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1. Spanning Configurations and Representation Stability

   https://doi.org/10.37236/11136  

   Pawlowski, Brendan; Ramos, Eric; Rhoades, Brendon (January 2023, The Electronic Journal of Combinatorics)

   Let $$V_1, V_2, V_3, \dots $$ be a sequence of $$\mathbb {Q}$$-vector spaces where $$V_n$$ carries an action of $$\mathfrak{S}_n$$. Representation stability and multiplicity stability are two related notions of when the sequence $$V_n$$ has a limit. An important source of stability phenomena arises when $$V_n$$ is the $$d^{th}$$ homology group (for fixed $$d$$) of the configuration space of $$n$$ distinct points in some fixed topological space $$X$$. We replace these configuration spaces with moduli spaces of tuples $$(W_1, \dots, W_n)$$ of subspaces of a fixed complex vector space $$\mathbb {C}^N$$ such that $$W_1 + \cdots + W_n = \mathbb {C}^N$$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory. 

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2. On lower bounds for Erdős-Szekeres products

   https://doi.org/10.1090/proc/15503  

   Billsborough, C.; Freedman, M.; Hart, S.; Kowalsky, G.; Lubinsky, D.; Pomeranz, A.; Sammel, A. (October 2021, Proceedings of the American Mathematical Society)

   Let { s j } j = 1 n \left \{ s_{j}\right \} _{j=1}^{n} be positive integers. We show that for any 1 ≤ L ≤ n , 1\leq L\leq n, ‖ ∏ j = 1 n ( 1 − z s j ) ‖ L ∞ ( | z | = 1 ) ≥ exp ⁡ ( 1 2 e L ( s 1 s 2 … s L ) 1 / L ) . \begin{equation*} \left \Vert \prod _{j=1}^{n}\left ( 1-z^{s_{j}}\right ) \right \Vert _{L_{\infty }\left ( \left \vert z\right \vert =1\right ) }\geq \exp \left ( \frac {1}{2e}\frac {L}{\left ( s_{1}s_{2}\ldots s_{L}\right ) ^{1/L}}\right ) . \end{equation*} In particular, this gives geometric growth if a positive proportion of the { s j } \left \{ s_{j}\right \} are bounded. We also show that when the { s j } \left \{ s_{j}\right \} grow regularly and faster than j ( log ⁡ j ) 2 + ε j\left ( \log j\right ) ^{2+\varepsilon } , some ε > 0 \varepsilon >0 , then the norms grow faster than exp ⁡ ( ( log ⁡ n ) 1 + δ ) \exp \left ( \left ( \log n\right ) ^{1+\delta }\right ) for some δ > 0 \delta >0 . 

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3. New Bounds for the Same-Type Lemma

   https://doi.org/10.37236/12414  

   Bukh, Boris; Vasileuski, Alexey (April 2024, The Electronic Journal of Combinatorics)

   Given finite sets $$X_1,\dotsc,X_m$$ in $$\mathbb{R}^d$$ (with $$d$$ fixed), we prove that there are respective subsets $$Y_1,\dotsc,Y_m$$ with $$\lvert Y_i\rvert \geq \frac{1}{poly(m)}\lvert X_i\rvert$$ such that, for $$y_1\in Y_1,\dotsc,y_m\in Y_m$$, the orientations of the\linebreak $(d+1)$-tuples from $$y_1,\dotsc,y_m$$ do not depend on the actual choices of points $$y_1,\dotsc,y_m$$. This generalizes previously known case when all the sets $$X_i$$ are equal. Furthermore, we give a construction showing that polynomial dependence on $$m$$ is unavoidable, as well as an algorithm that approximates the best-possible constants in this result. 

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4. A single-point Reshetnyak’s theorem

   https://doi.org/10.1090/tran/9415  

   Kangasniemi, Ilmari; Onninen, Jani (May 2025, Transactions of the American Mathematical Society)

   We prove a single-value version of Reshetnyak’s theorem. Namely, if a non-constant map $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$ from a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfies the estimate $\left|Df\right(x){|}^n\le K{J}_f(x)+\mathrm{\Sigma }(x\left)\right|f\left(x\right)-y_0{|}^n$ at almost every $x\in \mathrm{\Omega }$ for some $K\ge 1$, $y_0\in {\mathbb{R}}^n$ and $\mathrm{\Sigma }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\epsilon }\left(\mathrm{\Omega }\right)$, then $f^{-1}\left\{y_0\right\}$ is discrete, the local index $i(x,f)$ is positive in $f^{-1}\left\{y_0\right\}$, and every neighborhood of a point of $f^{-1}\left\{y_0\right\}$ is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$ at every $y_0\in {\mathbb{R}}^n$ is equivalent to assuming that the map $f$ is $K$-quasiregular, even if the choice of $\mathrm{\Sigma }$ is different for each $y_0$. Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$-quasiregularity. As a corollary of our single-value Reshetnyak’s theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calderón problem. 

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5. Removal lemmas and approximate homomorphisms

   https://doi.org/10.1017/S0963548321000572  

   Fox, Jacob; Zhao, Yufei (July 2022, Combinatorics, Probability and Computing)

   Abstract We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma , states that for each $$\epsilon>0$$ there exists M such that every triangle-free graph G has an $$\epsilon$$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $$\epsilon$$ -approximate homomorphism is a map $$V(G) \to V(F)$$ where all but at most $$\epsilon \left\lvert{V(G)}\right\rvert^2$$ edges of G are mapped to edges of F ). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $$\epsilon^{-1}$$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal. 

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