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1. ESKEMAP: exact sketch-based read mapping

   https://doi.org/10.1186/s13015-024-00261-7  

   Schulz, Tizian; Medvedev, Paul (December 2024, Algorithms for Molecular Biology)

   Abstract BackgroundGiven a sequencing read, the broad goal of read mapping is to find the location(s) in the reference genome that have a “similar sequence”. Traditionally, “similar sequence” was defined as having a high alignment score and read mappers were viewed as heuristic solutions to this well-defined problem. For sketch-based mappers, however, there has not been a problem formulation to capture what problem an exact sketch-based mapping algorithm should solve. Moreover, there is no sketch-based method that can find all possible mapping positions for a read above a certain score threshold. ResultsIn this paper, we formulate the problem of read mapping at the level of sequence sketches. We give an exact dynamic programming algorithm that finds all hits above a given similarity threshold. It runs in$$\mathcal {O} (|t| + |p| + \ell ^2)$$ $O\left(\right|t|+|p|+{\ell }^2)$time and$$\mathcal {O} (\ell \log \ell )$$ $O(\ell log\ell )$space, where |t| is the number of$$k$$ $k$-mers inside the sketch of the reference, |p| is the number of$$k$$ $k$-mers inside the read’s sketch and$$\ell$$ $\ell$is the number of times that$$k$$ $k$-mers from the pattern sketch occur in the sketch of the text. We evaluate our algorithm’s performance in mapping long reads to the T2T assembly of human chromosome Y, where ampliconic regions make it desirable to find all good mapping positions. For an equivalent level of precision as minimap2, the recall of our algorithm is 0.88, compared to only 0.76 of minimap2. 

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2. Carleson’s $$\varepsilon ^{2}$$ conjecture in higher dimensions

   https://doi.org/10.1007/s00222-025-01337-w  

   Fleschler, Ian; Tolsa, Xavier; Villa, Michele (July 2025, Inventiones mathematicae)

   Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

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3. A multiple-timing analysis of temporal ratcheting

   https://doi.org/10.1140/epje/s10189-024-00421-y  

   Hashemi, Aref; Gilman, Edward T.; Khair, Aditya S. (April 2024, The European Physical Journal E)

   AbstractWe develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies$$\omega $$ $\omega$and$$\alpha \omega $$ $\alpha \omega$, where$$\alpha $$ $\alpha$is a rational number. If$$\alpha $$ $\alpha$is a ratio of odd and even integers (e.g.,$$\tfrac{2}{1},\,\tfrac{3}{2},\,\tfrac{4}{3}$$ $\frac{2}{1},\frac{3}{2},\frac{4}{3}$), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for$$\alpha =2$$ $\alpha =2$. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for$$\alpha =\tfrac{3}{2}$$ $\alpha =\frac{3}{2}$and$$\tfrac{4}{3}$$ $\frac{4}{3}$appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these$$\alpha $$ $\alpha$values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems. Graphic abstract 

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4. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

   https://doi.org/10.1007/s10240-024-00144-y  

   De Lellis, Camillo; Nardulli, Stefano; Steinbrüchel, Simone (February 2024, Publications mathématiques de l'IHÉS)

   Abstract We consider integral area-minimizing 2-dimensional currents$$T$$ $T$in$$U\subset \mathbf {R}^{2+n}$$ $U\subset R^{2+n}$with$$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$, where$$Q\in \mathbf {N} \setminus \{0\}$$ $Q\in N\setminus \left\{0\right\}$and$$\Gamma $$ $\Gamma$is sufficiently smooth. We prove that, if$$q\in \Gamma $$ $q\in \Gamma$is a point where the density of$$T$$ $T$is strictly below$$\frac{Q+1}{2}$$ $\frac{Q+1}{2}$, then the current is regular at$$q$$ $q$. The regularity is understood in the following sense: there is a neighborhood of$$q$$ $q$in which$$T$$ $T$consists of a finite number of regular minimal submanifolds meeting transversally at$$\Gamma $$ $\Gamma$(and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$$Q=1$$ $Q=1$. As a corollary, if$$\Omega \subset \mathbf {R}^{2+n}$$ $\Omega \subset R^{2+n}$is a bounded uniformly convex set and$$\Gamma \subset \partial \Omega $$ $\Gamma \subset \partial \Omega$a smooth 1-dimensional closed submanifold, then any area-minimizing current$$T$$ $T$with$$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$is regular in a neighborhood of $$\Gamma $$ $\Gamma$. 

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5. Regularity of area minimizing currents mod p

   https://doi.org/10.1007/s00039-020-00546-0  

   De Lellis, Camillo; Hirsch, Jonas; Marchese, Andrea; Stuvard, Salvatore (October 2020, Geometric and Functional Analysis)

   Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$, for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$cannot be larger than$$m-1$$ $m-1$. Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ $(m-1)$-rectifiable with locally finite$$(m-1)$$ $(m-1)$-dimensional measure. 

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