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1. Complete positivity order and relative entropy decay

   https://doi.org/10.1017/fms.2024.117  

   Gao, Li; Junge, Marius; LaRacuente, Nicholas; Li, Haojian (January 2025, Forum of Mathematics, Sigma)

   Abstract We prove that for a GNS-symmetric quantum Markov semigroup, the complete modified logarithmic Sobolev constant is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this gives a short proof that every sub-Laplacian of a Hörmander system on a compact manifold satisfies a modified log-Sobolev inequality uniformly for scalar and matrix-valued functions. For quantum Markov semigroups, we show that the complete modified logarithmic Sobolev constant is comparable to the spectral gap up to the logarithm of the dimension. Such estimates are asymptotically tight for a quantum birth-death process. Our results, along with the consequence of concentration inequalities, are applicable to GNS-symmetric semigroups on general von Neumann algebras. 

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2. Proof of the Michael–Simon–Sobolev inequality using optimal transport

   https://doi.org/10.1515/crelle-2023-0054  

   Brendle, Simon; Eichmair, Michael (August 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2. 

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3. Sharp lower bounds on the manifold widths of Sobolev and Besov spaces

   https://doi.org/10.1016/j.jco.2024.101884  

   Siegel, Jonathan W (December 2024, Journal of Complexity)

   We consider the problem of determining the manifold $$n$$-widths of Sobolev and Besov spaces with error measured in the $$L_p$$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $$q$$ satisfies $$q\leq p$$ or $$1 \leq p \leq 2$$. We close this gap and obtain sharp lower bounds for all $$1 \leq p,q \leq \infty$$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $$\ell^M_q$$-balls in the $$\ell_p$$-norm when $$p\leq q$$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases. 

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4. Polynomial progressions in topological fields

   https://doi.org/10.1017/fms.2024.104  

   Krause, Ben; Mirek, Mariusz; Peluse, Sarah; Wright, James (January 2024, Forum of Mathematics, Sigma)

   Abstract Let$$P_1, \ldots , P_m \in \mathbb {K}[\mathrm {y}]$$be polynomials with distinct degrees, no constant terms and coefficients in a general local field$$\mathbb {K}$$. We give a quantitative count of the number of polynomial progressions$$x, x+P_1(y), \ldots , x + P_m(y)$$lying in a set$$S\subseteq \mathbb {K}$$of positive density. The proof relies on a general$$L^{\infty }$$inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex andp-adic analysis. 

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5. Aspherical complex surfaces, the Singer conjecture, and Gromov–Lück inequality $$\chi \ge |\sigma |$$

   https://doi.org/10.1017/S0305004125101400  

   ALBANESE, MICHAEL; DI_CERBO, LUCA F; LOMBARDI, LUIGI (July 2025, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We discuss the Singer conjecture and Gromov–Lück inequality$$\chi\geq |\sigma|$$for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group that does not rely on Gromov’s Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class$$\mathrm{VII}_0^+$$(a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen the Gromov-Lück inequality for aspherical complex surfaces that are not in Class$$\mathrm{VII}_0^+$$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid’s conjecture. 

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