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1. Almost Everywhere Behavior of Functions According to Partition Measures

   https://doi.org/10.1017/fms.2023.130  

   Chan, William; Jackson, Stephen; Trang, Nam (January 2024, Forum of Mathematics, Sigma)

   Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses.•If$$\kappa $$is a cardinal,$$\epsilon < \kappa $$,$${\mathrm {cof}}(\epsilon ) = \omega $$,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and a$$\delta < \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if$$f \upharpoonright \delta = g \upharpoonright \delta $$and$$\sup (f) = \sup (g)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$is a cardinal,$$\epsilon $$is countable,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$holds and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the strong almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and finitely many ordinals$$\delta _0, ..., \delta _k \leq \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$0 \leq i \leq k$$,$$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\kappa _2$$,$$\epsilon \leq \kappa $$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere monotonicity property: There is a club$$C \subseteq \kappa $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$\alpha < \epsilon $$,$$f(\alpha ) \leq g(\alpha )$$, then$$\Phi (f) \leq \Phi (g)$$.•Suppose dependent choice ($$\mathsf {DC}$$),$${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$$and the almost everywhere short length club uniformization principle for$${\omega _1}$$hold. Then every function$$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$$satisfies a finite continuity property with respect to closure points: Let$$\mathfrak {C}_f$$be the club of$$\alpha < {\omega _1}$$so that$$\sup (f \upharpoonright \alpha ) = \alpha $$. There is a club$$C \subseteq {\omega _1}$$and finitely many functions$$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$$so that for all$$f \in [C]^{\omega _1}_*$$, for all$$g \in [C]^{\omega _1}_*$$, if$$\mathfrak {C}_g = \mathfrak {C}_f$$and for all$$i < n$$,$$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$$, then$$\Phi (g) = \Phi (f)$$.•Suppose$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\epsilon _2$$for all$$\epsilon < \kappa $$. For all$$\chi < \kappa $$,$$[\kappa ]^{<\kappa }$$does not inject into$${}^\chi \mathrm {ON}$$, the class of$$\chi $$-length sequences of ordinals, and therefore,$$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$$. As a consequence, under the axiom of determinacy$$(\mathsf {AD})$$, these two cardinality results hold when$$\kappa $$is one of the following weak or strong partition cardinals of determinacy:$${\omega _1}$$,$$\omega _2$$,$$\boldsymbol {\delta }_n^1$$(for all$$1 \leq n < \omega $$) and$$\boldsymbol {\delta }^2_1$$(assuming in addition$$\mathsf {DC}_{\mathbb {R}}$$). 

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2. Surface layer response to heterogeneous tree canopy distributions: roughness regime regulates secondary flow polarity

   https://doi.org/10.1017/jfm.2022.583  

   Joshi, P.; Anderson, W. (September 2022, Journal of Fluid Mechanics)

   Large-eddy simulation was used to model turbulent atmospheric surface layer (ASL) flow over canopies composed of streamwise-aligned rows of synthetic trees of height,$$h$$, and systematically arranged to quantify the response to variable streamwise spacing,$$\delta _1$$, and spanwise spacing,$$\delta _2$$, between adjacent trees. The response to spanwise and streamwise heterogeneity has, indeed, been the topic of a sustained research effort: the former resulting in formation of Reynolds-averaged counter-rotating secondary cells, the latter associated with the$$k$$- and$$d$$-type response. No study has addressed the confluence of both, and results herein show secondary flow polarity reversal across ‘critical’ values of$$\delta _1$$and$$\delta _2$$. For$$\delta _2/\delta \lesssim 1$$and$$\gtrsim 2$$, where$$\delta$$is the flow depth, the counter-rotating secondary cells are aligned such that upwelling and downwelling, respectively, occurs above the elements. The streamwise spacing$$\delta _1$$regulates this transition, with secondary cell reversal occurring first for the largest$$k$$-type cases, as elevated turbulence production within the canopy necessitates entrainment of fluid from aloft. The results are interpreted through the lens of a benchmark prognostic closure for effective aerodynamic roughness,$$z_{0,{Eff.}} = \alpha \sigma _h$$, where$$\alpha$$is a proportionality constant and$$\sigma _h$$is height root mean square. We report$$\alpha \approx 10^{-1}$$, the value reported over many decades for a broad range of rough surfaces, for$$k$$-type cases at small$$\delta _2$$, whereas the transition to$$d$$-type arrangements necessitates larger$$\delta _2$$. Though preliminary, results highlight the non-trivial response to variation of streamwise and spanwise spacing. 

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3. On the ergodic theory of the real Rel foliation

   https://doi.org/10.1017/fmp.2024.6  

   Chaika, Jon; Weiss, Barak (January 2024, Forum of Mathematics, Pi)

   Abstract Let$${{\mathcal {H}}}$$be a stratum of translation surfaces with at least two singularities, let$$m_{{{\mathcal {H}}}}$$denote the Masur-Veech measure on$${{\mathcal {H}}}$$, and let$$Z_0$$be a flow on$$({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$$obtained by integrating a Rel vector field. We prove that$$Z_0$$is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces$$({\mathcal L}, m_{{\mathcal L}})$$, where$${\mathcal L} \subset {{\mathcal {H}}}$$is an orbit-closure for the action of$$G = \operatorname {SL}_2({\mathbb {R}})$$(i.e., an affine invariant subvariety) and$$m_{{\mathcal L}}$$is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of$$Z_0$$with respect to any of the measures$$m_{{{\mathcal L}}}$$is zero. 

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4. COVERING INTEGERS BY $x^2 + dy^2$

   https://doi.org/10.1017/S1474748024000513  

   Green, Ben; Soundararajan, Kannan (May 2025, Journal of the Institute of Mathematics of Jussieu)

   Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$. 

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5. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin; Harper, Alicia; Landsberg, J.M. (January 2023, Forum of Mathematics, Pi)

   Abstract Let$$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$$denote the matrix multiplication tensor (and write$$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$$), and let$$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$$denote the determinant polynomial considered as a tensor. For a tensorT, let$$\underline {\mathbf {R}}(T)$$denote its border rank. We (i) give the first hand-checkable algebraic proof that$$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$$, (ii) prove$$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$$and$$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$$M_{\langle 2\rangle }$$, (iii) prove$$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$$, (iv) prove$$\underline {\mathbf {R}}(\operatorname {det}_3)=17$$, improving the previous lower bound of$$12$$, (v) prove$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$$for all$$\mathbf {n}\geq 25$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$$was known, as well as lower bounds for$$4\leq \mathbf {n}\leq 25$$, and (vi) prove$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$$for all$$\mathbf {n} \ge 18$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 

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