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D ((⊕ n =1 ∞ X n ) p ) ≤ 1 + for every nested family of finite-dimensional Banach spaces { X n } n =1 ∞ and every 1 ≤ p ≤ ∞. (2) D ((⊕ n =1 ∞ ℓ ∞ n ) p ) = 1 + for 1 < p < ∞. (3) D ( X ) ≤ 4 + for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008). 

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4. 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.$\begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*}$with$\nu \ll 1$. We prove that for$\epsilon >0$sufficiently small (but independent of$\nu$), initial data which are$O(\epsilon \nu ^{1/3})$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as$t\to \infty$. The solutions exhibit uniform-in-$\nu$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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5. MHC‐restricted Ag85B‐specific CD8 + T cells are enhanced by recombinant BCG prime and DNA boost immunization in mice

   https://doi.org/10.1002/eji.201847988  

   Komine‐Aizawa, Shihoko ; Jiang, Jiansheng ; Mizuno, Satoru ; Hayakawa, Satoshi ; Matsuo, Kazuhiro ; Boyd, Lisa F. ; Margulies, David H. ; Honda, Mitsuo ( June 2019 , European Journal of Immunology)

   Despite efforts to develop effective treatments and vaccines,Mycobacterium tuberculosis(Mtb), particularly pulmonaryMtb, continues to provide major health challenges worldwide. To improve immunization against the persistent health challenge ofMtbinfection, we have studied the CD8⁺T cell response to Bacillus Calmette‐Guérin (BCG) and recombinant BCG (rBCG) in mice. Here, we generated CD8⁺T cells with an rBCG‐based vaccine encoding the Ag85B protein ofM. kansasii, termed rBCG‐Mkan85B, followed by boosting with plasmid DNA expressing the Ag85B gene (DNA‐Mkan85B). We identified two MHC‐I (H2‐K^d)‐restricted epitopes that induce cross‐reactive responses toMtband other related mycobacteria in both BALB/c (H2^d) and CB6F1 (H2^b/d) mice. The H2‐K^d‐restricted peptide epitopes elicited polyfunctional CD8⁺T cell responses that were also highly cross‐reactive with those of other proteins of the Ag85 complex. Tetramer staining indicated that the two H2‐K^d‐restricted epitopes elicit distinct CD8⁺T cell populations, a result explained by the X‐ray structure of the two peptide/H2‐K^dcomplexes. These results suggest that rBCG‐Mkan85B vector‐based immunization and DNA‐Mkan85B boost may enhance CD8⁺T cell response toMtb, and might help to overcome the limited effectiveness of the current BCG in eliciting tuberculosis immunity.

    

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