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1. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric; Müller-Hermes, Alexander (February 2023, Letters in Mathematical Physics)

   Abstract Let $$\phi $$ ϕ be a positive map from the $$n\times n$$ n × n matrices $$\mathcal {M}_n$$ M n to the $$m\times m$$ m × m matrices $$\mathcal {M}_m$$ M m . It is known that $$\phi $$ ϕ is 2-positive if and only if for all $$K\in \mathcal {M}_n$$ K ∈ M n and all strictly positive $$X\in \mathcal {M}_n$$ X ∈ M n , $$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ ϕ ( K ∗ X - 1 K ) ⩾ ϕ ( K ) ∗ ϕ ( X ) - 1 ϕ ( K ) . This inequality is not generally true if $$\phi $$ ϕ is merely a Schwarz map. We show that the corresponding tracial inequality $${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ Tr [ ϕ ( K ∗ X - 1 K ) ] ⩾ Tr [ ϕ ( K ) ∗ ϕ ( X ) - 1 ϕ ( K ) ] holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 

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2. Equal sums in random sets and the concentration of divisors

   https://doi.org/10.1007/s00222-022-01177-y  

   Ford, Kevin; Green, Ben; Koukoulopoulos, Dimitris (June 2023, Inventiones mathematicae)

   Abstract We study the extent to which divisors of a typical integer n are concentrated. In particular, defining $$\Delta (n) := \max _t \# \{d | n, \log d \in [t,t+1]\}$$ Δ ( n ) : = max t # { d | n , log d ∈ [ t , t + 1 ] } , we show that $$\Delta (n) \geqslant (\log \log n)^{0.35332277\ldots }$$ Δ ( n ) ⩾ ( log log n ) 0.35332277 … for almost all n , a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $${\textbf{A}} \subset {\mathbb {N}}$$ A ⊂ N by selecting i to lie in $${\textbf{A}}$$ A with probability 1/ i . What is the supremum of all exponents $$\beta _k$$ β k such that, almost surely as $$D \rightarrow \infty $$ D → ∞ , some integer is the sum of elements of $${\textbf{A}} \cap [D^{\beta _k}, D]$$ A ∩ [ D β k , D ] in k different ways? We characterise $$\beta _k$$ β k as the solution to a certain optimisation problem over measures on the discrete cube $$\{0,1\}^k$$ { 0 , 1 } k , and obtain lower bounds for $$\beta _k$$ β k which we believe to be asymptotically sharp. 

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3. Internal Sequential Commutation and Single Generation

   https://doi.org/10.1093/imrn/rnaf103  

   Gao, David; Kunnawalkam_Elayavalli, Srivatsav; Patchell, Gregory; Tan, Hui (April 2025, International Mathematics Research Notices)

   Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof. 

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4. Conditions for a Bigraph to be Super-Cyclic

   https://doi.org/10.37236/9683  

   Kostochka, Alexandr; Lavrov, Mikhail; Luo, Ruth; Zirlin, Dara (January 2021, The Electronic Journal of Combinatorics)

   A hypergraph $$\mathcal H$$ is super-pancyclic if for each $$A \subseteq V(\mathcal H)$$ with $$|A| \geqslant 3$$, $$\mathcal H$$ contains a Berge cycle with base vertex set $$A$$. We present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph $$\mathcal H$$ with $$ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$$. We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $$G$$ such that for each $$A \subseteq X$$ with $$|A| \geqslant 3$$, $$G$$ has a cycle $$C_A$$ such that $$V(C_A)\cap X=A$$. Such graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs. 

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5. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

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